diff --git a/-9AzT4oBgHgl3EQfhPxU/content/tmp_files/2301.01481v1.pdf.txt b/-9AzT4oBgHgl3EQfhPxU/content/tmp_files/2301.01481v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5c8b3d8e1323e52d68374fdb0b80c82c741abdd --- /dev/null +++ b/-9AzT4oBgHgl3EQfhPxU/content/tmp_files/2301.01481v1.pdf.txt @@ -0,0 +1,981 @@ +On Fairness of Medical Image Classification +with Multiple Sensitive Attributes via +Learning Orthogonal Representations +Wenlong Deng1∗, Yuan Zhong2∗, Qi Dou2, and Xiaoxiao Li1 +1 Department of Electrical and Computer Engineering, +The University of British Columbia, Vancouver, BC, Canada +2 Department of Computer Science and Engineering, +The Chinese University of Hong Kong, Hong Kong, China +Abstract. Mitigating the discrimination of machine learning models +has gained increasing attention in medical image analysis. However, rare +works focus on fair treatments for patients with multiple sensitive demo- +graphic ones, which is a crucial yet challenging problem for real-world +clinical applications. In this paper, we propose a novel method for fair +representation learning with respect to multi-sensitive attributes. We +pursue the independence between target and multi-sensitive representa- +tions by achieving orthogonality in the representation space. Concretely, +we enforce the column space orthogonality by keeping target information +on the complement of a low-rank sensitive space. Furthermore, in the row +space, we encourage feature dimensions between target and sensitive rep- +resentations to be orthogonal. The effectiveness of the proposed method +is demonstrated with extensive experiments on the CheXpert dataset. +To our best knowledge, this is the first work to mitigate unfairness with +respect to multiple sensitive attributes in the field of medical imaging. +The code will be available at https://github.com/vengdeng/FCRO. +1 +Introduction +With the increasing application of artificial intelligence systems for medical im- +age diagnosis, it is notably important to ensure fairness of image classification +models and investigate concealed model biases that are to-be-encountered in +complex real-world situations. Unfortunately, sensitive attributes (e.g., race and +gender) accompanied by medical images are prone to be inherently encoded by +machine learning models [5], and affect the model’s discrimination property [20]. +Recently, fair representation learning has shown great potential as it acts as a +group parity bottleneck that mitigates discrimination when generalized to down- +stream tasks. Existing methods [1,4,15,16,21] have studied the parity between +privileged and unprivileged groups upon just a single sensitive attribute, but +neglecting the flexibility with respect to multiple sensitive attributes, in which +the conjunctions of unprivileged attributes might also deteriorate discrimination. +* These authors contributed equally to this work. +arXiv:2301.01481v1 [cs.CV] 4 Jan 2023 + +2 +W. Deng et al. +Fig. 1. A t-SNE [10] visualization of (a) sensitive attribute and (b) target representa- +tions learned from our proposed methods FCRO on the CheXpert dataset [7]. Sensitive +embeddings capture subgroups’ variance. We claim FCRO enforces fair classification on +the target task by learning orthogonal target representations that are invariant over +different attributes. +This is a crucial yet challenging problem hindering the applicability of machine +learning models, especially for medical image classification where patients always +have many demographic attributes. +To date, it is still challenging to effectively learn target-related representa- +tions which are both fair and flexible to multiple sensitive attributes, regardless of +some promising investigations recently. For instance, adversarial methods [1,11] +produce robust representations by formulating a min-max game between an en- +coder that learns class-related representation and an adversary that removes sen- +sitive information from it. Disentanglement-based methods [4,21] achieve separa- +tion by minimizing the mutual information between target and sensitive attribute +representations. These methods typically gain efficacy by means of carefully de- +signing objectives. To extend them to the multi-attribute setting, additional loss +functions have to be explored, which should handle gradient conflict or interfer- +ence. Methods using variational autoencoder [3] decompose the latent distribu- +tions of target and sensitive and penalize their correlation for disentanglement. +However, aligning the distribution of the sensitive attributes is difficult or even +intractable given the complex combination of multiple factors. Besides, there +are some fairness methods based on causal inference [12] or bi-level optimization +[16], which also learn debiased while multi-attributes inflexible representations. +Recently, disentanglement is vigorously interpreted as the orthogonality of a de- +composed target-sensitive latent representation pair by [15], where they predefine +a pair of orthogonal subspaces for target and sensitive attribute representations. +In a multi-sensitive attributes setting, the dimension of the target space would +be continuously compressed and how to solve it is still an open problem. +In this paper, we propose a new method to achieve Fairness via Column- +Row space Orthogonality (called FCRO) by learning fair representations for med- +ical image classification with multiple sensitive attributes. FCRO considers multi- +sensitive attributes by encoding them into a unified attribute representation. It +achieves a best trade-off for fairness and data utility (see illustrations in Fig. 1) + +Race-Sex-Age +White, Male, 60- +White, Male, 18-60 +White, Female, 60- +福 +White, Female, 18-60 +non-White, Male, 60- +non-White, Male, 18-60 +non-White, Female, 60. +non-White, Female, 18-60 +(a) Sensitivie attribute representation +(b) Target representationOn Fairness of Image Classification with Multi-Sensitive Attributes +3 +Classi%ier +ℎ!! +𝑆! +𝑆! +" +𝑧̃# +$ +⋯ +Sensitive Encoder +𝜙! +𝑍! +Classifier +ℎ!" + 𝑋 + 𝑌 + 𝐴% + 𝑍! + 𝑍# + 𝐴& + ⋮ +𝜙!(𝑋) +𝜙"(𝑋) +× +(a) +∈ ℝ'×) +𝑋 +𝑧̂# +* +𝑧̂! ++ +Row space +(c) +𝑧̃! +𝐿!! +𝐿!" +𝑗 +Column space +(b) +𝐿#$%&' = 𝑐(𝑧̃" +(, 𝑆!) +𝐿)$)*+ = 𝑟(𝑧̂" +, , 𝑧̂! +-) +Classi%ier +ℎ" +Target Encoder +𝜙# +𝑍# +∈ ℝ'×) +𝑧̃# +𝐿# +𝑖 +𝑘 +𝑍# ⊥ 𝑍! +frozen modules +forward +backward +cause +correlated +Fig. 2. Overview of our proposed method FCRO. (a) The graphical model of orthogo- +nal representation learning for fair medical image classification with multiple sensitive +attributes. (b) The novel column-row space orthogonality. In the column space, we +encourage the target model to learn representations in the complement of a low-rank +sensitive space. In the row space, we enforce each row vector (feature dimension) of the +target and sensitive attribute representations to be orthogonal to each other (c) The +overall training pipeline. We use a pre-trained multi-sensitive branch, and propagate +orthogonal gradients to target encoder φT . +via orthogonality in both column and row spaces. Our contributions are summa- +rized as follows: (1) We tackle the practical and challenging problem of fairness +given multiple sensitive attributes for medical image classification. To the best +of our knowledge, this is the first work to study fairness with respect to multi- +sensitive attributes in the field of medical imaging. (2) We relax the independence +of target and sensitive attribute representations by orthogonality which can be +achieved by our proposed novel column and row losses. (3) We conduct exten- +sive experiments on the CheXpert [7] dataset with over 80,000 chest X-rays. +FCRO achieves a superior fairness-utility trade-off over state-of-the-art methods +regarding multiple sensitive attributes race, sex, and age. +2 +Methodology +2.1 +Problem Formulation +Notations. We consider group fairness in this work, group fairness articulates +the equality of some statistics like predictive rate between certain groups. Con- +sidering a binary classification problem with column vector inputs x ∈ X, + +AP PORT UPRICHT4 +W. Deng et al. +labels y ∈ Y = {0, 1}. Multi-sensitive attributes a ∈ A is vector of m at- +tributes sampled from the conjunction, i.e., Cartesian product, of sensitive at- +tributes A = � +i∈[m] Ai * where the i-th sensitive attribute Ai ∈ {0, 1}. Our +training data consist of tuples D = {(x, y, a)}. We denote the classification +model �y = f(x) = hT (φT (x)) that predicts a class label given an input x, +where φT : X �→ Rd is a feature encoder for target embeddings, and hT : +Rd �→ R is a scoring function. Similarly, we consider a sensitive attribute model +g(x) = {hA1(φA(x)), ..., hAm(φA(x))} that predicts sensitive attributes associ- +ated with input x. Given the number of samples n, the input data representa- +tion is X = [x1, . . . , xn] and we denote the feature representation ZT = φT (X), +ZA = φA(X) ∈ Rd×n. +Fair classifier on multiple sensitive attributes. A classifier predicts y given +an input x by estimating the posterior probability p(y|x). When inputs that +are affected by their associated attributes (i.e., {A1, . . . , Am} → X) are fed +into the network, the posterior probability is written as p(y|x, a). Since biased +information from A is encoded, this can lead to an unfair prediction by the +classifier. For example, in the diagnosis of a disease with sensitive attributes age, +sex, and race, a biased classifier will result in p(�y|A = male, old, black) ̸= p(�y|A = +female, young, white). In this work, we focus on equalized odds (ED), which is a +commonly used and crucial criterion of fair classification in the medical domain +[19]. In our case, ED regarding multiple sensitive attributes can be formulated +as follows: +P(�Y = y|A = π1, Y = y) = P(�Y = y|A = π2, Y = y), +∀π1, π2 ∈ A. +(1) +Recent methods [16] suggest achieving ED for a classifier by enforcing �Y ⊥ A|Y . +In other words, a fair classifier is expected to be independent of multi-sensitive +information: p(y|x) = p(y|x, a). +Fair representation. To enforce our aforementioned conditions, we follow [15] +and introduce target embedding zT and multi-attribute embedding zAi that is +generated from x. As in the causal structure graph for the classifier depicted +in Fig. 2 (a), if zT and zAi are independent, the probability of a fair classifier +p(y|x, a) is written as: +p(y, a|x) = p(y|x, a)p(x|a)p(a) +p(x) += p(y|x)p(a|x) +(2) += p(y|zT )p(zT |x) +� +i∈[m] +p(ai|zAi)p(zAi|x), +(3) +and we call zT fair representation for the target task (e.g., disease diagnosis). +To this end, we aim to maximize Eq. (3) with the conditional independence +constraint to train a fair classifier. It is noteworthy that in the multisensitive +attributes setting, forcing zT to be independent on all zAi, ∀i ∈ [m] is challenging +and even intractable when m is large. Therefore, we propose to encode multi- +sensitive attributes into a single compact encoding zA that is still predictive for +* [m] = {0, 1, .., m} + +On Fairness of Image Classification with Multi-Sensitive Attributes +5 +classifying attributes (i.e., zA → {a1, . . . , am}). Then we can rewrite Eq. (3) as +maximizing the likelihood with the independence constraint on zT and zA: +p(y, a|x) = p(y|zT )p(zT |x)p(a|zA)p(zA|x). +(4) +However, optimizing Eq. (4) brings two technical questions: +Q1: How to satisfy the independence constraint for zT and zA? +A1: We relax the independence by enforcing orthogonality. Different from pre- +defined orthogonal space in [15], we enforce orthogonality in both column spaces +(Sec. 2.2) and row spaces (Sec. 2.3) of ZT and ZA. +Q2: How to estimate p(y|zT ), p(zT |x), p(a|zA), p(zA|x)? +A2: We train two convolutional neural nets encoders zT = φT (x) and zA = +φA(x) to approximate p(zT |x) and p(zA|x) respectively; we train two multi- +layer perception classifier y = hT (zT ) and a = hA(zA) to approximate p(y|zT ) +and p(a|zA) respectively (Sec. 2.4). +2.2 +Column Space Orthogonality +First, we focus on the column space of the target and the sensitive attribute +representations. Column space orthogonality aims to learn target representations +ZT that fulfill the following two aims: 1) have the least projection onto the +sensitive space SA and 2) preserve the representation power to predict Y . +Denote the target representation ZT = [�z1 +T , �z2 +T , . . . , �zn +T ] and the sensitive at- +tribute representation ZA = [�z1 +A, �z2 +A, . . . , �zn +A], where �zi ∈ Rd×1 is a column vector +for i ∈ [n], we represent the column space for ZT and ZA as ST = span(ZT ) +and SA = span(ZA) respectively. Aim 1 can be achieved by forcing ST = S⊥ +A. +Although both �zT , �zA ∈ Rd, their coordinates may not be aligned as they are +generated from two separate encoders. As a result, if d ≪ ∞, then there is +no straightforward way to achieve ST ⊥ SA by directly constraining �zi +T , �zj +A +(e.g., forcing (�zi +T )⊤�zj +A = 0). Aim 2 can be achieved by seeking a low-rank rep- +resentation �SA for SA, whose rank is k such that k ≪ d, because we have +rank(ST ) + rank(SA) = d if ST = S⊥ +A holds. Then S⊥ +A would be a high- +dimensional space with sufficient representation power for target embeddings. +This is especially important when we face multiple sensitive attributes, as the +total size of the space is d, and increasing the number of sensitive attributes +would limit the capacity of ST to learn predictive �zT . To this end, we first pro- +pose to find the low rank sensitive attribute representation space �SA, and then +encourage ZT to be in �SA’s complement �S +⊥ +A. +Construct low-rank multi-sensitive space. We apply Singular Value De- +composition (SVD) on ZA = UAΣAVA to construct the low-rank space �SA, +where UA, VA ∈ Rd×d are orthogonal matrices with left and right singular vec- +tors ui ∈ Rd and vi +A ∈ Rn respectively. And ΣA ∈ Rd×n is a diagonal matrix with +descending non-negative singular values {δi +A}min{n,d} +i=1 +. Then we extract the most +important k left singular vectors to construct �SA = [u1 +A, ..., uk +A], where k controls +how much sensitive information to be captured in �SA. It is notable that �SA is + +6 +W. Deng et al. +agnostic to the number of sensitive attributes because they share the same ZA. +For situations that can not get the whole dataset at once, we follow [8] to select +most important bases from both bases of old iterations and newly constructed +ones. Thus providing an accumulative low-rank space construction variant to +update �SA iteratively. As we do not observe significant performance differences +between these two variants (see Fig. 4 (a)), we use and refer to the first one in +this paper if there is no special clarification. +Column orthogonal loss. With the low-rank space �SA for multiple sensitive +attributes, we encourage φT to learn representations in its complement �S⊥ +A. No- +tice that �S⊥ +A can also be interpreted as the kernel of the projection onto �SA, +i.e., �S⊥ +A = Ker(proj � +SA�zT ). Therefore, we achieve column orthogonal loss by +minimizing the projection of ZT to �SA, which can be defined as: +Lcorth = c(ZT , �SA) = +n +� +i=1 +��� �S⊤ +A �zi +T +��� +2 +2 +���zi +T +��2 +2 +. +(5) +As �SA is a low-rank space, �S⊥ +A will have abundant freedom for φT to extract +target information, thus reserving predictive ability. +2.3 +Row Space Orthogonality +Then, we study the row space of target and sensitive attribute representations. +Row space orthogonality aims to learn target representations ZT that have the +least projection onto the sensitive row space �SA. In other words, we want to +ensure orthogonality on each feature dimension between ZT and ZA. Denote +target representation ZT = [�z1 +T ; �z2 +T ; . . . ; �zd +T ] and sensitive attribute representation +ZA = [�z1 +A; �z2 +A; . . . ; �zd +A], where �zi ∈ R1×n is a row vector for i ∈ [d]. We represent +row space for target representations and sensitive attribute representations as +�ST = span(Z⊤ +T ) and �SA = span(Z⊤ +A) correspondingly. Different from column +space orthogonality, as the coordinates (i.e., the index of samples) of �zA and �zT +are aligned, forcing �ST = �S +⊥ +A can be directly applied by achieving ZT Z⊤ +A: +{(ZT Z⊤ +A)i,j = �zi +T (�zj +A)⊤, i, j ∈ d} = +n +� +t=1 +(�zi +T )t(�zj +A)t. +(6) +Unlike column space, the orthogonality here won’t affect the utility, as the row +vector �zT is not directly correlated to the target y. To be specific, we let pair-wise +row vectors ZT = [�z1 +T , �z2 +T , . . . , �zd +T ] and ZA = [�z1 +A, �z2 +A, . . . , �zd +A] have a small inner +product. Then for any i, j ∈ [d], we try to minimize < �zi +T , �zj +A >. Here we slightly +modify the orthogonality by extra subtracting the mean vector µA and µT from +ZA and ZT respectively, where µ = Ei∈[d]�zi ∈ R1×n. Then orthogonality loss +will naturally be integrated into a covariance loss: +Lrorth = r(ZT , ZA) = 1 +d2 +d +� +i=1 +d +� +j=1 +� +(�zi +T − µT )(�zj +A − µA)⊤�2 +. +(7) + +On Fairness of Image Classification with Multi-Sensitive Attributes +7 +Table 1. CheXpert dataset statistics and group positive rate p(y = 1|a) regarding +pleural effusion with three sensitive attributes race, sex, and age. +Dataset +#Sample +Group Positive Rate +Race +Sex +Age +(White/Non-white/gap) (Male/Female/gap) (>60/≤ 60/gap) +Original +127130 +.410/.393/.017 +.405/.408/.003 +.440/.359/.081 +Augmented 88215 +.264/.386/.122 +.254/.379/.125 +.264/.386/.122 +In this way, the resulting loss encourages each feature of ZT to be independent +of features in ZA thus suppressing the sensitive-encoded covariances that cause +the unfairness. +2.4 +Overall Training +In this section, we introduce the overall training schema as shown in Fig. 2 (c). +For the sensitive branch, since we observe that using a shared encoder may +threaten sensitive information leakage to classification [4] or obtain unsatisfied +sensitive attribute representations [15], we pretrain {φA, hA1, ..., hAm} for mul- +tiple sensitive attributes using the sensitive objective as Lsens = 1 +m +� +i∈[m] LAi. +Here we use cross-entropy loss as LAi for the i-th sensitive attribute. Hence +p(zA|x) and p(a|zA) in Eq. (4) can be obtained. Then, the multi-sensitive space +SA is constructed as in Section 2.2 over the training data. For the target branch, +we use cross-entropy loss as our classification objective LT to supervise the train- +ing of φT and hT and estimate p(zT |x) and p(y|zT ) in Eq. (4) respectively. Here +we do not make additional constraints to LT , which means it can be replaced +by any other task-specific losses. At last, we apply our column and row orthog- +onality losses Lcorth and Lrorth to representations as introduced in Section 2.2 +and Section 2.3 along with detached SA and ZA to approximate independence +between p(zA|x) and p(zT |x). The overall target objective is given as: +Ltarg = LT + λcLcorth + λrLrorth, +(8) +where λc and λr are hyper-parameters to weigh orthogonality and balance fair- +ness and utility. +3 +Experiments +3.1 +Setup +Dataset. We adopt CheXpert dataset [7] to predict Pleural Effusion in chest +X-rays, as it’s crucial for chronic obstructive pulmonary disease diagnosis with +high incidence. Subgroups are defined based on the following binarized sensitive +attributes: self-reported race and ethnicity, sex, and age. Note that data bias + +8 +W. Deng et al. +Table 2. Comparasion of predicting Pleural Effusion on CheXpert dataset. We report +the mean and standard deviation of 5-fold models trained with multi-sensitive +attributes. AUC is used as the utility metric, and fairness is evaluated using disparities +among subgroups defined on multi-sensitive attributes jointly and individually. +Methods +AUC (↑) +Subgroup Disparity (↓) +Joint +Race +Sex +Age +∆AUC +∆ED +∆AUC +∆ED +∆AUC +∆ED +∆AUC +∆ED +ERM [17] +0.863 +0.119 +0.224 +0.018 +0.055 +0.046 +0.142 +0.023 +0.038 +(.005) +(.017) +(.013) +(.009) +(.017) +(.008) +(.014) +(.004) +(.010) +G-DRO [14] +0.854 +0.101 +0.187 +0.015 +0.048 +0.034 +0.105 +0.035 +0.051 +(.004) +(.012) +(.034) +(.003) +(.014) +(.010) +(.025) +(.002) +(.010) +JTT [9] +0.834 +0.103 +0.166 +0.019 +0.056 +0.026 +0.079 +0.017 +0.030 +(.020) +(.017) +(.023) +(.008) +(.016) +(.002) +(.004) +(.006) +(.007) +Adv [18] +0.854 +0.089 +0.130 +0.017 +0.027 +0.022 +0.039 +0.016 +0.023 +(.002) +(.009) +(.018) +(.004) +(.009) +(.003) +(.008) +(.004) +(.004) +BR-Net [1] +0.849 +0.113 +0.200 +0.018 +0.051 +0.037 +0.109 +0.027 +0.039 +(.001) +(.025) +(.023) +(.008) +(.013) +(.012) +(.025) +(.006) +(.006) +PARADE [4] +0.857 +0.103 +0.193 +0.017 +0.052 +0.042 +0.104 +0.026 +0.031 +(.002) +(.022) +(.032) +(.002) +(.010) +(.006) +(.023) +(.006) +(.011) +Orth [15] +0.856 +0.084 +0.177 +0.011 +0.045 +0.022 +0.083 +0.025 +0.032 +(.007) +(.022) +(.016) +(.005) +(.012) +(.009) +(.012) +(.006) +(.005) +FCRO (ours) +0.858 +0.057 0.107 0.012 +0.033 +0.015 0.024 0.013 0.019 +(.001) +(.022) (.013) +(.003) +(.008) +(.004) (.008) +(.004) (.006) +(positive rate gap) is insignificant in the original dataset (see Table 1, row ’orig- +inal’). To demonstrate the effectiveness of bias mitigation methods, we amplify +the data bias by (1) firstly dividing the data into different groups according to the +conjunction of multi-sensitive labels; (2) secondly calculating the positive rate of +each subgroup; (3) sampling out patients and increase each subgroup’s positive +rate gap to 0.12 (see Table 1, row ‘augmented’). We resize all images to 224×224 +and split the dataset into a 15% test set, and an 85% 5-fold cross-validation set. +Evaluation metrics. We use the area under the ROC curve (AUC) to evaluate +the utility of classifiers. To measure fairness, we follow [13] and compute subgroup +disparity with respect to ED (denoted as ∆ED, which is based on true positive +rate (TPR) and true negative rate (TNR)) in (1). We quantify ED disparity as: +∆ED = +max +y∈Y,π1,π2∈A +���P(�Y = y|A = π1, Y = y) − P(�Y = y|A = π2, Y = y) +��� . (9) +We also follow [20] and compare subgroup disparity regarding AUC (denoted +as ∆AUC), which gives a threshold-free fairness metric. Note that we evaluate +disparities both jointly and individually. The joint disparities are calculated with +respect to the conjunction of multiple sensitive attributes A, and the individual +disparities are calculated with respect to a specific sensitive attribute Ai. +Implementation details. In our implementation, all methods use the same +training protocol. We choose DenseNet-121 [6] as the backbone, but replace the + +On Fairness of Image Classification with Multi-Sensitive Attributes +9 +(a) +(c) +(b) +INPUT +FCRO +ERM +(a) +(b) +(c) +INPUT +FCRO +ERM +(a) +(b) +Fig. 3. (a) Subgroup calibration curves. We report quantile calibration curves of the +mean (the line) and standard deviation (the shadow around it) of different subgroups +defined by the conjunction of race, sex, and age. Larger shadow areas correspond to +more severe unfairness. (b) Class activation map [2] generated from vanilla ERM [17] +and FCRO (ours). +final layer with a linear layer to extract 128-dimensional representations. The +optimizer is Adam with learning rate of 1e−4, and weight decay of 4e−4. We train +for 40 epochs with a batch size of 128. We sweep a range of hyper-parameters +for each method and empirically set λc = 80, λr = 500, and k = 3 for FCRO. We +train models in 5-fold with different random seeds. In each fold, we sort all the +validations according to AUC and select the best model with the lowest average +∆ED regarding each sensitive attribute among the top 5 utilities. +Baselines. We compare our method with (i) G-DRO [14] and (ii) JTT [9] – seek- +ing low worst-group error by minimax optimization on group fairness and target +task error, which can be naturally regarded as multi-sensitive fairness methods by +defining subgroups with multi-sensitive attributes conjunctions. We also extend +recent state-of-the-art fair representation learning methods on single sensitive +attributes to multiple ones and compare our method with them, including (iii) +Adv [18] and (iv) BR-Net [1] – achieve fair representation via disentanglement +using adversarial training, (v) PARADE [4] – a state-of-the-art method that ad- +versarially eliminates mutual information between target and sensitive attribute +representations and (vi) Orth [15] hard codes the means of both sensitive and +target prior distributions to orthogonal means and re-parameterize the encoder +output on the orthogonal priors. Besides, we give the result of (vii) ERM [17] – +vanilla classifier trained without any bias mitigation technique. +3.2 +Comparsion with Baselines +Quantitative results. We summarize quantitative comparisons in Table 2. +It can be observed that all the bias mitigation methods can improve fairness +compared to ERM [17] at the cost of utility. While ensuring considerable clas- +sification accuracy, FCRO achieves significant fairness improvement both jointly + +1.0 +0.861 +ERM +ERM +PARADE +0.25 +PARADE +Adv +0.860 +(个) +0.8 +Adv +FCRO (ours) +FCRO (ours) +/ Conjunctional +AUC ( +0.858 +0.15 +0.4 +★ +optimal +0.857 +moving space +0.10 +w/o column space +0.2 +0.856 +w/o row space +sweep ^c +0.855 +sweep 入, +0.0 + +0.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.09 +0.10 +0.11 +0.12 +0.13 +R +s +A +R×S +RxASxA +Mean Predicted Probability +Fairness - Conjunctional AeD (↓) +SensitiveAttributes0.8 +AUC(↑) +0.7 +k=1 +k=30 +k=3 +k=50 +0.6 +k=5 +k=100 +k=10 +0.5 +500 +750 +100012501500 1750 2000 22502500 +Iterations +% +0.115 +△ED +kept info +0.110 +0.9 +0.105 +0.8 +0.100 +ik= 3 +0 +20 +40 +60 +80 +100 +Rank of SA (k)10 +W. Deng et al. +Fig. 4. (a) Fairness-accuracy trade-off. The perfect point lies in the top left corner. +We report ablations and Pareto fronts of the sweep of hyperparameters. (b) Fairness +of models trained with various numbers and permutations of three sensitive +attributes: race (R), sex (S), and age (A). (c) AUC convergence with different rank k +of SA. (d) Fairness and total variance (the percentage of sensitive information captured +by SA) under different k. +and individually, demonstrating the effectiveness of our representation orthogo- +nality motivation. To summarize, compared with the best performance in each +metric, FCRO reduced classification disparity on subgroups with joint ∆AUC by +2.7% and joint ∆ED by 2.3% respectively, and experienced 0.5% ∆AUC and 0.4% +∆ED boosts regarding the average of three sensitive attributes. +As medical applications are sensitive to classification thresholds, we further +give calibration curves with the mean and standard deviation of subgroups de- +fined on the conjunction of multiple sensitive attributes in Fig. 3 (a). It can be +observed that the vanilla ERM [17] suffers from biased calibration among sub- +groups. Fairness algorithms can help mitigate this, while FCRO shows the most +harmonious deviation and the most trustworthy classification. +Qualitative results. We present the class activation map [2] in Fig. 3 (b). We +observe that the vanilla ERM [17] model tends to look for sensitive evidence +outside the lung regions, e.g., breast, which threatens unfairness. FCRO focuses +on the pathology-related part only for fair Pleural Effusion classification, which +visually confirms the validity of our method. +3.3 +Ablation Studies +Loss modules and hyperparameters. We further investigate the key com- +ponents of FCRO with reference to the fairness-utility trade-off. As shown in +Fig. 4 (a), we present the ablation of key components and the Pareto fronts +(i.e., the set of optimal points) curve of the sweep of a range of hyperparameters + +0.861 +ERM +0.25 +0.860 +(个) +PARADE +Adv +≤ 0.859 +FCRO(ours) +optimal +accumulative space +w/ocolumnspace +0.10 +w/orowspace +0.856 +sweep 入c +0.855 +sweep 入r +0.00 +0.09 +0.10 +0.11 +0.12 +0.13 +R +s +A +R×S +RXA +SxA +Fairness -Joint AED（↓） +Sensitive Attributes +(a) +(b) +0.118 +1.00 +0.80 +0.116 +AED +(%) +0.75 +total variance +0.95 +nation +0.114 +0.90 +Infori +0.60 +k=1 +k=30 +20.108 +k=3 +k=50 +0.55 +k=5 +k=100 +0.106 +0.80.0 +k=10 +0.104 +0.50 +500 +750 +1000 +1250 +1500 +1750 +2000 +2250 +2500 +0 3 +20 +40 +60 +80 +100 +Iterations +Rank of Sa (k) +(c) +(d)On Fairness of Image Classification with Multi-Sensitive Attributes +11 +λc and λr. We observe that removing either column or row space orthogonality +shows a decrease in joint ∆ED of 2.4% and 1.8% respectively, but still being +competitive. Besides, model utility is not sensitive to weights, which fulfills our +motivation of handling a large number of sensitive attributes. We also observe +that applying accumulative space introduced in Section 2.2 achieves a compara- +ble performance. +Training with different sensitive attributes. We present an in-depth abla- +tion study on multiple sensitive attributes in Fig. 4 (b), where models are trained +with various numbers and permutations of attributes. We show all methods per- +form reasonably better than ERM when trained with a single sensitive attribute +but FCRO brought significantly more benefits when trained with the union of +discriminated attributes (e.g., Sex × Age), which consolidate FCRO’s ability for +multi-sensitive attributes fairness. FCRO stand out among all methods. +Different rank k for �SA. We show the effect of choosing different k for column +space orthogonality. As shown in Fig. 4 (c), a lower rank k benefits convergence +of the model thus improving accuracy, which validates our insights in Section. 2.2 +that lower sensitive space rank will improve the utility of target representations. +In Fig. 4 (d), we show that k = 3 is enough to capture over 95% sensitive +information and keep increasing it does not bring too much benefit for fairness, +thus we choose k = 3 to achieve the best utility-fairness trade off. +4 +Conclusion and Future Work +This work studies an essential yet under-explored fairness problem in medical +image classification where samples are with sets of sensitive attributes. We for- +mulate this problem mathematically and propose a novel fair representation +learning algorithm named FCRO, which pursues orthogonality between sensitive +and target representations. Extensive experiments on a large public chest X- +rays demonstrate that FCRO significantly boosts the fairness-utility trade-off both +jointly and individually. Moreover, we show that FCRO performs stably under dif- +ferent situations with in-depth ablation studies. For future work, we plan to test +the scalability of FCRO on an extremely large number of sensitive attributes. +References +1. Adeli, E., Zhao, Q., Pfefferbaum, A., Sullivan, E.V., Fei-Fei, L., Niebles, J.C., Pohl, +K.M.: Representation learning with statistical independence to mitigate bias. In: +Proceedings of the IEEE/CVF Winter Conference on Applications of Computer +Vision. pp. 2513–2523 (2021) +2. Chattopadhay, A., Sarkar, A., Howlader, P., Balasubramanian, V.N.: Grad- +cam++: Generalized gradient-based visual explanations for deep convolutional +networks. In: IEEE winter conference on applications of computer vision (2018) +3. Creager, E., Madras, D., Jacobsen, J.H., Weis, M., Swersky, K., Pitassi, T., Zemel, +R.: Flexibly fair representation learning by disentanglement. In: International con- +ference on machine learning. pp. 1436–1445. PMLR (2019) + +12 +W. Deng et al. +4. Dullerud, N., Roth, K., Hamidieh, K., Papernot, N., Ghassemi, M.: Is fairness only +metric deep? evaluating and addressing subgroup gaps in deep metric learning. The +International Conference on Learning Representations (2022) +5. Glocker, B., Jones, C., Bernhardt, M., Winzeck, S.: Algorithmic encoding of pro- +tected characteristics in image-based models for disease detection. arXiv preprint +arXiv:2110.14755 (2021) +6. Huang, G., Liu, Z., Van Der Maaten, L., Weinberger, K.Q.: Densely connected +convolutional networks. In: Proceedings of the IEEE conference on computer vision +and pattern recognition. pp. 4700–4708 (2017) +7. Irvin, J., Rajpurkar, P., Ko, M., Yu, Y., Ciurea-Ilcus, S., Chute, C., Marklund, H., +Haghgoo, B., Ball, R., Shpanskaya, K., et al.: Chexpert: A large chest radiograph +dataset with uncertainty labels and expert comparison. In: Proceedings of the +AAAI conference on artificial intelligence. vol. 33, pp. 590–597 (2019) +8. Lin, S., Yang, L., Fan, D., Zhang, J.: Trgp: Trust region gradient projection for +continual learning. International Conference on Learning Representations (2022) +9. Liu, E.Z., Haghgoo, B., Chen, A.S., Raghunathan, A., Koh, P.W., Sagawa, S., +Liang, P., Finn, C.: Just train twice: Improving group robustness without training +group information. In: International Conference on Machine Learning (2021) +10. Van der Maaten, L., Hinton, G.: Visualizing data using t-sne. Journal of machine +learning research 9(11) (2008) +11. Madras, D., Creager, E., Pitassi, T., Zemel, R.: Learning adversarially fair and +transferable representations. In: International Conference on Machine Learning. +pp. 3384–3393. PMLR (2018) +12. Madras, D., Creager, E., Pitassi, T., Zemel, R.: Fairness through causal aware- +ness: Learning causal latent-variable models for biased data. In: Proceedings of the +conference on fairness, accountability, and transparency. pp. 349–358 (2019) +13. Roh, Y., Lee, K., Whang, S.E., Suh, C.: Fairbatch: Batch selection for model +fairness. International Conference on Learning Representations (2020) +14. Sagawa, S., Koh, P.W., Hashimoto, T.B., Liang, P.: Distributionally robust neu- +ral networks for group shifts: On the importance of regularization for worst-case +generalization. International Conference on Learning Representations (2019) +15. Sarhan, M.H., Navab, N., Eslami, A., Albarqouni, S.: Fairness by learning orthog- +onal disentangled representations. In: European Conference on Computer Vision. +pp. 746–761. Springer (2020) +16. Shui, C., Xu, G., Chen, Q., Li, J., Ling, C., Arbel, T., Wang, B., Gagné, C.: On +learning fairness and accuracy on multiple subgroups. Advances in Neural Infor- +mation Processing Systems (2022) +17. Vapnik, V.: Principles of risk minimization for learning theory. In: Advances in +Neural Information Processing Systems. vol. 4 (1991) +18. Wadsworth, C., Vera, F., Piech, C.: Achieving fairness through adversarial learning: +an application to recidivism prediction. arXiv preprint arXiv:1807.00199 (2018) +19. Xu, Z., Li, J., Yao, Q., Li, H., Shi, X., Zhou, S.K.: A survey of fairness in medical +image analysis: Concepts, algorithms, evaluations, and challenges. arXiv preprint +arXiv:2209.13177 (2022) +20. Zhang, H., Dullerud, N., Roth, K., Oakden-Rayner, L., Pfohl, S., Ghassemi, M.: +Improving the fairness of chest x-ray classifiers. In: Conference on Health, Inference, +and Learning. pp. 204–233. PMLR (2022) +21. Zhu, W., Zheng, H., Liao, H., Li, W., Luo, J.: Learning bias-invariant represen- +tation by cross-sample mutual information minimization. In: Proceedings of the +IEEE/CVF International Conference on Computer Vision. pp. 15002–15012 (2021) + diff --git a/-9AzT4oBgHgl3EQfhPxU/content/tmp_files/load_file.txt b/-9AzT4oBgHgl3EQfhPxU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a2c4120d0c262c60f14abc534dc98db68ffd1da --- /dev/null +++ b/-9AzT4oBgHgl3EQfhPxU/content/tmp_files/load_file.txt @@ -0,0 +1,756 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf,len=755 +page_content='On Fairness of Medical Image Classification with Multiple Sensitive Attributes via Learning Orthogonal Representations Wenlong Deng1∗, Yuan Zhong2∗, Qi Dou2, and Xiaoxiao Li1 1 Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada 2 Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong, China Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Mitigating the discrimination of machine learning models has gained increasing attention in medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' However, rare works focus on fair treatments for patients with multiple sensitive demo- graphic ones, which is a crucial yet challenging problem for real-world clinical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In this paper, we propose a novel method for fair representation learning with respect to multi-sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We pursue the independence between target and multi-sensitive representa- tions by achieving orthogonality in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Concretely, we enforce the column space orthogonality by keeping target information on the complement of a low-rank sensitive space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Furthermore, in the row space, we encourage feature dimensions between target and sensitive rep- resentations to be orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' The effectiveness of the proposed method is demonstrated with extensive experiments on the CheXpert dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To our best knowledge, this is the first work to mitigate unfairness with respect to multiple sensitive attributes in the field of medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' The code will be available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='com/vengdeng/FCRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 1 Introduction With the increasing application of artificial intelligence systems for medical im- age diagnosis, it is notably important to ensure fairness of image classification models and investigate concealed model biases that are to-be-encountered in complex real-world situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Unfortunately, sensitive attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', race and gender) accompanied by medical images are prone to be inherently encoded by machine learning models [5], and affect the model’s discrimination property [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Recently, fair representation learning has shown great potential as it acts as a group parity bottleneck that mitigates discrimination when generalized to down- stream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Existing methods [1,4,15,16,21] have studied the parity between privileged and unprivileged groups upon just a single sensitive attribute, but neglecting the flexibility with respect to multiple sensitive attributes, in which the conjunctions of unprivileged attributes might also deteriorate discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='01481v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='CV] 4 Jan 2023 2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' A t-SNE [10] visualization of (a) sensitive attribute and (b) target representa- tions learned from our proposed methods FCRO on the CheXpert dataset [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Sensitive embeddings capture subgroups’ variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We claim FCRO enforces fair classification on the target task by learning orthogonal target representations that are invariant over different attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' This is a crucial yet challenging problem hindering the applicability of machine learning models, especially for medical image classification where patients always have many demographic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To date, it is still challenging to effectively learn target-related representa- tions which are both fair and flexible to multiple sensitive attributes, regardless of some promising investigations recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' For instance, adversarial methods [1,11] produce robust representations by formulating a min-max game between an en- coder that learns class-related representation and an adversary that removes sen- sitive information from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Disentanglement-based methods [4,21] achieve separa- tion by minimizing the mutual information between target and sensitive attribute representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' These methods typically gain efficacy by means of carefully de- signing objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To extend them to the multi-attribute setting, additional loss functions have to be explored, which should handle gradient conflict or interfer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Methods using variational autoencoder [3] decompose the latent distribu- tions of target and sensitive and penalize their correlation for disentanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' However, aligning the distribution of the sensitive attributes is difficult or even intractable given the complex combination of multiple factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Besides, there are some fairness methods based on causal inference [12] or bi-level optimization [16], which also learn debiased while multi-attributes inflexible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Recently, disentanglement is vigorously interpreted as the orthogonality of a de- composed target-sensitive latent representation pair by [15], where they predefine a pair of orthogonal subspaces for target and sensitive attribute representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In a multi-sensitive attributes setting, the dimension of the target space would be continuously compressed and how to solve it is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In this paper, we propose a new method to achieve Fairness via Column- Row space Orthogonality (called FCRO) by learning fair representations for med- ical image classification with multiple sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' FCRO considers multi- sensitive attributes by encoding them into a unified attribute representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' It achieves a best trade-off for fairness and data utility (see illustrations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 1) Race-Sex-Age White, Male, 60- White, Male, 18-60 White, Female, 60- 福 White, Female, 18-60 non-White, Male, 60- non-White, Male, 18-60 non-White, Female, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' non-White, Female, 18-60 (a) Sensitivie attribute representation (b) Target representationOn Fairness of Image Classification with Multi-Sensitive Attributes 3 Classi%ier ℎ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' " 𝑧̃# $ ⋯ Sensitive Encoder 𝜙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 𝑍!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Classifier ℎ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='" 𝑋 𝑌 𝐴% 𝑍!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 𝑍# 𝐴& ⋮ 𝜙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (𝑋) 𝜙"(𝑋) × (a) ∈ ℝ\'×) 𝑋 𝑧̂# 𝑧̂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' + Row space (c) 𝑧̃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 𝐿!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='" 𝑗 Column space (b) 𝐿#$%&\' = 𝑐(𝑧̃" (, 𝑆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=') 𝐿)$)*+ = 𝑟(𝑧̂" , , 𝑧̂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' ) Classi%ier ℎ" Target Encoder 𝜙# 𝑍# ∈ ℝ\'×) 𝑧̃# 𝐿# 𝑖 𝑘 𝑍# ⊥ 𝑍!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' frozen modules forward backward cause correlated Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Overview of our proposed method FCRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (a) The graphical model of orthogo- nal representation learning for fair medical image classification with multiple sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (b) The novel column-row space orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In the column space, we encourage the target model to learn representations in the complement of a low-rank sensitive space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In the row space, we enforce each row vector (feature dimension) of the target and sensitive attribute representations to be orthogonal to each other (c) The overall training pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We use a pre-trained multi-sensitive branch, and propagate orthogonal gradients to target encoder φT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' via orthogonality in both column and row spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Our contributions are summa- rized as follows: (1) We tackle the practical and challenging problem of fairness given multiple sensitive attributes for medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To the best of our knowledge, this is the first work to study fairness with respect to multi- sensitive attributes in the field of medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (2) We relax the independence of target and sensitive attribute representations by orthogonality which can be achieved by our proposed novel column and row losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (3) We conduct exten- sive experiments on the CheXpert [7] dataset with over 80,000 chest X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' FCRO achieves a superior fairness-utility trade-off over state-of-the-art methods regarding multiple sensitive attributes race, sex, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='1 Problem Formulation Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We consider group fairness in this work, group fairness articulates the equality of some statistics like predictive rate between certain groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Con- sidering a binary classification problem with column vector inputs x ∈ X, AP PORT UPRICHT4 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' labels y ∈ Y = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Multi-sensitive attributes a ∈ A is vector of m at- tributes sampled from the conjunction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Cartesian product, of sensitive at- tributes A = � i∈[m] Ai * where the i-th sensitive attribute Ai ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Our training data consist of tuples D = {(x, y, a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We denote the classification model �y = f(x) = hT (φT (x)) that predicts a class label given an input x, where φT : X �→ Rd is a feature encoder for target embeddings, and hT : Rd �→ R is a scoring function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Similarly, we consider a sensitive attribute model g(x) = {hA1(φA(x)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', hAm(φA(x))} that predicts sensitive attributes associ- ated with input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Given the number of samples n, the input data representa- tion is X = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' , xn] and we denote the feature representation ZT = φT (X), ZA = φA(X) ∈ Rd×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Fair classifier on multiple sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' A classifier predicts y given an input x by estimating the posterior probability p(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' When inputs that are affected by their associated attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' , Am} → X) are fed into the network, the posterior probability is written as p(y|x, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Since biased information from A is encoded, this can lead to an unfair prediction by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' For example, in the diagnosis of a disease with sensitive attributes age, sex, and race, a biased classifier will result in p(�y|A = male, old, black) ̸= p(�y|A = female, young, white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In this work, we focus on equalized odds (ED), which is a commonly used and crucial criterion of fair classification in the medical domain [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In our case, ED regarding multiple sensitive attributes can be formulated as follows: P(�Y = y|A = π1, Y = y) = P(�Y = y|A = π2, Y = y), ∀π1, π2 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (1) Recent methods [16] suggest achieving ED for a classifier by enforcing �Y ⊥ A|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In other words, a fair classifier is expected to be independent of multi-sensitive information: p(y|x) = p(y|x, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Fair representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To enforce our aforementioned conditions, we follow [15] and introduce target embedding zT and multi-attribute embedding zAi that is generated from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' As in the causal structure graph for the classifier depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2 (a), if zT and zAi are independent, the probability of a fair classifier p(y|x, a) is written as: p(y, a|x) = p(y|x, a)p(x|a)p(a) p(x) = p(y|x)p(a|x) (2) = p(y|zT )p(zT |x) � i∈[m] p(ai|zAi)p(zAi|x), (3) and we call zT fair representation for the target task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', disease diagnosis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To this end, we aim to maximize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (3) with the conditional independence constraint to train a fair classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' It is noteworthy that in the multisensitive attributes setting, forcing zT to be independent on all zAi, ∀i ∈ [m] is challenging and even intractable when m is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Therefore, we propose to encode multi- sensitive attributes into a single compact encoding zA that is still predictive for [m] = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='., m} On Fairness of Image Classification with Multi-Sensitive Attributes 5 classifying attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', zA → {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' , am}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Then we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (3) as maximizing the likelihood with the independence constraint on zT and zA: p(y, a|x) = p(y|zT )p(zT |x)p(a|zA)p(zA|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (4) However, optimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (4) brings two technical questions: Q1: How to satisfy the independence constraint for zT and zA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' A1: We relax the independence by enforcing orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Different from pre- defined orthogonal space in [15], we enforce orthogonality in both column spaces (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2) and row spaces (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='3) of ZT and ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Q2: How to estimate p(y|zT ), p(zT |x), p(a|zA), p(zA|x)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' A2: We train two convolutional neural nets encoders zT = φT (x) and zA = φA(x) to approximate p(zT |x) and p(zA|x) respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' we train two multi- layer perception classifier y = hT (zT ) and a = hA(zA) to approximate p(y|zT ) and p(a|zA) respectively (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 Column Space Orthogonality First, we focus on the column space of the target and the sensitive attribute representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Column space orthogonality aims to learn target representations ZT that fulfill the following two aims: 1) have the least projection onto the sensitive space SA and 2) preserve the representation power to predict Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Denote the target representation ZT = [�z1 T , �z2 T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' , �zn T ] and the sensitive at- tribute representation ZA = [�z1 A, �z2 A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' , �zn A], where �zi ∈ Rd×1 is a column vector for i ∈ [n], we represent the column space for ZT and ZA as ST = span(ZT ) and SA = span(ZA) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Aim 1 can be achieved by forcing ST = S⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Although both �zT , �zA ∈ Rd, their coordinates may not be aligned as they are generated from two separate encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' As a result, if d ≪ ∞, then there is no straightforward way to achieve ST ⊥ SA by directly constraining �zi T , �zj A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', forcing (�zi T )⊤�zj A = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Aim 2 can be achieved by seeking a low-rank rep- resentation �SA for SA, whose rank is k such that k ≪ d, because we have rank(ST ) + rank(SA) = d if ST = S⊥ A holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Then S⊥ A would be a high- dimensional space with sufficient representation power for target embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' This is especially important when we face multiple sensitive attributes, as the total size of the space is d, and increasing the number of sensitive attributes would limit the capacity of ST to learn predictive �zT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To this end, we first pro- pose to find the low rank sensitive attribute representation space �SA, and then encourage ZT to be in �SA’s complement �S ⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Construct low-rank multi-sensitive space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We apply Singular Value De- composition (SVD) on ZA = UAΣAVA to construct the low-rank space �SA, where UA, VA ∈ Rd×d are orthogonal matrices with left and right singular vec- tors ui ∈ Rd and vi A ∈ Rn respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' And ΣA ∈ Rd×n is a diagonal matrix with descending non-negative singular values {δi A}min{n,d} i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Then we extract the most important k left singular vectors to construct �SA = [u1 A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', uk A], where k controls how much sensitive information to be captured in �SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' It is notable that �SA is 6 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' agnostic to the number of sensitive attributes because they share the same ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' For situations that can not get the whole dataset at once, we follow [8] to select most important bases from both bases of old iterations and newly constructed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Thus providing an accumulative low-rank space construction variant to update �SA iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' As we do not observe significant performance differences between these two variants (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4 (a)), we use and refer to the first one in this paper if there is no special clarification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Column orthogonal loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' With the low-rank space �SA for multiple sensitive attributes, we encourage φT to learn representations in its complement �S⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' No- tice that �S⊥ A can also be interpreted as the kernel of the projection onto �SA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', �S⊥ A = Ker(proj � SA�zT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Therefore, we achieve column orthogonal loss by minimizing the projection of ZT to �SA, which can be defined as: Lcorth = c(ZT , �SA) = n � i=1 ��� �S⊤ A �zi T ��� 2 2 ���zi T ��2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (5) As �SA is a low-rank space, �S⊥ A will have abundant freedom for φT to extract target information, thus reserving predictive ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='3 Row Space Orthogonality Then, we study the row space of target and sensitive attribute representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Row space orthogonality aims to learn target representations ZT that have the least projection onto the sensitive row space �SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In other words, we want to ensure orthogonality on each feature dimension between ZT and ZA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Denote target representation ZT = [�z1 T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' �z2 T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' �zd T ] and sensitive attribute representation ZA = [�z1 A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' �z2 A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' �zd A], where �zi ∈ R1×n is a row vector for i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We represent row space for target representations and sensitive attribute representations as �ST = span(Z⊤ T ) and �SA = span(Z⊤ A) correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Different from column space orthogonality, as the coordinates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', the index of samples) of �zA and �zT are aligned, forcing �ST = �S ⊥ A can be directly applied by achieving ZT Z⊤ A: {(ZT Z⊤ A)i,j = �zi T (�zj A)⊤, i, j ∈ d} = n � t=1 (�zi T )t(�zj A)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (6) Unlike column space, the orthogonality here won’t affect the utility, as the row vector �zT is not directly correlated to the target y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To be specific, we let pair-wise row vectors ZT = [�z1 T , �z2 T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' , �zd T ] and ZA = [�z1 A, �z2 A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' , �zd A] have a small inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Then for any i, j ∈ [d], we try to minimize < �zi T , �zj A >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Here we slightly modify the orthogonality by extra subtracting the mean vector µA and µT from ZA and ZT respectively, where µ = Ei∈[d]�zi ∈ R1×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Then orthogonality loss will naturally be integrated into a covariance loss: Lrorth = r(ZT , ZA) = 1 d2 d � i=1 d � j=1 � (�zi T − µT )(�zj A − µA)⊤�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (7) On Fairness of Image Classification with Multi-Sensitive Attributes 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' CheXpert dataset statistics and group positive rate p(y = 1|a) regarding pleural effusion with three sensitive attributes race, sex, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Dataset #Sample Group Positive Rate Race Sex Age (White/Non-white/gap) (Male/Female/gap) (>60/≤ 60/gap) Original 127130 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='410/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='393/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='017 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='405/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='408/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='003 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='440/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='359/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='081 Augmented 88215 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='264/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='386/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='122 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='254/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='379/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='125 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='264/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='386/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='122 In this way, the resulting loss encourages each feature of ZT to be independent of features in ZA thus suppressing the sensitive-encoded covariances that cause the unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='4 Overall Training In this section, we introduce the overall training schema as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' For the sensitive branch, since we observe that using a shared encoder may threaten sensitive information leakage to classification [4] or obtain unsatisfied sensitive attribute representations [15], we pretrain {φA, hA1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', hAm} for mul- tiple sensitive attributes using the sensitive objective as Lsens = 1 m � i∈[m] LAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Here we use cross-entropy loss as LAi for the i-th sensitive attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Hence p(zA|x) and p(a|zA) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (4) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Then, the multi-sensitive space SA is constructed as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 over the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' For the target branch, we use cross-entropy loss as our classification objective LT to supervise the train- ing of φT and hT and estimate p(zT |x) and p(y|zT ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Here we do not make additional constraints to LT , which means it can be replaced by any other task-specific losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' At last, we apply our column and row orthog- onality losses Lcorth and Lrorth to representations as introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='3 along with detached SA and ZA to approximate independence between p(zA|x) and p(zT |x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' The overall target objective is given as: Ltarg = LT + λcLcorth + λrLrorth, (8) where λc and λr are hyper-parameters to weigh orthogonality and balance fair- ness and utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 3 Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='1 Setup Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We adopt CheXpert dataset [7] to predict Pleural Effusion in chest X-rays, as it’s crucial for chronic obstructive pulmonary disease diagnosis with high incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Subgroups are defined based on the following binarized sensitive attributes: self-reported race and ethnicity, sex, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Note that data bias 8 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Comparasion of predicting Pleural Effusion on CheXpert dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We report the mean and standard deviation of 5-fold models trained with multi-sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' AUC is used as the utility metric, and fairness is evaluated using disparities among subgroups defined on multi-sensitive attributes jointly and individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Methods AUC (↑) Subgroup Disparity (↓) Joint Race Sex Age ∆AUC ∆ED ∆AUC ∆ED ∆AUC ∆ED ∆AUC ∆ED ERM [17] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='038 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='005) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='017) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='013) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='009) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='017) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='008) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='014) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='010) G-DRO [14] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='051 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='012) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='034) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='003) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='014) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='010) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='025) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='002) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='010) JTT [9] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='030 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='020) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='017) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='023) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='008) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='016) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='002) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='006) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='007) Adv [18] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='023 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='002) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='009) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='018) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='009) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='003) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='008) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) BR-Net [1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='039 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='001) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='025) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='023) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='008) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='013) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='012) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='025) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='006) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='006) PARADE [4] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='031 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='002) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='022) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='032) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='002) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='010) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='006) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='023) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='006) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='011) Orth [15] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='032 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='007) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='022) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='016) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='005) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='012) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='009) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='012) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='006) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='005) FCRO (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='019 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='001) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='022) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='013) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='003) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='008) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='008) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='004) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='006) (positive rate gap) is insignificant in the original dataset (see Table 1, row ’orig- inal’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To demonstrate the effectiveness of bias mitigation methods, we amplify the data bias by (1) firstly dividing the data into different groups according to the conjunction of multi-sensitive labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (2) secondly calculating the positive rate of each subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (3) sampling out patients and increase each subgroup’s positive rate gap to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='12 (see Table 1, row ‘augmented’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We resize all images to 224×224 and split the dataset into a 15% test set, and an 85% 5-fold cross-validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We use the area under the ROC curve (AUC) to evaluate the utility of classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To measure fairness, we follow [13] and compute subgroup disparity with respect to ED (denoted as ∆ED, which is based on true positive rate (TPR) and true negative rate (TNR)) in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We quantify ED disparity as: ∆ED = max y∈Y,π1,π2∈A ���P(�Y = y|A = π1, Y = y) − P(�Y = y|A = π2, Y = y) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (9) We also follow [20] and compare subgroup disparity regarding AUC (denoted as ∆AUC), which gives a threshold-free fairness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Note that we evaluate disparities both jointly and individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' The joint disparities are calculated with respect to the conjunction of multiple sensitive attributes A, and the individual disparities are calculated with respect to a specific sensitive attribute Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In our implementation, all methods use the same training protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We choose DenseNet-121 [6] as the backbone, but replace the On Fairness of Image Classification with Multi-Sensitive Attributes 9 (a) (c) (b) INPUT FCRO ERM (a) (b) (c) INPUT FCRO ERM (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (a) Subgroup calibration curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We report quantile calibration curves of the mean (the line) and standard deviation (the shadow around it) of different subgroups defined by the conjunction of race, sex, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Larger shadow areas correspond to more severe unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (b) Class activation map [2] generated from vanilla ERM [17] and FCRO (ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' final layer with a linear layer to extract 128-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' The optimizer is Adam with learning rate of 1e−4, and weight decay of 4e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We train for 40 epochs with a batch size of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We sweep a range of hyper-parameters for each method and empirically set λc = 80, λr = 500, and k = 3 for FCRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We train models in 5-fold with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In each fold, we sort all the validations according to AUC and select the best model with the lowest average ∆ED regarding each sensitive attribute among the top 5 utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We compare our method with (i) G-DRO [14] and (ii) JTT [9] – seek- ing low worst-group error by minimax optimization on group fairness and target task error, which can be naturally regarded as multi-sensitive fairness methods by defining subgroups with multi-sensitive attributes conjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We also extend recent state-of-the-art fair representation learning methods on single sensitive attributes to multiple ones and compare our method with them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' including (iii) Adv [18] and (iv) BR-Net [1] – achieve fair representation via disentanglement using adversarial training,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (v) PARADE [4] – a state-of-the-art method that ad- versarially eliminates mutual information between target and sensitive attribute representations and (vi) Orth [15] hard codes the means of both sensitive and target prior distributions to orthogonal means and re-parameterize the encoder output on the orthogonal priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Besides, we give the result of (vii) ERM [17] – vanilla classifier trained without any bias mitigation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 Comparsion with Baselines Quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We summarize quantitative comparisons in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' It can be observed that all the bias mitigation methods can improve fairness compared to ERM [17] at the cost of utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' While ensuring considerable clas- sification accuracy, FCRO achieves significant fairness improvement both jointly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='861 ERM ERM PARADE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='25 PARADE Adv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='860 (个) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='8 Adv FCRO (ours) FCRO (ours) / Conjunctional AUC ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='4 ★ optimal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='857 moving space 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='10 w/o column space 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='856 w/o row space sweep ^c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='855 sweep 入, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='0 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='13 R s A R×S RxASxA Mean Predicted Probability Fairness - Conjunctional AeD (↓) SensitiveAttributes0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='8 AUC(↑) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='7 k=1 k=30 k=3 k=50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='6 k=5 k=100 k=10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='5 500 750 100012501500 1750 2000 22502500 Iterations % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='115 △ED kept info 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='100 ik= 3 0 20 40 60 80 100 Rank of SA (k)10 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (a) Fairness-accuracy trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' The perfect point lies in the top left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We report ablations and Pareto fronts of the sweep of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (b) Fairness of models trained with various numbers and permutations of three sensitive attributes: race (R), sex (S), and age (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (c) AUC convergence with different rank k of SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' (d) Fairness and total variance (the percentage of sensitive information captured by SA) under different k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' and individually, demonstrating the effectiveness of our representation orthogo- nality motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' To summarize, compared with the best performance in each metric, FCRO reduced classification disparity on subgroups with joint ∆AUC by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='7% and joint ∆ED by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='3% respectively, and experienced 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='5% ∆AUC and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='4% ∆ED boosts regarding the average of three sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' As medical applications are sensitive to classification thresholds, we further give calibration curves with the mean and standard deviation of subgroups de- fined on the conjunction of multiple sensitive attributes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' It can be observed that the vanilla ERM [17] suffers from biased calibration among sub- groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Fairness algorithms can help mitigate this, while FCRO shows the most harmonious deviation and the most trustworthy classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We present the class activation map [2] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We observe that the vanilla ERM [17] model tends to look for sensitive evidence outside the lung regions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', breast, which threatens unfairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' FCRO focuses on the pathology-related part only for fair Pleural Effusion classification, which visually confirms the validity of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='3 Ablation Studies Loss modules and hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We further investigate the key com- ponents of FCRO with reference to the fairness-utility trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4 (a), we present the ablation of key components and the Pareto fronts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', the set of optimal points) curve of the sweep of a range of hyperparameters 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='861 ERM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='860 (个) PARADE Adv ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='859 FCRO(ours) optimal accumulative space w/ocolumnspace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='10 w/orowspace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='856 sweep 入c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='855 sweep 入r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='13 R s A R×S RXA SxA Fairness -Joint AED（↓） Sensitive Attributes (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='118 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='116 AED (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='75 total variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='95 nation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='90 Infori 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='60 k=1 k=30 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='108 k=3 k=50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='55 k=5 k=100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='0 k=10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='50 500 750 1000 1250 1500 1750 2000 2250 2500 0 3 20 40 60 80 100 Iterations Rank of Sa (k) (c) (d)On Fairness of Image Classification with Multi-Sensitive Attributes 11 λc and λr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We observe that removing either column or row space orthogonality shows a decrease in joint ∆ED of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='4% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='8% respectively, but still being competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Besides, model utility is not sensitive to weights, which fulfills our motivation of handling a large number of sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We also observe that applying accumulative space introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 achieves a compara- ble performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Training with different sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We present an in-depth abla- tion study on multiple sensitive attributes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4 (b), where models are trained with various numbers and permutations of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We show all methods per- form reasonably better than ERM when trained with a single sensitive attribute but FCRO brought significantly more benefits when trained with the union of discriminated attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Sex × Age), which consolidate FCRO’s ability for multi-sensitive attributes fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' FCRO stand out among all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Different rank k for �SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We show the effect of choosing different k for column space orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4 (c), a lower rank k benefits convergence of the model thus improving accuracy, which validates our insights in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='2 that lower sensitive space rank will improve the utility of target representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4 (d), we show that k = 3 is enough to capture over 95% sensitive information and keep increasing it does not bring too much benefit for fairness, thus we choose k = 3 to achieve the best utility-fairness trade off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4 Conclusion and Future Work This work studies an essential yet under-explored fairness problem in medical image classification where samples are with sets of sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' We for- mulate this problem mathematically and propose a novel fair representation learning algorithm named FCRO, which pursues orthogonality between sensitive and target representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Extensive experiments on a large public chest X- rays demonstrate that FCRO significantly boosts the fairness-utility trade-off both jointly and individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Moreover, we show that FCRO performs stably under dif- ferent situations with in-depth ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' For future work, we plan to test the scalability of FCRO on an extremely large number of sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Adeli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Pfefferbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Sullivan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Fei-Fei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Niebles, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Pohl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' : Representation learning with statistical independence to mitigate bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 2513–2523 (2021) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Chattopadhay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Howlader, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Balasubramanian, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' : Grad- cam++: Generalized gradient-based visual explanations for deep convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: IEEE winter conference on applications of computer vision (2018) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Creager, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Madras, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Jacobsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Weis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Swersky, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Pitassi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Zemel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Flexibly fair representation learning by disentanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: International con- ference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 1436–1445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' PMLR (2019) 12 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Dullerud, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Roth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Hamidieh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Papernot, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Ghassemi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Is fairness only metric deep?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' evaluating and addressing subgroup gaps in deep metric learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' The International Conference on Learning Representations (2022) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Glocker, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Jones, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Bernhardt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Winzeck, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Algorithmic encoding of pro- tected characteristics in image-based models for disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='14755 (2021) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Van Der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Weinberger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' : Densely connected convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4700–4708 (2017) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Irvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Rajpurkar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Ko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Ciurea-Ilcus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Chute, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Marklund, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Haghgoo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Ball, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Shpanskaya, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' : Chexpert: A large chest radiograph dataset with uncertainty labels and expert comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: Proceedings of the AAAI conference on artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 590–597 (2019) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Fan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Trgp: Trust region gradient projection for continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' International Conference on Learning Representations (2022) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Liu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Haghgoo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Raghunathan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Koh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Sagawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Liang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Finn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Just train twice: Improving group robustness without training group information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: International Conference on Machine Learning (2021) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Van der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Visualizing data using t-sne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Journal of machine learning research 9(11) (2008) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Madras, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Creager, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Pitassi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Zemel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Learning adversarially fair and transferable representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 3384–3393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' PMLR (2018) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Madras, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Creager, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Pitassi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Zemel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Fairness through causal aware- ness: Learning causal latent-variable models for biased data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: Proceedings of the conference on fairness, accountability, and transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 349–358 (2019) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Roh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Whang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Suh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Fairbatch: Batch selection for model fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' International Conference on Learning Representations (2020) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Sagawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Koh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Hashimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Liang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Distributionally robust neu- ral networks for group shifts: On the importance of regularization for worst-case generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' International Conference on Learning Representations (2019) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Sarhan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Navab, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Eslami, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Albarqouni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Fairness by learning orthog- onal disentangled representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: European Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 746–761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Springer (2020) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Shui, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Ling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Arbel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Gagné, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': On learning fairness and accuracy on multiple subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Advances in Neural Infor- mation Processing Systems (2022) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Vapnik, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Principles of risk minimization for learning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: Advances in Neural Information Processing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 4 (1991) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Wadsworth, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Vera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Piech, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Achieving fairness through adversarial learning: an application to recidivism prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' arXiv preprint arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='00199 (2018) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Yao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' : A survey of fairness in medical image analysis: Concepts, algorithms, evaluations, and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content='13177 (2022) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Dullerud, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Roth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Oakden-Rayner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Pfohl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Ghassemi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Improving the fairness of chest x-ray classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: Conference on Health, Inference, and Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 204–233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' PMLR (2022) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Zheng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Liao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=', Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=': Learning bias-invariant represen- tation by cross-sample mutual information minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' In: Proceedings of the IEEE/CVF International Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} +page_content=' 15002–15012 (2021)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf'} diff --git a/.gitattributes b/.gitattributes index f80db8ff288837265d923559412d177ab69aa407..38297ed21a027579c2d757261a67e9e7b933f60d 100644 --- a/.gitattributes +++ b/.gitattributes @@ -170,3 +170,5 @@ GdE1T4oBgHgl3EQfrAWz/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex htA0T4oBgHgl3EQfIP9i/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text JdAyT4oBgHgl3EQffviy/content/2301.00347v1.pdf filter=lfs diff=lfs merge=lfs -text p9FLT4oBgHgl3EQfiS8Z/content/2301.12106v1.pdf filter=lfs diff=lfs merge=lfs -text +7dAzT4oBgHgl3EQfgPzo/content/2301.01467v1.pdf filter=lfs diff=lfs merge=lfs -text +FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/1dA0T4oBgHgl3EQfMv9X/content/tmp_files/2301.02136v1.pdf.txt b/1dA0T4oBgHgl3EQfMv9X/content/tmp_files/2301.02136v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..95623847cb90a0bf769e1413864bbfe2934c73d0 --- /dev/null +++ b/1dA0T4oBgHgl3EQfMv9X/content/tmp_files/2301.02136v1.pdf.txt @@ -0,0 +1,1597 @@ +MULTISCALE TRANSFORMS FOR SIGNALS ON SIMPLICIAL COMPLEXES +NAOKI SAITO∗, STEFAN C. SCHONSHECK †, AND EUGENE SHVARTS ‡ +Abstract. Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar- +Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Pack- +ets (NGWPs); and their relatives—were developed for analyzing data recorded on nodes of a given graph. In this +article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more gen- +erally κ-dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to +use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ-dimensional simplices in +a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demon- +strate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial +complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset. +Key words. Simplicial complexes, graph basis dictionaries, hierarchical partitioning, Fiedler vectors, Hodge +Laplacians, Haar-Walsh wavelet packets +1. Introduction. For conventional digital signals and images sampled on regular lat- +tices, multiscale basis dictionaries, i.e., wavelet packet dictionaries including wavelet bases, +local cosine dictionaries, and their variants (see, e.g., [50, Chap. 4, 7], [23, Chap. 6, 7], [30, +Chap. 8]), have a proven track record of success: JPEG 2000 Image Compression Stan- +dard [41, Sec. 15.9]; Modified Discrete Cosine Transform (MDCT) in MP3 [41, Sec. 16.3]; dis- +criminant feature extraction for signal classification [37, 38, 39], just to name a few. Consid- +ering the abundance of data measured on graphs and networks and the increasing impor- +tance to analyze such data (see, e.g., [11, 31, 6, 29, 46]), it is quite natural to lift/generalize +these dictionaries to the graph setting. Our group have developed the graph versions of the +block/local cosine and wavelet packet dictionaries for analysis of graph signals sampled +at nodes so far. These include the Generalized Haar-Walsh Transform (GHWT) [17], the +Hierarchical Graph Laplacian Eigen Transform (HGLET) [18], the Natural Graph Wavelet +Packets (NGWPs) [7], and their relatives [20, 45, 40]; see also [19, 21]. Some of these will be +briefly reviewed in the later sections. +In this article, we propose their generalization for analyzing data recorded on edges, +faces (i.e., triangles), or more generally cells (i.e., polytopes) of a class of special graphs +called simplicial complexes (e.g., a triangle mesh of a manifold). The key idea is to use the +Hodge Laplacians and their variants for hierarchical partitioning of a set of κ-dimensional +simplices in a given simplicial complex, and then build localized basis functions on these +partitioned subsets. We demonstrate their usefulness for data representation on both il- +lustrative synthetic examples and real-world simplicial complexes generated from a co- +authorship/citation dataset and an ocean current/flow dataset. +1.1. Related work. Graph-based methods for analyzing data have been widely adopted +in many domains, [2, 32, 10]. Often, these graphs are fully defined by data (such as a graph +of social media “friends"), but they can also be induced through the persistence homology +of generic point clouds [4]. In either case, the vast majority of these analytical techniques +deal with signals which are defined on the nodes of a given graph. More recently, there +has been a surge in interest in studying signals defined on edges, triangles, and higher- +dimensional substructures within the graph [4, 47, 14, 1, 5]. The fundamental tool em- +ployed for analyzing these signals, the Hodge Laplacian, has been studied in the context +of differential geometry for over half a century but has only recently entered the toolbox +∗Department of Mathematics, University of California, Davis (saito@math.ucdavis.edu, ). +†Department of Mathematics, University of California, Davis (scschonsheck@ucdavis.edu). +‡Department of Mathematics, University of California, Davis (eshvarts@ucdavis.edu) +1 +arXiv:2301.02136v1 [cs.SI] 28 Dec 2022 + +of applied mathematics. This rise in popularity is largely due to the adaptation of discrete +differential geometry [9] in applications in computer vision [28, 36], statistics [24], topo- +logical data analysis [5, 43], and network analysis [42]. +One of the key challenges to applying wavelets and similar constructions to node- +based graph signals is that graphs lack a natural translation operator, which prevents the +construction of convolutional operators and traditional Littlewood-Paley theory [19, 25, +44]. This challenge is also present for general κ-dimensional simplices. One method for +overcoming this difficulty is to perform convolution solely in the “frequency” domain and +define wavelet-like bases entirely in the coefficient space of the Laplacian (or in this case +Hodge Laplacian) transform. Following this line of research, there have been several ap- +proaches to defining wavelets [35] and convolutional neural networks [12] in which the +input signal is transformed in a series of coefficients in the eigenspace of the Hodge Lapla- +cian. Unfortunately, the atoms (or basis vectors) generated by these methods are not al- +ways locally supported, and can be difficult to interpret their role in analyzing a given +graph signal. +An alternative path to the creation of wavelet-like dictionaries and transforms is to +first develop a hierarchical block decomposition of the domain and then use this to de- +velop multiscale transforms [18, 17, 40]. These techniques rely on recursively computing +bipartitions of the domain and then generating localized bases on the subsets of the do- +main. In this work, we propose a simplex analog to the Fielder vector [16] to solve a relaxed +version of the simplex-normalized-cut problem, which we can apply iteratively to develop +a hierarchical bipartition of the κ-dimensional simplices in a simplicial complex. From +here, we are able to apply the general scheme of [18] and [17] to develop the Hierarchi- +cal Graph Laplacian Eigen Transform and the Generalized Haar-Walsh Transform, respec- +tively, for a given collection of simplices of an arbitrarily high order. As a result, we can +also generate orthonormal Haar bases, orthonormal Walsh bases, as well as data-adaptive +orthonormal bases using the best-basis selection method [8]. +1.2. Outline. This article is organized as follows: In Section 2 we formally describe +simplicial complexes and how their geometry leads to notions of adjacency and orienta- +tion. This allows us to define boundary operators, which admits a map between the κ and +κ ± 1 degree faces of the complex as well discrete differential operators acting on signals +defined on the complex. In Section 3 we use these boundary operators to describe the +Hodge Laplacian and discuss several different variants, some analogous to different nor- +malizations of the graph Laplacian and some more novel. In Section 4 we show how the +eigenvectors of the Hodge Laplacian can be use to solve relaxed-cut-like problems to parti- +tion a complex. We also develop hierarchical bipartitions, which decompose a given com- +plex roughly in half at each level until we are left with a division into individual elements. +In Section 5 we use these bipartitions to develop orthonormal Haar bases. In Section 6, we +create overcomplete dictionaries based on given bipartitions and, as a consequence, are +also able to define a canonical orthonormal Walsh Basis. In Section 7, we present numeri- +cal experiments on both illustrative synthetic examples and real-world problems in signal +approximation, clustering, and supervised classification. Finally, we conclude this article +with Section 8 discussing our potential future work. +2. Simplicial Complexes. In this section we review concepts from algebraic topology +to formally define simplicial complexes and introduce some notions of how two simplices +can be “adjacent", for a more thorough review see [4, 14]. Given a vertex set V = {v1,...,vn}, +a κ-simplex σ is a (κ+1)-subset of V . A face of σ is a κ-subset of σ, and so σ has κ+1 faces. +A co-face of σ is a (κ+1)-simplex, of which σ is a face. +Suppose σ = {vi1,...,viκ+1}, i1 < ··· < iκ+1, and α ⊂ σ is its face. Then, σ\α consists of a +2 + +Fig. 1: In this small 2-complex C, e1 ∼ e4 because they share the face v2, and e1 ∼ e2 be- +cause they share the face v1. Further e1 ≃ e2 because their hull t1 ∈ C, but e1 � e4, so that +e1 ∼ +1 e4. We have t1 ∼ t2 because they share the face e3, and also t1 ∼ +2 t2. +single vertex; let viℓ∗ be that vertex where 1 ≤ ℓ∗ ≤ κ+1. Then the natural parity of σ with +respect to its face α is defined as +nat(σ,α) := (−1)ℓ∗ . +When α is not a face of σ, nat(σ,α) = 0. The natural parity of κ-simplices with respect to +their faces generalizes the idea of a directed edge having a head vertex and a tail vertex, and +is “natural” because it disallows situations analogous to a directed edge with two heads or +two tails. +A simplicial complex C is a collection of simplices closed under subsets, where if σ ∈ C, +then α ⊂ σ =⇒ α ∈ C. In particular, if σ ∈ C, so does each face of σ. Let κmax(C) � +max +� +κ|σ ∈ C is a κ-simplex +� +, and let Cκ denote the set of κ-simplices in C for each κ = +1,...,κmax. When κ > κmax, Cκ = �. We also refer toC as a κ-complex to note that κmax(C) = +κ. Let a κ-region of C refer to any non-empty subset of Cκ. +Let C be a simplicial complex, and σ,τ ∈ Cκ, for some κ > 0. When σ,τ share a +face, they are weakly adjacent, denoted by σ ∼ τ. Their shared boundary face is denoted +bd(σ,τ). When σ ∼ τ, additionally they both share a co-face, their hull, denoted by hl(σ,τ). +If σ,τ ∈ C, σ ∼ τ, and hl(σ,τ) ∈ C, then σ,τ are strongly adjacent, denoted by σ ≃ τ. If σ ∼ τ, +but σ � τ in C, then σ,τ are κ-adjacent, denoted σ ∼ +κ τ. +2.1. Oriented Simplicial Complexes and Boundary Operators. An oriented simplex +σ further has an orientation pσ ∈ {±1}, which indicates whether its parity with its faces +is the same as, or opposite to, its natural parity. When pσ = +1, we say σ is in natural +orientation. For example, a directed edge e = (vi,v j ) for i < j is in natural orientation, +while if i > j, pe = −1. An oriented simplicial complex contains at most one orientation +for any given simplex. +Let Xκ be the space of real-valued functions on Cκ for each κ ∈ {0,1,...,κmax(C)}. In +the case of graphs, X0 consists of functions taking values on vertices, or graph signals. +X1 consists of functions on edges, or edge flows. A function in X1 is positive when the +corresponding flow direction agrees with the edge orientation, and negative when the flow +disagrees. X2 consists of functions on oriented triangles. +Given an oriented simplicial complex C, for each κ ∈ {0,1,...,κmax}, the boundary op- +erator is a linear operator Bκ : Xκ+1 �→ Xκ, where for σ ∈ Cκ+1, α ∈ Cκ, the corresponding +matrix entries are [Bκ]ασ = pσpα nat(σ,α). Likewise, the coboundary operator for each +κ ∈ {0,1,...,κmax} is just BκT : Xκ → Xκ+1, the adjoint to Bκ. The entries of Bκ express +relative orientation between simplex and face, and they are a natural way to construct +functions taking local signed averages, according to adjacency in the simplicial complex. +3 + +2 +e4 +e3 +t2 +t1 +V1 +V +4 +e2 +e5Fig. 2: Pairs of κ-simplices demonstrating consistency at their boundary face, for κ = 1,2. +The mixed-color pairs are consistent, and the same-color pairs are inconsistent. +2.2. Data on Simplicial Complexes. Signal processing on simplicial complexes arises +as a natural problem in the setting where richer structure is incorporated in data, than just +scalar functions and pairwise relationships. In this article, we assume the input data is +given on an existing simplicial complex. +A simple directed graphG = (V,E) can always be represented as an oriented 1-complex +˜G, with each directed edge e = (vi,v j ) inserted as a 1-simplex having orientation pe = +sign(i − j). With this convention, natural orientation corresponds to the agreement of the +edge direction with the global ordering of the vertices. +3. Hodge Laplacian. The boundary operators just introduced represent discrete dif- +ferential operators encoding the structure of κ-regions in a simplicial complex, and so can +be building blocks towards a spectral analysis of functions on those regions. For analyzing +functions on κ-simplices with κ > 0, we will construct operators based on the Hodge Lapla- +cian, or κ-Laplacian. As in [28], the combinatorial κ-Laplacian is defined for κ-simplices +as +Lκ � BT +κ−1Bκ−1 +BκBT +κ . +We refer to L ∨ +κ � BT +κ−1Bκ−1 and L ∧ +κ � BκBT +κ as the lower and upper κ-Laplacians, respec- +tively. +3.1. Simplex consistency. Let C be an oriented simplicial complex, and σ ∼ τ ∈ Cκ, +with α = bd(σ,τ). Then we may write Lκ as diag(Lκ)−Sκ, where for κ > 0, Sκ is the signed +adjacency matrix +[Sκ]στ � +� +−pσpτ nat(σ,α)nat(τ,α) +σ ∼ +κ τ +0 +otherwise +. +When Sκ > 0, we say σ,τ are consistent, and otherwise they are inconsistent. A consistent +pair of simplices view their shared boundary face in opposite ways; one as a head face, +and the other as a tail face. An inconsistent pair of simplices view their shared boundary +face identically. In the case of κ = 1, two directed edges are consistent when they flow +into each other at their boundary vertex, and are inconsistent when they collide at the +boundary vertex, either both pointing toward it, or both pointing away. Cases for κ = 1,2 +are demonstrated in Figure 2. +The combinatorial κ-Laplacian represents signed adjacency between κ-adjacent sim- +plices via their consistency. In particular, this means that Lκ depends only on the ori- +entations of simplices in Cκ. Naively, constructing the boundary matrices Bκ−1,Bκ then +additionally requires superfluous sign information – the orientation of each member of +both Cκ−1 and Cκ+1. This situation exactly mirrors that of the graph Laplacian L0: in order +to construct L0 for an undirected graph via the product B0BT +0 , one must assign an arbi- +trary direction to each edge, and the resulting Laplacian is independent of that choice of +directions. +4 + +Swt +1 = 1 +4 +� +����� +0 +2 +−1 +1 +0 +2 +0 +2 +0 +1 +−2 +2 +0 +2 +−2 +1 +0 +2 +0 +2 +0 +1 +−2 +2 +0 +� +����� +Fig. 3: The complex from Figure 1 on the left, with natural orientation displayed as directed +edges, together with its weighted, unnormalized signed adjacency matrix Swt +1 , with D2 = I. +Notice that weights differ depending on consistency and presence or lack of hull, and that +the presence of a hull can switch the expected sign. +3.2. Weighted and Normalized Hodge Laplacian. In order to introduce a weighted +simplicial complex, consider the symmetrically normalized graph Laplacian +Lsym +0 +� D−1/2 +0 +B0D1BT +0 D−1/2 +0 += +� +D−1/2 +0 +B0D1/2 +1 +�� +D−1/2 +0 +B0D1/2 +1 +�T , +where D0 = diag(|B0|1), the diagonal matrix of node degrees, and D1 is the diagonal ma- +trix of edge weights. Letting Dκ generally refer to a diagonal matrix containing κ-simplex +weights, we proceed as in [5] and define the symmetrically normalized κ-Laplacian as +Lsym +κ +� BT +κ−1Bκ−1 +BκBT +κ , +where Bκ � D−1/2 +κ +BκD1/2 +κ+1. Here Dℓ = diag(|Bℓ|1) for ℓ = κ−1,κ, and Dκ+1 is the diagonal +matrix of (κ+1)-hull weights. +From Lsym +κ +we may define the usual weighted unnormalized, and random-walk nor- +malized κ-Laplacians Lwt +κ and Lrw +κ , whose eigenvectors will be the basis for our bipartition- +ing: +Lwt +κ � D1/2 +κ Lsym +κ +D1/2 +κ +and +Lrw +κ � D−1 +κ Lwt +κ +. +While in the combinatorial case, Lκ vanishes for pairs σ ≃ τ, each of the weighted +Laplacians may be nonzero whenever σ ∼ τ. Finally, we define the weighted analogues of +the signed adjacency matrices, Swt +κ ,Ssym +κ +,Srw +κ , as the off-diagonal parts of their respective +Laplacians. +4. Cuts, Fielder Vectors, and Hierarchical Bipartitions. +4.1. Fielder Vector. Let C be a simplicial complex, such that G = (C0,C1) is a con- +nected graph. For a given κ, let p be a vector of orientations over Cκ, with each [p]σ = +pσ ∈ ±1, and let P = diag(p). Let Lwt +κ , ˜Lwt +κ denote the weighted κ-Laplacian of Cκ with nat- +ural orientations, and with orientations given by p, respectively. Let λ0 ≤ ··· ≤ λn−1 be the +eigenvalues of Lwt +κ and φ0,φ1,...,φn−1 be the corresponding eigenvectors where n = |C0|. +Then, let ( ˜λi, ˜φi) be the eigenpairs for ˜Lwt +κ . Because ˜Lwt +κ = PLwt +κ P, ˜λi = λi and ˜φi = Pφi for +0 ≤ i < n. +For κ = 0, with the vertices of G in natural orientation, we have that λ0 = 0, λ1 > 0, +φ0 = 1 and in particular is non-oscillatory, and that φ1 acts as a single global oscillation, +appropriate to partition the vertices of G with. Considering ˜Lwt +0 for nontrivial p � ±1, ˜φ0 is +oscillatory, and ˜φ1 is no longer appropriate for clustering; this is one reason that oriented +0-simplices are always considered to be in natural orientation. +5 + +e4 +t2 +e3 +ti +V1 +V +4 +e2 +e5 +3For κ > 0 however, it is no longer true that φ0 will be non-oscillatory. Let p∗ be a vector +of orientations such that where [φ0]σ � 0, [p∗]σ = sign([φ0]σ). Then the corresponding +˜φ0 is non-oscillatory, and acts as a DC component. This motivates taking sign(φ0) · φ1 +(element-wise) as the Fiedler vector of Lwt +κ , with which to partition Cκ. +We will aim to bipartition κ-regions by following a standard strategy in spectral clus- +tering, of minimizing a relaxation of a combinatorial cut function over possible partitions. +Just as a graph cut is typically defined as the volume of edge weight which crosses a parti- +tion of the nodes, we can define the consistency cut of Cκ into subregions A,B as +Ccut(A,B) � +� +σ∈A,τ∈B +σ∼τ +[Swt +κ ]στ . +Because of the signs introduced by consistency, we consider Swt +κ as the signed, weighted +adjacency matrix for a signed graph over Cκ, and so can utilize the framework of signed +Laplacians [26]. Let [S+ +κ]στ � max(0,[Swt +κ ]στ) and [S− +κ]στ � min(0,−[Swt +κ ]στ), i.e., indicator +functions for consistent/inconsistent pairs, respectively. Then, we can define the consis- +tency volume Cvol±(A) � Ccut±(A, A) and the signed κ-cut +κCut(A,B) � 2Ccut+(A,B)+Cvol−(A)+Cvol−(B) . +In the κ = 0 case, with all vertices in natural orientation, Swt +0 +is just the usual adjacency +matrix, and so S− +0 = 0; hence κCut = 2Ccut, yielding the traditional cut objective. For +κ > 0, κCut increases with the number of consistent pairs of κ-adjacent simplices across +the partition, and with the number of inconsistent pairs within each κ-region. Equiva- +lently, minimizing κCut requires maximizing consistent pairs within each κ-region, and +maximizing inconsistent pairs across the partition. +Let Lκ be the signed Laplacian with signed adjacency Swt +κ . Let A be a κ-region, r A � +1A − 1Cκ\A, and define RA(L) � r T +ALr A. Then because Lκ differs from Lwt +κ only on the di- +agonal, RA(Lκ) differs from RA(Lwt +κ ) by a constant independent of A. From [26], we know +that RA(Lκ) ∝ κCut(A,Cκ \ A). Hence, minA⊂Cκ RA(Lwt +κ ) = minA⊂Cκ κCut(A,Cκ \ A), and we +obtain φ0 as a relaxed solution to κ-cut minimization. +Now, notice that if the orientations of Cκ were changed according to some p, this +would be equivalent to a different choice of A; namely, if [p]σ = −1, then σ moves to the +other side of the partition, either into or out of A. As all orientations are available to us, +this includes one for which ˜φ0 is non-oscillatory, so that its sign does not partition Cκ. We +then instead take ˜φ1 as our relaxed solution, which we may compute via sign(φ0)·φ1. +An improved cut objective is the signed Ratio Cut, which encourages more balanced +partitions: +SignedRatioCut(A) � +� 1 +|A| + +1 +|Cκ \ A| +� +κCut(A,Cκ \ A) . +From [26], we know that with rA above scaled by a factor of cA � �|A|/|Cκ \ A|, the analo- +gous result holds, that the eigenvectors of Lκ yield a relaxed solution to minA⊂Cκ SignedRatioCut(A). +However, the new dependence on A means the resulting objective is slightly different for +Lκ, so the relaxation is only approximate. +Finally, the signed Normalized Cut balances the partitions by degree rather than sim- +plex count: +SignedNormalizedCut(A) � +� +1 +Cvol(A) + +1 +Cvol(Cκ \ A) +� +κCut(A,Cκ \ A). +Here, the eigenvectors of diag(Lκ)−1Lκ yield a relaxed solution to minA⊂Cκ SignedNormalizedCut(A), +and an approximate relaxed solution is given by the eigenvectors of Lrw +κ . In our numeri- +6 + +Fig. 4: One possible hierarchical bipartitioning of a simple 2-complex, from j = 0 with no +partition on the left, to j = 5 on the right, where each of the 27 triangles form their own +subregion. Colors indicate distinct subregions. +cal experiments, we use the random-walk κ-Laplacian for bipartitioning simplicial com- +plexes. +4.2. Hierarchical Bipartitions. The foundation upon which our multiscale transforms +on a κ-simplices Cκ of a given simplicial complex C are constructed is a hierarchical bi- +partition tree (also known as a binary partition tree) of Cκ, a set of tree-structured κ- +subregions of Cκ constructed by recursively bipartitioning Cκ. This bipartitioning opera- +tion ideally splits each κ-subregion into two smaller κ-subregions that are roughly equal in +size while keeping tightly-connected κ-simplices grouped together. More specifically, let +C j +k denote the kth κ-subregion on level j of the binary partition tree of Cκ and n j +k � +���C j +k +���, +where j,k ∈ Z≥0. Note C 0 +0 = Cκ, n0 +0 = n, i.e., level j = 0 represents the root node of this +tree. Then the two children of C j +k in the tree, C j+1 +k′ +and C j+1 +k′+1, are obtained through parti- +tioning C j +k using the Fiedler vector of Lrw +κ (C j +k). This partitioning is recursively performed +until each subregion corresponding to the leaf contains only a simplex singleton. Note +that k′ = 2k if the resulting binary partition tree is a perfect binary tree. We note that even +other (non-spectral) partitioning methods can be used to form the binary partition tree, +but in this article, we stick with the spectral clustering using the Fielder vectors. For more +details see on hierarchical partitioning, (specifically for the κ = 0 case), see [22, Chap. 3] +and [40]. Figure 4 demonstrates such a hierarchical bipartition tree for a simple 2-complex +consisting of triangles. +5. Orthonormal κ-Haar Bases. The classical Haar basis [15] was introduced in 1909 +as a piecewise-constant compactly-supported multiscale orthonormal basis (ONB) for square- +integrable functions but has since been recognized as a wavelet family and adapted to +many domains. In one dimension, the family of Haar wavelets on the interval [0,1] can be +generated by the following mother and scaling (or father) functions: +ψ(x) = +� +� +� +� +� +1, +0 ≤ x < 1 +2; +−1, +1 +2 ≤ x < 1; +0, +otherwise. +φ(x) = +� +1, +0 ≤ x < 1; +0, +otherwise. +Unfortunately, these definitions do not generalize to non-homogeneous domains due to +the lack of appropriate translation operators and dilation operators [44]. Instead, several +methods have been proposed to generate similar bases, and overcomplete dictionaries to +apply more abstract domains such as graphs and discretized manifolds [17, 45, 40]. Here, +we describe a method to compute similar, piecewise-constant locally supported bases for +κ-simplex valued functional spaces, which we call the (orthonormal) κ-Haar bases. +Rather than basing our construction on some kind of translation or transportation +schemes, we instead employ the hierarchical bipartition, as we discussed in Section 4.2, to +7 + +Fig. 5: The 2-Haar basis vectors on the same simple 2-complex shown in Figure 4. The +yellow, dark green, violet regions in each vector indicate its positive, zero, and negative +components. +divide the domain, i.e., the κ-simplicesCκ of a given simplicial complexC into appropriate +locally-supported κ-regions. For each κ-region in the bipartition tree, if that region has two +children in the tree, then we create a vector that is positive on one child, negative on the +other, and zero elsewhere. To avoid sign ambiguity, we dictate that the positive portion is +on the region whose region index is smaller among these two. +Several remarks on this basis are in order. First, since the division is not symmetri- +cally dyadic, we need to compute the scaling factor for each region separately. For each +given basis vector ξ except the scaling vector, we break it into positive and negative parts +ξ+ and ξ− and ensure that � +i([ξ+]i + [ξ−]i) = 0 and ∥ξ∥ = 1. If the members of κ-region +are weighted, then this sum and norm can be computed with respect to those weights. Fi- +nally, we note that different hierarchical bipartition schemes may arise from the different +weighting of the Hodge Laplacian, which will correspond to bases with different supports. +Figure 5 demonstrates the 2-Haar basis on the simple 2-complex used in Figure 4, which +has a hole in the center. +6. Overcomplete Dictionaries. In this section, we introduce two overcomplete dic- +tionaries for analyzing real-valued functions defined on κ-simplices in a given simplicial +complex: the κ-Hierarchical Graph Laplacian Eigen Transform (κ-HGLET), based on the +Hierarchical Graph Laplacian Eigen Transform (HGLET) [18] and the κ-Generalized Haar- +Walsh Transform (κ-GHWT), based on the Generalized Haar-Walsh Transform (GHWT) [17] +for graph signals. +6.1. κ-Hierarchical Graph Laplacian Eigen Transform (κ-HGLET). The first over- +complete transform we describe can be viewed as a generalization of the Hierarchical +Block Discrete Cosine Transform (HBDCT). The classical HBDCT is generated by creat- +ing a hierarchical bipartition of the signal domain and computing the DCT of the local +signal supported on each subdomain. We note that a specific version of the HBDCT (i.e., a +homogeneous split of an input image into a set of blocks of size 8×8 pixels) has been used +in the JPEG image compression standard [34]. This process was generalized to the graph +case in [18], i.e., the Hierarchical Graph Laplacian Eigen Transform (HGLET), from which +we base our algorithm and notation. The basis given by the set {φj +k,l} where j denotes the +level of the partition (with j = 0 being the root), k indicates the partition within the level, +and l indexes the elements within each partition in increasing frequency. +To compute the transform, we first compute the complete set of eigenvectors {φ0 +0,l}l=1:n +of the Hodge Laplacian of the entire κ-simplices Cκ of a given simplicial complex C and or- +8 + +Fig. 6: 2-HGLET dictionary on the 2-complex shown in Figure 4. Here, the color scale is +consistent across each row (which corresponds to the level) to better visualize the smooth- +ness of the elements +der them by nondecreasing eigenvalues. We then partition Cκ into two disjoint κ-regions +C 1 +0 and C 1 +1 as described in Section 4. We then compute the complete set of eigenvectors of +the Hodge Laplacian on C 1 +0 and C 1 +1. We again order each set by nondecreasing frequency +(i.e., eigenvalue) and label these {φ1 +0,l}l=1:n1 +0 and {φ1 +1,l}l=1:n1 +1 Note that n1 +0 + n1 +1 = n0 +0 = n, +and that all of the elements in {φ1 +0,l} are orthogonal to those in {φ1 +1,l} since their supports +are disjoint. Then the set {φ1 +0,l}l=1:n1 +0 ∪ {φ1 +1,l}l=1:n1 +1 form an orthonormal basis for vectors +on Cκ. From here, we apply this process recursively, generating an orthonormal basis for +each level in the given hierarchical bipartition tree. +If the hierarchical bipartition tree terminates at every region containing only a κ- +simplex singleton, then the final level will simply be the standard basis of Rn. Each level +of the dictionary contains an ONB whose vectors have the support of roughly half the size +of the previous level. There are roughly (1.5)n possible ONBs formed by selecting differ- +ent covering sets of regions from the hierarchical bipartition tree; see, e.g., [49, 40] for more +about the number of possible ONBs in such a hierarchical bipartition tree. Finally, we note +that the computational cost of generating the entire dictionary is O(n3) and that any valid +hierarchical bipartition tree can be used to create a similar dictionary. Figure 6 shows the +2-HGLET constructed on the same 2-complex shown in Figure 4. +6.2. κ-Generalized Haar-Walsh Transform (κ-GHWT). The second transform we present +here is based on the Generalized Haar-Walsh Transform (GHWT) [17], which can itself be +viewed as a generalization of the Wash-Hadamard transform. This basis is formed by first +generating a hierarchical bipartition tree of Cκ. We then work in a bottom-up manner, be- +ginning with the finest level in which each region only contains a single element. We call +these functions scaling vectors and label them {ψjmax +k,0 }k=0:n−1. For the next level, we first +assign a constant scaling vector for support on each region. Then, for each region that con- +tains two children in the partition tree, we form a Haar-like basis element by subtracting +the scaling function associated with the child element with a higher index from that child +element with a lower index. This procedure will form an ONB {ψjmax−1 +k,l +}k=0:k′−1,l=0:l(k)−1 +(where k′ is the number of κ-subregions at level jmax − 1 and l(k) = 1 or 2 depending on +the partition k) whose vectors have support of at most 2. For the next level, we begin by +computing the scaling and Haar-like vectors as before. Next, for any region that contains +three or more elements, we also compute Walsh-like vectors by adding and subtracting the +Haar-like vectors in the children’s regions. From here, we form the rest of the dictionary +recursively. A full description of this algorithm (for the κ = 0 case) is given in [18]. Figure 7 +9 + +Fig. 7: Course-to-Fine (C2F) 2-GHWT dictionary. The yellow, dark green, and violet regions +in each vector indicate its positive, zero, and negative components, respectively. +displays the 2-GHWT dictionary on the same 2-complex used in Figures 5 and 7. We make +several observations about this dictionary. First, like the κ-HGLET, each level of the dic- +tionary forms an ONB, and at each level, basis vectors have the support of roughly half the +size of the previous level. These basis vectors also have the same support as the κ-HGLET +basis vectors (that is, supp(φj +k,l) = supp(ψj +k,l) for all j,k,l). However, the computational +cost of computing the κ-GHWT is only O(n logn) compared to the O(n3) of the κ-HGLET. +Finally, we note that at the coarsest level (j = 0) the κ-GHWT dictionary contains +globally-supported piecewise-constant basis vectors, which are ordered by increasing os- +cillation (or “sequency”). This forms an ONB analogous to the classical Walsh Basis. This +allows us to define an associated Walsh transform and conduct Walsh analysis on signals +defined on simplicial complexes. Although not the primary focus of this article, we con- +duct some numerical experiments using the Walsh bases explicitly in Section 7. +6.3. Organizing the Dictionaries. For many downstream applications, it is impor- +tant to organize the order of these bases. In general, the κ-HGLET dictionary is naturally +ordered in a Coarse-to-Fine (C2F) fashion. In each region, the basis vectors are ordered +by frequency (i.e., eigenvalue). Similarly, the GHWT dictionary is also naturally ordered +in a C2F fashion, with increasing “sequency” within each subgraph. Another useful way +to order the GHWT is in a Fine-to-Coarse (F2C) ordering, which approximates “sequency” +domain partitioning. See, e.g., Figure 8, which shows the F2C 2-GHWT dictionary on the +triangle graph. We also note that the F2C ordering is not possible for the κ-HGLET dictio- +nary because some parent subspaces and the direct sum of their children subspaces are +not equivalent; see, e.g., [22, Eq. (5.6)] for the details. Other relabeling schemes, such as +those proposed in [45, 40] may also be useful but are outside the scope of this article and +will be explored further in our future work. . +6.4. Basis and Frame Selection. Once we have established these arrangements of ba- +sis vectors, we can efficiently apply the best-basis algorithm [8] to select an ONB that is op- +timal for a task at hand for a given input signal or a class of input signals; see also our previ- +ous work of applying the best-basis algorithm in the graph setting [18, 17, 19, 21, 45, 7, 40]. +Given some cost function F and signal x, we traverse the partition tree and select the basis +that minimizes F restricted to each region. For the C2F dictionary, we initialize the best +basis as the finest (j = jmax) level of the GHWT dictionary. We then proceed upward one +level at a time and compute the cost of each subspace at that level and compare it to the +10 + +Fig. 8: Fine-to-Coarse (F2C) 2-GHWT dictionary. Note that this dictionary is not generated +by simply reversing the row indices of the C2F dictionary, but instead by arranging each +level (row) by “sequency”. +cost of the union of its children subspaces. If the latter cost is lower, the basis is updated; +if not, the children subspaces (and their basis vectors) are propagated to the current level. +This algorithm yields the C2F best basis. The F2C best basis is performed similarly, i.e., we +begin with the globally-supported basis (j = 0) at the bottom of the rearranged tree and +proceed in the same bottom-up direction. As for the HGLET dictionary, it has only a C2F +basis as we discussed earlier. +In some contexts, it is not necessary to generate a complete ONB, but rather some +sparse set of vectors in the dictionary (also known as atoms) that most accurately approx- +imate a given signal or class of signals. In this case, we can directly apply the orthogo- +nal matching pursuit of [3] to find the best m-dimensional orthogonal subframe (m ≤ n) +selected from the dictionary. Additionally, for some downstream tasks, such as sparse ap- +proximation or sparse feature selection, generating orthogonal sets of atoms is not critical. +In these cases, we can employ a greedy algorithm to generate efficient approximation. This +algorithm simply selects the atoms in the dictionary with the largest coefficient, removes +it, then computes the transform of the residual and proceeds so forth. These basis and +subframe algorithms are studied intensively in the subsequent section. +7. Numerical Experiments. We demonstrate the efficacy of our proposed partition- +ing techniques and basis constructions by conducting a series of experiments. In Sec- +tion 7.1 we show how our multiscale bases and overcomplete dictionaries can be used +to sparsely approximate signals defined on κ-simplices. In Section 7.2 we show how these +representations can be used in supervised classification and unsupervised clustering prob- +lems. +7.1. Approximation and Signal Compression. We begin with an illustrative example +by creating some synthetic data for 1- and 2-simplices by triangulating a digital image. We +start with a 512 × 512 “peppers” image and map it to a Cartesian grid on the unit square +[0,1]2. We then randomly sample 1028 points within this square (not necessarily on a grid). +We then create a triangular mesh from these points using Delaunay triangulation. Next, +we interpolate the image from the Cartesian grid to the sampled vertices by computing the +barycentric coordinate of each vertex from the square inside the Cartesian grid. Finally, +we interpolate the signal to the edges and triangles of the triangulation by averaging the +values of the vertices that they contain. The result, for our random seed, is a signal defined +11 + +Fig. 9: Nonlinear approximation of the peppers image for κ = 2 +on the 3050 edges of the triangulation and another on the 2067 triangles. We now consider +the sparse representation of these signals. Figure 9 shows the nonlinear approximation +(i.e., using the largest expansion coefficients in magnitude) of the triangle-based signals +in the Hodge Laplacian eigenbasis (Fourier), the orthonormal Haar basis, orthonormal +Walsh basis as well as the approximation prescribed by applying the best-basis and greedy +algorithms to the HGLET and GHWT dictionaries. Figure 10 shows the approximation +error vs the number of terms used for both the edge-based and triangle-based functions. +A number of observations are in order. First, the multiscale dictionary-based meth- +ods consistently outperformed the generic orthonormal bases. The greedy approximation +algorithm achieved the best approximation results, but it is also more costly to compute +than any of the other methods, and the set of atoms used in the approximation may not +be orthogonal. This may be detrimental to downstream tasks. Overall the GHWT-based +method performed best, with the F2C best basis performing much better than the C2F +best basis, which suggests that the fine-scale features of this signal are the most impor- +tant. Similarly, the Walsh basis achieved much better results than the Haar basis, again +emphasizing the necessity of capturing details at the fine scale. +Next, we apply our approach to real-world data for higher degree signals for κ = 0,...,5. +The citation complex [33, 12] is a simplicial complex derived from the Cora citation com- +plex [48], which models the interactions between multiple authors of scientific papers. A +paper with κ authors is represented by a (κ − 1)-simplex. We first build a graph whose +vertices represent the authors in this Cora database. Then, the vertices are connected by +edges that represent co-authored papers. Note that if two authors co-authored multiple +12 + +1% +5% +10% +25% +50% +75% +90% +Delta +Fourier +Haar +Walsh +HGLET (BB) +GHWT (BB)Fig. 10: Nonlinear approximation errors of the peppers image, Left: L2 error, Right: log(L2 +error) for up to 50% of the terms retained. Top κ = 1, Bottom: κ = 2. +papers, these two vertices are connected by a single edge. Next, we assign each edge the +sum of the citation numbers of all the co-authored papers by the authors, forming this +edge as its weight (or value). Finally, we assign each higher-order simplex the sum of the +values of its lower-order simplices as its value. See [12] for a more thorough description +of the construction of this complex. Table 1 reports some basic information about the +number of simplices of different degrees in this citation complex. Figure 11 shows the +approximation of this signal(i.e., a vector of citation numbers) for κ = 0,1,...,5 with the +Delta, Fourier, Haar, HGLET, and GHWT bases. Figure 12 shows the log error. The HGLET +and GHWT bases were selected with the best-basis algorithm using the C2F ordering for +the GHWT dictionary. +In these experiments, we observe that the best bases (GHWT and HGLET) outper- +formed the canonical bases, with the GHWT being the most efficient basis for each κ. Ad- +ditionally, for κ > 0, the orthonormal Haar basis performed best in the semi-sparse regime +(1 and 10% of terms retrained). This suggests that the signals on each degree of the citation +complex are similar in that they are all close to being piecewise constant. However, when +13 + +K=l, Approximation Error +K=1, Log Approximation Error +1.0 +Delta Basis +0.0 +Frourier Basis +Orthogonal Haar +Orthogonal Walsh +-0.5 +BB HGLET +0.8 +HGLET (Greedy) +BB GHWT C2F +Log L2 Approximation Error + Approximation Error +BB GHWT F2C +-1.0 +GHWT (Greedy) +0.6 +1.5 +0.4 +Delta Basis +Frourier Basis +Orthogonal Haar +-2.0- +Orthogonal Walsh +0.2 +BB HGLET +HGLET (Greedy) +-2.5 - +BB GHWT C2F +BB GHWT F2C +0.0 +GHWT (Greedy) +0 +500 +1000 +1500 +2000 +2500 +3000 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +# of Terms +# of TermsK=2, Approximation Error +K=2, Log Approximation Error +Delta Basis +1.0 +0.0 +Frourier Basis +Orthogonal Haar +Orthogonal Walsh +-0.5 +BB HGLET +0.8 +HGLET (Greedy) +BB GHWT C2F +Log L2 Approximation Error +Approximation Error +BB GHWT F2C +-1.0 +GHWT (Greedy) +0.6 +-1.5 +0.4 +Delta Basis +-2.0 +Frourier Basis +Orthogonal Haar +Orthogonal Walsh +0.2 +BB HGLET +-2.5 +HGLET (Greedy) +BB GHWT C2F +BB GHWT F2C +-3.0 - +0.0 - +GHWT (Greedy) +750 +0 +200 +400 +600 +800 +0 +250 +500 +100012501500 1750 2000 +1000 +# of Terms +# of Termsκ +0 +1 +2 +3 +4 +5 +# of elements +1126 +5059 +11840 +18822 +21472 +17896 +Table 1: The number of element in the κ-simplices in the Cora complex for κ = 0,1,...,5 +Fig. 11: Approximation of the Citation Complex for κ = 0,...,5. +more terms are considered, the HGLET best basis achieved a lower approximation error +than the orthonormal Haar basis achieved. +7.2. Signal Clustering and Classification. Since the basis (and dictionary) vectors we +present are both multiscale and built from the Hodge Laplacians that are aware of both +topological and geometric properties of the domain [5], they can function as very powerful +feature extractors for general data science applications. In this section, we present two +clustering-type applications—one supervised and one unsupervised. For baselines, we +compare our proposed dictionaries with Fourier and Delta (indicator function) bases and +with the Hodgelets proposed in [35] for cases when κ = 1. +7.2.1. Supervised Classification. First, we present our study in supervised classifi- +cation. We begin by computing edge-valued signals for 1000 handwritten digits from the +MNIST dataset [27] by sampling 500 points in the unit square and following the interpola- +tion method presented for the peppers image in Section 7.1. We then compute the features +of these images using the proposed orthogonal transforms and best bases from the over- +complete dictionaries. Next, we train a support vector machine (SVM) to classify the digits +for each of the transformed representations using the 1000 training examples. Finally, we +test these SVMs on the rest of the whole MNIST dataset. +We repeat this experiment for the FMNIST dataset [51], again using only 1000 exam- +ples for training data. Results are presented in Table 2. We remark that these tests are not +meant to achieve state-of-the-art results for image classification but rather to showcase +the effectiveness of these representations for downstream tasks. Unsurprisingly, the dic- +tionary methods outperformed the basis methods. Again, the piecewise constant meth- +ods (GHWT, Haar) achieved better approximations than the smoother methods (Fourier, +14 + +Approximation k=0 +1.0 +Delta +Fourier +Haar +HGLET +0.8 +GHWT +0.6 +ux +0.4 +0.2 +0.0 +0 +50 +100 +150 +200 +250 +300 +350 +# of elementsApproximation k=1 +1.0 +Delta +Fourier +Haar +0.8 +HGLET +GHWT +0.6 +x +- +X +0.4 +0.2 +0.0 +0 +200 +400 +600 +800 +1000 +1200 +1400 +# of elementsApproximation k=2 +1.0 +Delta +Fourier +Haar +0.8 +HGLET +GHWT +0.6 +x +- +0.4 +0.2 +0.0 +0 +500 +1000 +1500 +2000 +2500 +3000 +# of elementsApproximation k=3 +1.0 +Delta +Fourier +Haar +HGLET +0.8 +GHWT +0.6 +x +- +0.4 +0.2 +0.0 +0 +1000 +2000 +3000 +4000 +5000 +# of elementsApproximation k=4 +1.0 +Delta +Fourier +Haar +HGLET +0.8 +GHWT +0.6 +x +- +0.4 +0.2 +0.0 +0 +1000 +2000 +3000 +4000 +5000 +# of elementsApproximation k=5 +1.0 +Delta +Fourier +Haar +HGLET +0.8 +GHWT +0.6 +x +- +0.4 +0.2 +0.0 +0 +1000 +2000 +3000 +4000 +# of elementsFig. 12: Top: Approximation of the Citation Complex for κ = 0,...,5. Bottom: Log of the +error for up to 50% of the terms retained. +Basis Methods +Dictionary Methods +Delta +Fourier +Haar +Walsh +HGLET +(BB) +GHWT +(BB C2F) +GHWT +(BB F2C) +Joint +Separate +HGLET +GHWT +# of terms +661 +661 +661 +661 +661 +661 +661 +5288 +5288 +9254 +9254 +MNIST +68.675 +77.053 +75.388 +77.011 +77.991 +78.779 +77.156 +79.202 +80.038 +80.001 +81.089 +FMNIST +64.370 +76.753 +76.779 +75.230 +76.117 +76.991 +76.121 +78.761 +78.738 +79.739 +80.789 +Table 2: Test Accuracy for SVMs trained on transforms of MNIST signals interpolated to a +random triangulation +HGLET, Joint, and Separate Hodgelets). This is likely due to the near-binary nature of im- +ages in both datasets. +7.2.2. Unsupervised Clustering. A natural setting for studying κ = 1 valued signals is +the analysis of trajectories [5, 36, 35]. Of particular interest is the case where the domain +has nontrivial topological features. Such is the case of the Global Drifter Program dataset, +which tracks the positions of 334 buoys dropped into the ocean at various points around +the island of Madagascar [35]. +We begin by dividing the dataset into three subsets, train (|Xtr| = 176), test (|Xte| = +83) and validation (|Xvl| = 84). We then use orthogonal matching pursuit [3] (OMP) to +compute the m significant features of the training set. Next, we extract these features for +the test set and use them to compute the centroids {c j }d +j=1 for each cluster. To evaluate +these clusters K -score (i.e. the standard k-means objective) on the transformed features +of the validation set: +K −score := 1 +N +N +� +i=1 +min +1≤j≤d ∥f (xi)−c j ∥2, +xi ∈ Xvl. +where f (·) represents the feature extraction prescribed by applying OMP to the test set. We +repeat this experiment for m = 5,10,15,20,25 (number of features) and d = 2,...,7 (num- +ber of clusters). Figure 13 summarizes the results of this test, while Table 3 shows the full +15 + +Approximationk=0 +0.0 +Delta +Fourier +0.5 +Haar +HGLET +GHWT +1.0 +(llux +-1.5 +- +)601 +-2.0 +2.5 +3.0 +3.5 +0 +25 +50 +75 +100 +125 +150 +175 +#ofelementsApproximation k=1 +0.0 +Delta +Fourier +Haar +-0.5 +HGLET +GHWT +-1.0 +-1.5 +-2.0 +-2.5 +-3.0 +-3.5 +0 +100 +200 +300 +400 +500 +600 +700 +# of elementsApproximation k=2 +0.0 +Delta +Fourier +Haar +-0.5 +HGLET +GHWT +-1.0 +-1.5 +-2.0 +-2.5 +-3.0 +0 +250 +500 +750 +1000 +1250 +1500 +# of elementsApproximation k=3 +0.0 +Delta +Fourier +Haar +-0.5 +HGLET +GHWT +-1.0 : +- xII)60| +-1.5 +-2.0 +-2.5 +-3.0 +0 +500 +1000 +1500 +2000 +2500 +# of elementsApproximation k=4 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +Delta +-2.5 +Fourier +Haar +HGLET +-3.0 +GHWT +0 +500 +1000 +1500 +2000 +2500 +# of elementsApproximation k=5 +0.0 +-0.5 +-1.0 +-1.5 +-2.0 +Delta +-2.5 +Fourier +Haar +-3.0 +HGLET +GHWT +0 +500 +1000 +1500 +2000 +# of elementsFig. 13: Extensive results for buoy cluster test. Leftmost figure shows which method pre- +formed best, the second to the left shows the second best and so on. The x-axis in each +subplot indicates the number of coefficients used and the y-axis is the number of clusters. +Full numerical results are presented in Table 3. +numerical results. In this experiment, the GHWT outperformed all other bases because +the trajectories are roughly constant and locally supported. The orthogonal matching pur- +suit scheme can select elements with the correct support size, and the piecewise constant +nature of the GHWT atoms can capture the action of the trajectory with very few elements. +8. Conclusions and Future work. In this article, we have developed several general- +izations of orthonormal bases and overcomplete transforms/dictionaries for signals de- +fined on κ-simplices, and demonstrated their usefulness for data representation on both +illustrative synthetic examples and real-world simplicial complexes generated from a co- +authorship/citation dataset and an ocean current/flow dataset. However, there are many +more tools from harmonic analysis that we have not addressed in this article. From a +theoretical standpoint, future work may involve 1) defining additional families of mul- +tiscale transforms such as the extended Generalized Haar-Walsh Transform(eGHWT) [45] +and Natural Graph Wavelet Packets (NGWPs) [7]; 2) exploring different best-basis selection +criteria tailored for classification and regression problems such as the Local Discriminant +Basis [37, 39] and the Local Regression Basis [38] on simplicial complexes; and 3) inves- +tigating nonlinear feature extraction techniques such as the Geometric Scattering Trans- +form [13]. From an application standpoint, we look forward to applying the techniques +presented here to data science problems in computational chemistry, weather forecasting, +and genetic analysis, all of which have elements that are naturally modeled with simplicial +complexes. +Acknowledgments. This research was partially supported by the US National Science +Foundation grants DMS-1418779, DMS-1912747, CCF-1934568; the US Office of Naval Re- +search grant N00014-20-1-2381. +16 + +Validation set Best +Validation set Second +Validation set Third +6 +6 +5 +5+ +num_clusters +4 +GHWT +3 - +3 + HGLET +2 - +2 +5 +5 +5 +num coefs +num coefs + Haar +num coefs +Validation set Fourth +Validation set Fifth +Validation set Sixth +Separate +6 +-Joint +4 +- Fourier +3 +3 . +2 +num coefs +num coefs +9 +num coefsREFERENCES +[1] S. BARBAROSSA AND S. SARDELLITTI, Topological signal processing over simplicial complexes, IEEE Trans. +Signal Process., 68 (2020), pp. 2992–3007. +[2] J. BRUNA, W. ZAREMBA, A. SZLAM, AND Y. LECUN, Spectral networks and locally connected networks on +graphs, arXiv preprint arXiv:1312.6203, (2013). +[3] T. T. CAI AND L. WANG, Orthogonal matching pursuit for sparse signal recovery with noise, IEEE Trans. +Inform. Theory, 57 (2011), pp. 4680–4688. +[4] G. CARLSSON, Topology and data, Bull. Amer. Math. Soc., 46 (2009), pp. 255–308. +[5] Y.-C. CHEN, M. MEIL ˘A, AND I. G. KEVREKIDIS, Helmholtzian eigenmap: Topological feature discovery & +edge flow learning from point cloud data, arXiv preprint arXiv:2103.07626, (2021). +[6] F. CHUNG AND L. LU, Complex Graphs and Networks, no. 107 in CBMS Regional Conference Series in Math- +ematics, Amer. Math. Soc., Providence, RI, 2006, https://doi.org/10.1090/cbms/107. +[7] A. CLONINGER, H. LI, AND N. SAITO, Natural graph wavelet packet dictionaries, J. Fourier Anal. Appl., 27 +(2021), p. 41, https://doi.org/10.1007/s00041-021-09832-3. a part of “Topical Collection: Harmonic +Analysis on Combinatorial Graphs”. +[8] R. R. COIFMAN AND M. V. WICKERHAUSER, Entropy-based algorithms for best basis selection, IEEE Trans. +Inform. Theory, 38 (1992), pp. 713–718. +[9] K. CRANE, Discrete differential geometry: An applied introduction, Notices of the AMS, Communication, +(2018), pp. 1153–1159. +[10] X. DONG, D. THANOU, L. TONI, M. BRONSTEIN, AND P. FROSSARD, Graph signal processing for machine +learning: A review and new perspectives, IEEE Signal Processing Magazine, 37 (2020), pp. 117–127. +[11] D. EASLEY AND J. KLEINBERG, Networks, Crowds, and Markets: Reasoning and a Highly Connected World, +Cambridge Univ. Press, New York, 2010. +[12] S. EBLI, M. DEFFERRARD, AND G. SPREEMANN, Simplicial neural networks, arXiv preprint arXiv:2010.03633, +(2020). +[13] F. GAO, G. WOLF, AND M. HIRN, Geometric scattering for graph data analysis, in International Conference +on Machine Learning, PMLR, 2019, pp. 2122–2131. +[14] C. GIUSTI, R. GHRIST, AND D. S. BASSETT, Two’s company, three (or more) is a simplex, J. Comput. Neurosci., +41 (2016), pp. 1–14. +[15] A. HAAR, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann., 69 (1910), pp. 331–371. +[16] M. HOLZRICHTER AND S. OLIVEIRA, A graph based method for generating the fiedler vector of irregular prob- +lems, in International Parallel Processing Symposium, Springer, 1999, pp. 978–985. +[17] J. IRION AND N. SAITO, The generalized Haar-Walsh transform, in 2014 IEEE Workshop on Statistical Signal +Processing (SSP), IEEE, 2014, pp. 472–475. +[18] J. IRION AND N. SAITO, Hierarchical graph Laplacian eigen transforms, JSIAM Letters, 6 (2014), pp. 21–24. +[19] J. IRION AND N. SAITO, Applied and computational harmonic analysis on graphs and networks, in Wavelets +and Sparsity XVI, Proc. SPIE 9597, M. Papadakis, V. K. Goyal, and D. Van De Ville, eds., 2015, https: +//doi.org/10.1117/12.2186921. Paper # 95971F. +[20] J. IRION AND N. SAITO, Learning sparsity and structure of matrices with multiscale graph basis dictionaries, +in Proc. 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP), +A. Uncini, K. Diamantaras, F. A. N. Palmieri, and J. Larsen, eds., 2016, https://doi.org/10.1109/MLSP. +2016.7738892. +[21] J. IRION AND N. SAITO, Efficient approximation and denoising of graph signals using the multiscale basis +dictionaries, IEEE Trans. Signal Inform. Process. Netw., 3 (2017), pp. 607–616, https://doi.org/10.1109/ +TSIPN.2016.2632039. +[22] J. L. IRION, Multiscale Transforms for Signals on Graphs: +Methods and Applications, PhD thesis, +Appl. Math., Univ. California, Davis, Dec. 2015, https://github.com/JeffLIrion/MTSG_Toolbox/blob/ +master/Publications/irion_dissertation.pdf. +[23] S. JAFFARD, Y. MEYER, AND R. D. RYAN, Wavelets: Tools for Science & Technology, SIAM, Philadelphia, PA, +2001. +[24] X. JIANG, L.-H. LIM, Y. YAO, AND Y. YE, Statistical ranking and combinatorial Hodge theory, Math. Pro- +gram., 127 (2011), pp. 203–244. +[25] R. KONDOR AND S. TRIVEDI, On the generalization of equivariance and convolution in neural networks +to the action of compact groups, in International Conference on Machine Learning, PMLR, 2018, +pp. 2747–2755. +[26] J. KUNEGIS, S. SCHMIDT, A. LOMMATZSCH, J. LERNER, E. W. D. LUCA, AND S. ALBAYRAK, Spectral analy- +sis of signed graphs for clustering, prediction and visualization, in Proceedings of the 2010 SIAM In- +ternational Conference on Data Mining (SDM), SIAM, 2012, pp. 559–570, https://doi.org/10.1137/1. +9781611972801.49, https://arxiv.org/abs/https://epubs.siam.org/doi/pdf/10.1137/1.9781611972801. +49. +[27] Y. LECUN AND C. CORTES, MNIST handwritten digit database. http://yann.lecun.com/exdb/mnist/, 2010. +17 + +[28] L.-H. LIM, Hodge Laplacians on graphs, SIAM Review, 62 (2020), pp. 685–715. +[29] L. LOVÁSZ, Large Networks and Graph Limits, vol. 60 of Colloquium Publications, Amer. Math. Soc., Provi- +dence, RI, 2012. +[30] S. MALLAT, A Wavelet Tour of Signal Processing, Academic Press, Burlington, MA, third ed., 2009. +[31] M. NEWMAN, Networks, Oxford Univ. Press, Oxford, UK, 2nd ed., 2018. +[32] A. ORTEGA, P. FROSSARD, J. KOVA ˇCEVI ´C, J. M. MOURA, AND P. VANDERGHEYNST, Graph signal processing: +Overview, challenges, and applications, Proc. IEEE, 106 (2018), pp. 808–828. +[33] A. PATANIA, G. PETRI, AND F. VACCARINO, The shape of collaborations, EPJ Data Science, 6 (2017), pp. 1–16. +[34] W. B. PENNEBAKER AND J. L. MITCHELL, JPEG Still Image Data Compression Standard, Van Nostrand Rein- +hold, New York, 1993. +[35] T. M. RODDENBERRY, F. FRANTZEN, M. T. SCHAUB, AND S. SEGARRA, Hodgelets: Localized spectral repre- +sentations of flows on simplicial complexes, in ICASSP 2022-2022 IEEE International Conference on +Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2022, pp. 5922–5926. +[36] T. M. RODDENBERRY, M. T. SCHAUB, AND M. HAJIJ, Signal processing on cell complexes, in ICASSP 2022- +2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2022, +pp. 8852–8856. +[37] N. SAITO AND R. R. COIFMAN, Local discriminant bases and their applications, J. Math. Imaging Vis., 5 +(1995), pp. 337–358. Invited paper. +[38] N. SAITO AND R. R. COIFMAN, Extraction of geological information from acoustic well-logging wave- +forms using time-frequency wavelets, Geophysics, 62 (1997), pp. 1921–1930, https://doi.org/10.1190/ +1.1444292. +[39] N. SAITO, R. R. COIFMAN, F. B. GESHWIND, AND F. WARNER, Discriminant feature extraction using empir- +ical probability density estimation and a local basis library, Pattern Recognition, 35 (2002), pp. 2841– +2852. +[40] N. SAITO AND Y. SHAO, eGHWT: The Extended Generalized Haar–Walsh Transform, J. Math. Imaging Vis., +64 (2022), pp. 261–283. +[41] K. SAYOOD, Introduction to Data Compression, Morgan Kaufmann Publishers, Inc., San Francisco, CA, +3rd ed., 2006. +[42] M. T. SCHAUB, A. R. BENSON, P. HORN, G. LIPPNER, AND A. JADBABAIE, Random walks on simplicial com- +plexes and the normalized Hodge 1-laplacian, SIAM Review, 62 (2020), pp. 353–391. +[43] N. C. SCHONSHECK AND S. C. SCHONSHECK, Spherical coordinates from persistent cohomology, arXiv +preprint arXiv:2209.02791, (2022). +[44] S. C. SCHONSHECK, B. DONG, AND R. LAI, Parallel transport convolution: Deformable convolutional net- +works on manifold-structured data, SIAM Journal on Imaging Sciences, 15 (2022), pp. 367–386. +[45] Y. SHAO AND N. SAITO, The extended generalized Haar-Walsh transform and applications, in Wavelets and +Sparsity XVIII, Proc. SPIE 11138, Paper #111380C, 2019. +[46] D. I. SHUMAN, S. K. NARANG, P. FROSSARD, A. ORTEGA, AND P. VANDERGHEYNST, The emerging field of +signal processing on graphs, IEEE Signal Processing Magazine, 30 (2013), pp. 83–98, https://doi.org/ +10.1109/MSP.2012.2235192. +[47] D. I. SHUMAN, S. K. NARANG, P. FROSSARD, A. ORTEGA, AND P. VANDERGHEYNST, The emerging field of +signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular +domains, IEEE Signal Processing Magazine, 30 (2013), pp. 83–98. +[48] L. ŠUBELJ AND M. BAJEC, Model of complex networks based on citation dynamics, in Proceedings of the +22nd international conference on World Wide Web, 2013, pp. 527–530. +[49] C. M. THIELE AND L. F. VILLEMOES, A fast algorithm for adapted time-frequency tilings, Appl. Comput. +Harmon. Anal., 3 (1996), pp. 91–99, https://doi.org/10.1006/acha.1996.0009. +[50] M. V. WICKERHAUSER, Adapted Wavelet Analysis from Theory to Software, A K Peters, Ltd., Wellesley, MA, +1994. +[51] H. XIAO, K. RASUL, AND R. VOLLGRAF, Fashion-MNIST: a novel image dataset for benchmarking ma- +chine learning algorithms, arXiv preprint arXiv:/1708.07747, (2017), https://arxiv.org/abs/cs.LG/1708. +07747. +18 + +Appendix A. Full Results for Buoy Clustering. +Clusters +# Feat. +Fourier +Joint +Separate +Haar +HGLET +GHWT +5 +0.174 +0.183 +0.122 +0.115 +0.154 +0.024 +10 +0.150 +0.151 +0.109 +0.110 +0.124 +0.023 +2 +15 +0.129 +0.129 +0.120 +0.093 +0.119 +0.021 +20 +0.118 +0.113 +0.108 +0.084 +0.107 +0.023 +25 +0.104 +0.099 +0.096 +03073 +0.103 +0.024 +5 +0.174 +0.163 +0.110 +.0115 +0.126 +0.026 +10 +0.143 +0.137 +0.100 +0.108 +0.103 +0.023 +3 +15 +0.126 +0.112 +0.113 +0.095 +0.118 +0.021 +20 +0.114 +0.104 +0.100 +0.081 +0.095 +0.019 +25 +0.099 +0.092 +0.089 +0.069 +0.093 +0.021 +5 +0.139 +0.135 +0.096 +0.091 +0.101 +0.023 +10 +0.137 +0.120 +0.090 +0.096 +0.082 +0.019 +4 +15 +0.116 +0.099 +0.083 +0.079 +0.097 +0.018 +20 +0.111 +0.094 +0.084 +0.072 +0.090 +0.021 +25 +0.094 +0.083 +0.076 +0.062 +0.087 +0.022 +5 +0.135 +0.116 +0.087 +0.081 +0.074 +0.014 +10 +0.118 +0.109 +0.083 +0.090 +0.062 +0.018 +5 +15 +0.110 +0.090 +0.078 +0.074 +0.083 +0.017 +20 +0.103 +0.090 +0.075 +0.068 +0.079 +0.020 +25 +0.083 +0.079 +0.069 +0.058 +0.083 +0.019 +5 +0.135 +0.116 +0.087 +0.081 +0.074 +0.014 +10 +0.118 +0.109 +0.083 +0.090 +0.062 +0.018 +6 +15 +0.110 +0.090 +0.078 +0.074 +0.083 +0.017 +20 +0.103 +0.092 +0.075 +0.068 +0.073 +0.020 +25 +0.083 +0.073 +0.069 +0.058 +0.083 +0.019 +5 +0.116 +0.137 +0.084 +0.082 +0.065 +0.014 +10 +0.115 +0.106 +0.089 +0.092 +0.055 +0.013 +7 +15 +0.097 +0.088 +0.069 +0.074 +0.067 +0.013 +20 +0.095 +0.080 +0.055 +0.068 +0.067 +0.014 +25 +0.087 +0.070 +0.051 +0.058 +0.076 +0.013 +Table 3: K -score for buoys tests, smaller is better +19 + diff --git a/1dA0T4oBgHgl3EQfMv9X/content/tmp_files/load_file.txt b/1dA0T4oBgHgl3EQfMv9X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e633254a5af3003bf71362c5d373f6bcdaab0a96 --- /dev/null +++ b/1dA0T4oBgHgl3EQfMv9X/content/tmp_files/load_file.txt @@ -0,0 +1,1183 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf,len=1182 +page_content='MULTISCALE TRANSFORMS FOR SIGNALS ON SIMPLICIAL COMPLEXES NAOKI SAITO∗, STEFAN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHONSHECK †, AND EUGENE SHVARTS ‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar- Walsh Transform (GHWT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Hierarchical Graph Laplacian Eigen Transform (HGLET);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Natural Graph Wavelet Pack- ets (NGWPs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' and their relatives—were developed for analyzing data recorded on nodes of a given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this article, we propose their generalization for analyzing data recorded on edges, faces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', triangles), or more gen- erally κ-dimensional simplices of a simplicial complex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', a triangle mesh of a manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ-dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We demon- strate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Simplicial complexes, graph basis dictionaries, hierarchical partitioning, Fiedler vectors, Hodge Laplacians, Haar-Walsh wavelet packets 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For conventional digital signals and images sampled on regular lat- tices, multiscale basis dictionaries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', wavelet packet dictionaries including wavelet bases, local cosine dictionaries, and their variants (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', [50, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 4, 7], [23, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 6, 7], [30, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 8]), have a proven track record of success: JPEG 2000 Image Compression Stan- dard [41, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Modified Discrete Cosine Transform (MDCT) in MP3 [41, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' dis- criminant feature extraction for signal classification [37, 38, 39], just to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Consid- ering the abundance of data measured on graphs and networks and the increasing impor- tance to analyze such data (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', [11, 31, 6, 29, 46]), it is quite natural to lift/generalize these dictionaries to the graph setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Our group have developed the graph versions of the block/local cosine and wavelet packet dictionaries for analysis of graph signals sampled at nodes so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' These include the Generalized Haar-Walsh Transform (GHWT) [17], the Hierarchical Graph Laplacian Eigen Transform (HGLET) [18], the Natural Graph Wavelet Packets (NGWPs) [7], and their relatives [20, 45, 40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' see also [19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Some of these will be briefly reviewed in the later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this article, we propose their generalization for analyzing data recorded on edges, faces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', triangles), or more generally cells (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', polytopes) of a class of special graphs called simplicial complexes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', a triangle mesh of a manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ-dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We demonstrate their usefulness for data representation on both il- lustrative synthetic examples and real-world simplicial complexes generated from a co- authorship/citation dataset and an ocean current/flow dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Graph-based methods for analyzing data have been widely adopted in many domains, [2, 32, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Often, these graphs are fully defined by data (such as a graph of social media “friends"), but they can also be induced through the persistence homology of generic point clouds [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In either case, the vast majority of these analytical techniques deal with signals which are defined on the nodes of a given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' More recently, there has been a surge in interest in studying signals defined on edges, triangles, and higher- dimensional substructures within the graph [4, 47, 14, 1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The fundamental tool em- ployed for analyzing these signals, the Hodge Laplacian, has been studied in the context of differential geometry for over half a century but has only recently entered the toolbox ∗Department of Mathematics, University of California, Davis (saito@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='edu, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' †Department of Mathematics, University of California, Davis (scschonsheck@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' ‡Department of Mathematics, University of California, Davis (eshvarts@ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='edu) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='02136v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='SI] 28 Dec 2022 of applied mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This rise in popularity is largely due to the adaptation of discrete differential geometry [9] in applications in computer vision [28, 36], statistics [24], topo- logical data analysis [5, 43], and network analysis [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' One of the key challenges to applying wavelets and similar constructions to node- based graph signals is that graphs lack a natural translation operator, which prevents the construction of convolutional operators and traditional Littlewood-Paley theory [19, 25, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This challenge is also present for general κ-dimensional simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' One method for overcoming this difficulty is to perform convolution solely in the “frequency” domain and define wavelet-like bases entirely in the coefficient space of the Laplacian (or in this case Hodge Laplacian) transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Following this line of research, there have been several ap- proaches to defining wavelets [35] and convolutional neural networks [12] in which the input signal is transformed in a series of coefficients in the eigenspace of the Hodge Lapla- cian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Unfortunately, the atoms (or basis vectors) generated by these methods are not al- ways locally supported, and can be difficult to interpret their role in analyzing a given graph signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' An alternative path to the creation of wavelet-like dictionaries and transforms is to first develop a hierarchical block decomposition of the domain and then use this to de- velop multiscale transforms [18, 17, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' These techniques rely on recursively computing bipartitions of the domain and then generating localized bases on the subsets of the do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this work, we propose a simplex analog to the Fielder vector [16] to solve a relaxed version of the simplex-normalized-cut problem, which we can apply iteratively to develop a hierarchical bipartition of the κ-dimensional simplices in a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From here, we are able to apply the general scheme of [18] and [17] to develop the Hierarchi- cal Graph Laplacian Eigen Transform and the Generalized Haar-Walsh Transform, respec- tively, for a given collection of simplices of an arbitrarily high order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' As a result, we can also generate orthonormal Haar bases, orthonormal Walsh bases, as well as data-adaptive orthonormal bases using the best-basis selection method [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This article is organized as follows: In Section 2 we formally describe simplicial complexes and how their geometry leads to notions of adjacency and orienta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This allows us to define boundary operators, which admits a map between the κ and κ ± 1 degree faces of the complex as well discrete differential operators acting on signals defined on the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In Section 3 we use these boundary operators to describe the Hodge Laplacian and discuss several different variants, some analogous to different nor- malizations of the graph Laplacian and some more novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In Section 4 we show how the eigenvectors of the Hodge Laplacian can be use to solve relaxed-cut-like problems to parti- tion a complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We also develop hierarchical bipartitions, which decompose a given com- plex roughly in half at each level until we are left with a division into individual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In Section 5 we use these bipartitions to develop orthonormal Haar bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In Section 6, we create overcomplete dictionaries based on given bipartitions and, as a consequence, are also able to define a canonical orthonormal Walsh Basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In Section 7, we present numeri- cal experiments on both illustrative synthetic examples and real-world problems in signal approximation, clustering, and supervised classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, we conclude this article with Section 8 discussing our potential future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Simplicial Complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this section we review concepts from algebraic topology to formally define simplicial complexes and introduce some notions of how two simplices can be “adjacent", for a more thorough review see [4, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Given a vertex set V = {v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',vn}, a κ-simplex σ is a (κ+1)-subset of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A face of σ is a κ-subset of σ, and so σ has κ+1 faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A co-face of σ is a (κ+1)-simplex, of which σ is a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Suppose σ = {vi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',viκ+1}, i1 < ··· < iκ+1, and α ⊂ σ is its face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then, σ\\α consists of a 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 1: In this small 2-complex C, e1 ∼ e4 because they share the face v2, and e1 ∼ e2 be- cause they share the face v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Further e1 ≃ e2 because their hull t1 ∈ C, but e1 � e4, so that e1 ∼ 1 e4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We have t1 ∼ t2 because they share the face e3, and also t1 ∼ 2 t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' single vertex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' let viℓ∗ be that vertex where 1 ≤ ℓ∗ ≤ κ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then the natural parity of σ with respect to its face α is defined as nat(σ,α) := (−1)ℓ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' When α is not a face of σ, nat(σ,α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The natural parity of κ-simplices with respect to their faces generalizes the idea of a directed edge having a head vertex and a tail vertex, and is “natural” because it disallows situations analogous to a directed edge with two heads or two tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A simplicial complex C is a collection of simplices closed under subsets, where if σ ∈ C, then α ⊂ σ =⇒ α ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In particular, if σ ∈ C, so does each face of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let κmax(C) � max � κ|σ ∈ C is a κ-simplex � , and let Cκ denote the set of κ-simplices in C for each κ = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',κmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' When κ > κmax, Cκ = �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We also refer toC as a κ-complex to note that κmax(C) = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let a κ-region of C refer to any non-empty subset of Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let C be a simplicial complex, and σ,τ ∈ Cκ, for some κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' When σ,τ share a face, they are weakly adjacent, denoted by σ ∼ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Their shared boundary face is denoted bd(σ,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' When σ ∼ τ, additionally they both share a co-face, their hull, denoted by hl(σ,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' If σ,τ ∈ C, σ ∼ τ, and hl(σ,τ) ∈ C, then σ,τ are strongly adjacent, denoted by σ ≃ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' If σ ∼ τ, but σ � τ in C, then σ,τ are κ-adjacent, denoted σ ∼ κ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Oriented Simplicial Complexes and Boundary Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' An oriented simplex σ further has an orientation pσ ∈ {±1}, which indicates whether its parity with its faces is the same as, or opposite to, its natural parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' When pσ = +1, we say σ is in natural orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For example, a directed edge e = (vi,v j ) for i < j is in natural orientation, while if i > j, pe = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' An oriented simplicial complex contains at most one orientation for any given simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let Xκ be the space of real-valued functions on Cκ for each κ ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',κmax(C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In the case of graphs, X0 consists of functions taking values on vertices, or graph signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' X1 consists of functions on edges, or edge flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A function in X1 is positive when the corresponding flow direction agrees with the edge orientation, and negative when the flow disagrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' X2 consists of functions on oriented triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Given an oriented simplicial complex C, for each κ ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',κmax}, the boundary op- erator is a linear operator Bκ : Xκ+1 �→ Xκ, where for σ ∈ Cκ+1, α ∈ Cκ, the corresponding matrix entries are [Bκ]ασ = pσpα nat(σ,α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Likewise, the coboundary operator for each κ ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',κmax} is just BκT : Xκ → Xκ+1, the adjoint to Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The entries of Bκ express relative orientation between simplex and face, and they are a natural way to construct functions taking local signed averages, according to adjacency in the simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 3 2 e4 e3 t2 t1 V1 V 4 e2 e5Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2: Pairs of κ-simplices demonstrating consistency at their boundary face, for κ = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The mixed-color pairs are consistent, and the same-color pairs are inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Data on Simplicial Complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Signal processing on simplicial complexes arises as a natural problem in the setting where richer structure is incorporated in data, than just scalar functions and pairwise relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this article, we assume the input data is given on an existing simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A simple directed graphG = (V,E) can always be represented as an oriented 1-complex ˜G, with each directed edge e = (vi,v j ) inserted as a 1-simplex having orientation pe = sign(i − j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' With this convention, natural orientation corresponds to the agreement of the edge direction with the global ordering of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Hodge Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The boundary operators just introduced represent discrete dif- ferential operators encoding the structure of κ-regions in a simplicial complex, and so can be building blocks towards a spectral analysis of functions on those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For analyzing functions on κ-simplices with κ > 0, we will construct operators based on the Hodge Lapla- cian, or κ-Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' As in [28], the combinatorial κ-Laplacian is defined for κ-simplices as Lκ � BT κ−1Bκ−1 +BκBT κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We refer to L ∨ κ � BT κ−1Bκ−1 and L ∧ κ � BκBT κ as the lower and upper κ-Laplacians, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Simplex consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let C be an oriented simplicial complex, and σ ∼ τ ∈ Cκ, with α = bd(σ,τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then we may write Lκ as diag(Lκ)−Sκ, where for κ > 0, Sκ is the signed adjacency matrix [Sκ]στ � � −pσpτ nat(σ,α)nat(τ,α) σ ∼ κ τ 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' When Sκ > 0, we say σ,τ are consistent, and otherwise they are inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A consistent pair of simplices view their shared boundary face in opposite ways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' one as a head face, and the other as a tail face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' An inconsistent pair of simplices view their shared boundary face identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In the case of κ = 1, two directed edges are consistent when they flow into each other at their boundary vertex, and are inconsistent when they collide at the boundary vertex, either both pointing toward it, or both pointing away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Cases for κ = 1,2 are demonstrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The combinatorial κ-Laplacian represents signed adjacency between κ-adjacent sim- plices via their consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In particular, this means that Lκ depends only on the ori- entations of simplices in Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Naively, constructing the boundary matrices Bκ−1,Bκ then additionally requires superfluous sign information – the orientation of each member of both Cκ−1 and Cκ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This situation exactly mirrors that of the graph Laplacian L0: in order to construct L0 for an undirected graph via the product B0BT 0 , one must assign an arbi- trary direction to each edge, and the resulting Laplacian is independent of that choice of directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 4 Swt 1 = 1 4 � ����� 0 2 −1 1 0 2 0 2 0 1 −2 2 0 2 −2 1 0 2 0 2 0 1 −2 2 0 � ����� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 3: The complex from Figure 1 on the left, with natural orientation displayed as directed edges, together with its weighted, unnormalized signed adjacency matrix Swt 1 , with D2 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Notice that weights differ depending on consistency and presence or lack of hull, and that the presence of a hull can switch the expected sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Weighted and Normalized Hodge Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In order to introduce a weighted simplicial complex, consider the symmetrically normalized graph Laplacian Lsym 0 � D−1/2 0 B0D1BT 0 D−1/2 0 = � D−1/2 0 B0D1/2 1 �� D−1/2 0 B0D1/2 1 �T , where D0 = diag(|B0|1), the diagonal matrix of node degrees, and D1 is the diagonal ma- trix of edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Letting Dκ generally refer to a diagonal matrix containing κ-simplex weights, we proceed as in [5] and define the symmetrically normalized κ-Laplacian as Lsym κ � BT κ−1Bκ−1 +BκBT κ , where Bκ � D−1/2 κ BκD1/2 κ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Here Dℓ = diag(|Bℓ|1) for ℓ = κ−1,κ, and Dκ+1 is the diagonal matrix of (κ+1)-hull weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From Lsym κ we may define the usual weighted unnormalized, and random-walk nor- malized κ-Laplacians Lwt κ and Lrw κ , whose eigenvectors will be the basis for our bipartition- ing: Lwt κ � D1/2 κ Lsym κ D1/2 κ and Lrw κ � D−1 κ Lwt κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' While in the combinatorial case, Lκ vanishes for pairs σ ≃ τ, each of the weighted Laplacians may be nonzero whenever σ ∼ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, we define the weighted analogues of the signed adjacency matrices, Swt κ ,Ssym κ ,Srw κ , as the off-diagonal parts of their respective Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Cuts, Fielder Vectors, and Hierarchical Bipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Fielder Vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let C be a simplicial complex, such that G = (C0,C1) is a con- nected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For a given κ, let p be a vector of orientations over Cκ, with each [p]σ = pσ ∈ ±1, and let P = diag(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let Lwt κ , ˜Lwt κ denote the weighted κ-Laplacian of Cκ with nat- ural orientations, and with orientations given by p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let λ0 ≤ ··· ≤ λn−1 be the eigenvalues of Lwt κ and φ0,φ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',φn−1 be the corresponding eigenvectors where n = |C0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then, let ( ˜λi, ˜φi) be the eigenpairs for ˜Lwt κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Because ˜Lwt κ = PLwt κ P, ˜λi = λi and ˜φi = Pφi for 0 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For κ = 0, with the vertices of G in natural orientation, we have that λ0 = 0, λ1 > 0, φ0 = 1 and in particular is non-oscillatory, and that φ1 acts as a single global oscillation, appropriate to partition the vertices of G with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Considering ˜Lwt 0 for nontrivial p � ±1, ˜φ0 is oscillatory, and ˜φ1 is no longer appropriate for clustering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' this is one reason that oriented 0-simplices are always considered to be in natural orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 5 e4 t2 e3 ti V1 V 4 e2 e5 3For κ > 0 however, it is no longer true that φ0 will be non-oscillatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let p∗ be a vector of orientations such that where [φ0]σ � 0, [p∗]σ = sign([φ0]σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then the corresponding ˜φ0 is non-oscillatory, and acts as a DC component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This motivates taking sign(φ0) · φ1 (element-wise) as the Fiedler vector of Lwt κ , with which to partition Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We will aim to bipartition κ-regions by following a standard strategy in spectral clus- tering, of minimizing a relaxation of a combinatorial cut function over possible partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Just as a graph cut is typically defined as the volume of edge weight which crosses a parti- tion of the nodes, we can define the consistency cut of Cκ into subregions A,B as Ccut(A,B) � � σ∈A,τ∈B σ∼τ [Swt κ ]στ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Because of the signs introduced by consistency, we consider Swt κ as the signed, weighted adjacency matrix for a signed graph over Cκ, and so can utilize the framework of signed Laplacians [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let [S+ κ]στ � max(0,[Swt κ ]στ) and [S− κ]στ � min(0,−[Swt κ ]στ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', indicator functions for consistent/inconsistent pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then, we can define the consis- tency volume Cvol±(A) � Ccut±(A, A) and the signed κ-cut κCut(A,B) � 2Ccut+(A,B)+Cvol−(A)+Cvol−(B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In the κ = 0 case, with all vertices in natural orientation, Swt 0 is just the usual adjacency matrix, and so S− 0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' hence κCut = 2Ccut, yielding the traditional cut objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For κ > 0, κCut increases with the number of consistent pairs of κ-adjacent simplices across the partition, and with the number of inconsistent pairs within each κ-region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Equiva- lently, minimizing κCut requires maximizing consistent pairs within each κ-region, and maximizing inconsistent pairs across the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let Lκ be the signed Laplacian with signed adjacency Swt κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Let A be a κ-region, r A � 1A − 1Cκ\\A, and define RA(L) � r T ALr A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then because Lκ differs from Lwt κ only on the di- agonal, RA(Lκ) differs from RA(Lwt κ ) by a constant independent of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From [26], we know that RA(Lκ) ∝ κCut(A,Cκ \\ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Hence, minA⊂Cκ RA(Lwt κ ) = minA⊂Cκ κCut(A,Cκ \\ A), and we obtain φ0 as a relaxed solution to κ-cut minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Now, notice that if the orientations of Cκ were changed according to some p, this would be equivalent to a different choice of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' namely, if [p]σ = −1, then σ moves to the other side of the partition, either into or out of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' As all orientations are available to us, this includes one for which ˜φ0 is non-oscillatory, so that its sign does not partition Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then instead take ˜φ1 as our relaxed solution, which we may compute via sign(φ0)·φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' An improved cut objective is the signed Ratio Cut, which encourages more balanced partitions: SignedRatioCut(A) � � 1 |A| + 1 |Cκ \\ A| � κCut(A,Cκ \\ A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From [26], we know that with rA above scaled by a factor of cA � �|A|/|Cκ \\ A|, the analo- gous result holds, that the eigenvectors of Lκ yield a relaxed solution to minA⊂Cκ SignedRatioCut(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' However, the new dependence on A means the resulting objective is slightly different for Lκ, so the relaxation is only approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, the signed Normalized Cut balances the partitions by degree rather than sim- plex count: SignedNormalizedCut(A) � � 1 Cvol(A) + 1 Cvol(Cκ \\ A) � κCut(A,Cκ \\ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Here, the eigenvectors of diag(Lκ)−1Lκ yield a relaxed solution to minA⊂Cκ SignedNormalizedCut(A), and an approximate relaxed solution is given by the eigenvectors of Lrw κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In our numeri- 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 4: One possible hierarchical bipartitioning of a simple 2-complex, from j = 0 with no partition on the left, to j = 5 on the right, where each of the 27 triangles form their own subregion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Colors indicate distinct subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' cal experiments, we use the random-walk κ-Laplacian for bipartitioning simplicial com- plexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Hierarchical Bipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The foundation upon which our multiscale transforms on a κ-simplices Cκ of a given simplicial complex C are constructed is a hierarchical bi- partition tree (also known as a binary partition tree) of Cκ, a set of tree-structured κ- subregions of Cκ constructed by recursively bipartitioning Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This bipartitioning opera- tion ideally splits each κ-subregion into two smaller κ-subregions that are roughly equal in size while keeping tightly-connected κ-simplices grouped together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' More specifically, let C j k denote the kth κ-subregion on level j of the binary partition tree of Cκ and n j k � ���C j k ���, where j,k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Note C 0 0 = Cκ, n0 0 = n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', level j = 0 represents the root node of this tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then the two children of C j k in the tree, C j+1 k′ and C j+1 k′+1, are obtained through parti- tioning C j k using the Fiedler vector of Lrw κ (C j k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This partitioning is recursively performed until each subregion corresponding to the leaf contains only a simplex singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Note that k′ = 2k if the resulting binary partition tree is a perfect binary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We note that even other (non-spectral) partitioning methods can be used to form the binary partition tree, but in this article, we stick with the spectral clustering using the Fielder vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For more details see on hierarchical partitioning, (specifically for the κ = 0 case), see [22, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 3] and [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 4 demonstrates such a hierarchical bipartition tree for a simple 2-complex consisting of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Orthonormal κ-Haar Bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The classical Haar basis [15] was introduced in 1909 as a piecewise-constant compactly-supported multiscale orthonormal basis (ONB) for square- integrable functions but has since been recognized as a wavelet family and adapted to many domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In one dimension, the family of Haar wavelets on the interval [0,1] can be generated by the following mother and scaling (or father) functions: ψ(x) = � � � � � 1, 0 ≤ x < 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' −1, 1 2 ≤ x < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' φ(x) = � 1, 0 ≤ x < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Unfortunately, these definitions do not generalize to non-homogeneous domains due to the lack of appropriate translation operators and dilation operators [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Instead, several methods have been proposed to generate similar bases, and overcomplete dictionaries to apply more abstract domains such as graphs and discretized manifolds [17, 45, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Here, we describe a method to compute similar, piecewise-constant locally supported bases for κ-simplex valued functional spaces, which we call the (orthonormal) κ-Haar bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Rather than basing our construction on some kind of translation or transportation schemes, we instead employ the hierarchical bipartition, as we discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2, to 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 5: The 2-Haar basis vectors on the same simple 2-complex shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The yellow, dark green, violet regions in each vector indicate its positive, zero, and negative components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' divide the domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', the κ-simplicesCκ of a given simplicial complexC into appropriate locally-supported κ-regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For each κ-region in the bipartition tree, if that region has two children in the tree, then we create a vector that is positive on one child, negative on the other, and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' To avoid sign ambiguity, we dictate that the positive portion is on the region whose region index is smaller among these two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Several remarks on this basis are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' First, since the division is not symmetri- cally dyadic, we need to compute the scaling factor for each region separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For each given basis vector ξ except the scaling vector, we break it into positive and negative parts ξ+ and ξ− and ensure that � i([ξ+]i + [ξ−]i) = 0 and ∥ξ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' If the members of κ-region are weighted, then this sum and norm can be computed with respect to those weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Fi- nally, we note that different hierarchical bipartition schemes may arise from the different weighting of the Hodge Laplacian, which will correspond to bases with different supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 5 demonstrates the 2-Haar basis on the simple 2-complex used in Figure 4, which has a hole in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Overcomplete Dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this section, we introduce two overcomplete dic- tionaries for analyzing real-valued functions defined on κ-simplices in a given simplicial complex: the κ-Hierarchical Graph Laplacian Eigen Transform (κ-HGLET), based on the Hierarchical Graph Laplacian Eigen Transform (HGLET) [18] and the κ-Generalized Haar- Walsh Transform (κ-GHWT), based on the Generalized Haar-Walsh Transform (GHWT) [17] for graph signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' κ-Hierarchical Graph Laplacian Eigen Transform (κ-HGLET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The first over- complete transform we describe can be viewed as a generalization of the Hierarchical Block Discrete Cosine Transform (HBDCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The classical HBDCT is generated by creat- ing a hierarchical bipartition of the signal domain and computing the DCT of the local signal supported on each subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We note that a specific version of the HBDCT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', a homogeneous split of an input image into a set of blocks of size 8×8 pixels) has been used in the JPEG image compression standard [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This process was generalized to the graph case in [18], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', the Hierarchical Graph Laplacian Eigen Transform (HGLET), from which we base our algorithm and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The basis given by the set {φj k,l} where j denotes the level of the partition (with j = 0 being the root), k indicates the partition within the level, and l indexes the elements within each partition in increasing frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' To compute the transform, we first compute the complete set of eigenvectors {φ0 0,l}l=1:n of the Hodge Laplacian of the entire κ-simplices Cκ of a given simplicial complex C and or- 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 6: 2-HGLET dictionary on the 2-complex shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Here, the color scale is consistent across each row (which corresponds to the level) to better visualize the smooth- ness of the elements der them by nondecreasing eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then partition Cκ into two disjoint κ-regions C 1 0 and C 1 1 as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then compute the complete set of eigenvectors of the Hodge Laplacian on C 1 0 and C 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We again order each set by nondecreasing frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', eigenvalue) and label these {φ1 0,l}l=1:n1 0 and {φ1 1,l}l=1:n1 1 Note that n1 0 + n1 1 = n0 0 = n, and that all of the elements in {φ1 0,l} are orthogonal to those in {φ1 1,l} since their supports are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then the set {φ1 0,l}l=1:n1 0 ∪ {φ1 1,l}l=1:n1 1 form an orthonormal basis for vectors on Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From here, we apply this process recursively, generating an orthonormal basis for each level in the given hierarchical bipartition tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' If the hierarchical bipartition tree terminates at every region containing only a κ- simplex singleton, then the final level will simply be the standard basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Each level of the dictionary contains an ONB whose vectors have the support of roughly half the size of the previous level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' There are roughly (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5)n possible ONBs formed by selecting differ- ent covering sets of regions from the hierarchical bipartition tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', [49, 40] for more about the number of possible ONBs in such a hierarchical bipartition tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, we note that the computational cost of generating the entire dictionary is O(n3) and that any valid hierarchical bipartition tree can be used to create a similar dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 6 shows the 2-HGLET constructed on the same 2-complex shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' κ-Generalized Haar-Walsh Transform (κ-GHWT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The second transform we present here is based on the Generalized Haar-Walsh Transform (GHWT) [17], which can itself be viewed as a generalization of the Wash-Hadamard transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This basis is formed by first generating a hierarchical bipartition tree of Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then work in a bottom-up manner, be- ginning with the finest level in which each region only contains a single element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We call these functions scaling vectors and label them {ψjmax k,0 }k=0:n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For the next level, we first assign a constant scaling vector for support on each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then, for each region that con- tains two children in the partition tree, we form a Haar-like basis element by subtracting the scaling function associated with the child element with a higher index from that child element with a lower index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This procedure will form an ONB {ψjmax−1 k,l }k=0:k′−1,l=0:l(k)−1 (where k′ is the number of κ-subregions at level jmax − 1 and l(k) = 1 or 2 depending on the partition k) whose vectors have support of at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For the next level, we begin by computing the scaling and Haar-like vectors as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Next, for any region that contains three or more elements, we also compute Walsh-like vectors by adding and subtracting the Haar-like vectors in the children’s regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From here, we form the rest of the dictionary recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A full description of this algorithm (for the κ = 0 case) is given in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 7 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 7: Course-to-Fine (C2F) 2-GHWT dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The yellow, dark green, and violet regions in each vector indicate its positive, zero, and negative components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' displays the 2-GHWT dictionary on the same 2-complex used in Figures 5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We make several observations about this dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' First, like the κ-HGLET, each level of the dic- tionary forms an ONB, and at each level, basis vectors have the support of roughly half the size of the previous level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' These basis vectors also have the same support as the κ-HGLET basis vectors (that is, supp(φj k,l) = supp(ψj k,l) for all j,k,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' However, the computational cost of computing the κ-GHWT is only O(n logn) compared to the O(n3) of the κ-HGLET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, we note that at the coarsest level (j = 0) the κ-GHWT dictionary contains globally-supported piecewise-constant basis vectors, which are ordered by increasing os- cillation (or “sequency”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This forms an ONB analogous to the classical Walsh Basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This allows us to define an associated Walsh transform and conduct Walsh analysis on signals defined on simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Although not the primary focus of this article, we con- duct some numerical experiments using the Walsh bases explicitly in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Organizing the Dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For many downstream applications, it is impor- tant to organize the order of these bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In general, the κ-HGLET dictionary is naturally ordered in a Coarse-to-Fine (C2F) fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In each region, the basis vectors are ordered by frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', eigenvalue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Similarly, the GHWT dictionary is also naturally ordered in a C2F fashion, with increasing “sequency” within each subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Another useful way to order the GHWT is in a Fine-to-Coarse (F2C) ordering, which approximates “sequency” domain partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', Figure 8, which shows the F2C 2-GHWT dictionary on the triangle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We also note that the F2C ordering is not possible for the κ-HGLET dictio- nary because some parent subspaces and the direct sum of their children subspaces are not equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', [22, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6)] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Other relabeling schemes, such as those proposed in [45, 40] may also be useful but are outside the scope of this article and will be explored further in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Basis and Frame Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Once we have established these arrangements of ba- sis vectors, we can efficiently apply the best-basis algorithm [8] to select an ONB that is op- timal for a task at hand for a given input signal or a class of input signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' see also our previ- ous work of applying the best-basis algorithm in the graph setting [18, 17, 19, 21, 45, 7, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Given some cost function F and signal x, we traverse the partition tree and select the basis that minimizes F restricted to each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For the C2F dictionary, we initialize the best basis as the finest (j = jmax) level of the GHWT dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then proceed upward one level at a time and compute the cost of each subspace at that level and compare it to the 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 8: Fine-to-Coarse (F2C) 2-GHWT dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Note that this dictionary is not generated by simply reversing the row indices of the C2F dictionary, but instead by arranging each level (row) by “sequency”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' cost of the union of its children subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' If the latter cost is lower, the basis is updated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' if not, the children subspaces (and their basis vectors) are propagated to the current level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This algorithm yields the C2F best basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The F2C best basis is performed similarly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', we begin with the globally-supported basis (j = 0) at the bottom of the rearranged tree and proceed in the same bottom-up direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' As for the HGLET dictionary, it has only a C2F basis as we discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In some contexts, it is not necessary to generate a complete ONB, but rather some sparse set of vectors in the dictionary (also known as atoms) that most accurately approx- imate a given signal or class of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this case, we can directly apply the orthogo- nal matching pursuit of [3] to find the best m-dimensional orthogonal subframe (m ≤ n) selected from the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Additionally, for some downstream tasks, such as sparse ap- proximation or sparse feature selection, generating orthogonal sets of atoms is not critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In these cases, we can employ a greedy algorithm to generate efficient approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This algorithm simply selects the atoms in the dictionary with the largest coefficient, removes it, then computes the transform of the residual and proceeds so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' These basis and subframe algorithms are studied intensively in the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Numerical Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We demonstrate the efficacy of our proposed partition- ing techniques and basis constructions by conducting a series of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In Sec- tion 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1 we show how our multiscale bases and overcomplete dictionaries can be used to sparsely approximate signals defined on κ-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 we show how these representations can be used in supervised classification and unsupervised clustering prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Approximation and Signal Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We begin with an illustrative example by creating some synthetic data for 1- and 2-simplices by triangulating a digital image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We start with a 512 × 512 “peppers” image and map it to a Cartesian grid on the unit square [0,1]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then randomly sample 1028 points within this square (not necessarily on a grid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then create a triangular mesh from these points using Delaunay triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Next, we interpolate the image from the Cartesian grid to the sampled vertices by computing the barycentric coordinate of each vertex from the square inside the Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, we interpolate the signal to the edges and triangles of the triangulation by averaging the values of the vertices that they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The result, for our random seed, is a signal defined 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 9: Nonlinear approximation of the peppers image for κ = 2 on the 3050 edges of the triangulation and another on the 2067 triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We now consider the sparse representation of these signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 9 shows the nonlinear approximation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', using the largest expansion coefficients in magnitude) of the triangle-based signals in the Hodge Laplacian eigenbasis (Fourier), the orthonormal Haar basis, orthonormal Walsh basis as well as the approximation prescribed by applying the best-basis and greedy algorithms to the HGLET and GHWT dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 10 shows the approximation error vs the number of terms used for both the edge-based and triangle-based functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A number of observations are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' First, the multiscale dictionary-based meth- ods consistently outperformed the generic orthonormal bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The greedy approximation algorithm achieved the best approximation results, but it is also more costly to compute than any of the other methods, and the set of atoms used in the approximation may not be orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This may be detrimental to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Overall the GHWT-based method performed best, with the F2C best basis performing much better than the C2F best basis, which suggests that the fine-scale features of this signal are the most impor- tant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Similarly, the Walsh basis achieved much better results than the Haar basis, again emphasizing the necessity of capturing details at the fine scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Next, we apply our approach to real-world data for higher degree signals for κ = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The citation complex [33, 12] is a simplicial complex derived from the Cora citation com- plex [48], which models the interactions between multiple authors of scientific papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A paper with κ authors is represented by a (κ − 1)-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We first build a graph whose vertices represent the authors in this Cora database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Then, the vertices are connected by edges that represent co-authored papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Note that if two authors co-authored multiple 12 1% 5% 10% 25% 50% 75% 90% Delta Fourier Haar Walsh HGLET (BB) GHWT (BB)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 10: Nonlinear approximation errors of the peppers image, Left: L2 error, Right: log(L2 error) for up to 50% of the terms retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Top κ = 1, Bottom: κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' papers, these two vertices are connected by a single edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Next, we assign each edge the sum of the citation numbers of all the co-authored papers by the authors, forming this edge as its weight (or value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, we assign each higher-order simplex the sum of the values of its lower-order simplices as its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' See [12] for a more thorough description of the construction of this complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Table 1 reports some basic information about the number of simplices of different degrees in this citation complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 11 shows the approximation of this signal(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', a vector of citation numbers) for κ = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',5 with the Delta, Fourier, Haar, HGLET, and GHWT bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 12 shows the log error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The HGLET and GHWT bases were selected with the best-basis algorithm using the C2F ordering for the GHWT dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In these experiments, we observe that the best bases (GHWT and HGLET) outper- formed the canonical bases, with the GHWT being the most efficient basis for each κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Ad- ditionally, for κ > 0, the orthonormal Haar basis performed best in the semi-sparse regime (1 and 10% of terms retrained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This suggests that the signals on each degree of the citation complex are similar in that they are all close to being piecewise constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' However, when 13 K=l, Approximation Error K=1, Log Approximation Error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Basis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Frourier Basis Orthogonal Haar Orthogonal Walsh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 BB HGLET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 HGLET (Greedy) BB GHWT C2F Log L2 Approximation Error Approximation Error BB GHWT F2C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 GHWT (Greedy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 Delta Basis Frourier Basis Orthogonal Haar 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0- Orthogonal Walsh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 BB HGLET HGLET (Greedy) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 - BB GHWT C2F BB GHWT F2C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 GHWT (Greedy) 0 500 1000 1500 2000 2500 3000 0 200 400 600 800 1000 1200 1400 1600 # of Terms # of TermsK=2, Approximation Error K=2, Log Approximation Error Delta Basis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Frourier Basis Orthogonal Haar Orthogonal Walsh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 BB HGLET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 HGLET (Greedy) BB GHWT C2F Log L2 Approximation Error Approximation Error BB GHWT F2C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 GHWT (Greedy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 Delta Basis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Frourier Basis Orthogonal Haar Orthogonal Walsh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 BB HGLET 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 HGLET (Greedy) BB GHWT C2F BB GHWT F2C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 - GHWT (Greedy) 750 0 200 400 600 800 0 250 500 100012501500 1750 2000 1000 # of Terms # of Termsκ 0 1 2 3 4 5 # of elements 1126 5059 11840 18822 21472 17896 Table 1: The number of element in the κ-simplices in the Cora complex for κ = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 11: Approximation of the Citation Complex for κ = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' more terms are considered, the HGLET best basis achieved a lower approximation error than the orthonormal Haar basis achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Signal Clustering and Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Since the basis (and dictionary) vectors we present are both multiscale and built from the Hodge Laplacians that are aware of both topological and geometric properties of the domain [5], they can function as very powerful feature extractors for general data science applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this section, we present two clustering-type applications—one supervised and one unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' For baselines, we compare our proposed dictionaries with Fourier and Delta (indicator function) bases and with the Hodgelets proposed in [35] for cases when κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Supervised Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' First, we present our study in supervised classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We begin by computing edge-valued signals for 1000 handwritten digits from the MNIST dataset [27] by sampling 500 points in the unit square and following the interpola- tion method presented for the peppers image in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then compute the features of these images using the proposed orthogonal transforms and best bases from the over- complete dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Next, we train a support vector machine (SVM) to classify the digits for each of the transformed representations using the 1000 training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Finally, we test these SVMs on the rest of the whole MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We repeat this experiment for the FMNIST dataset [51], again using only 1000 exam- ples for training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Results are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We remark that these tests are not meant to achieve state-of-the-art results for image classification but rather to showcase the effectiveness of these representations for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Unsurprisingly, the dic- tionary methods outperformed the basis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Again, the piecewise constant meth- ods (GHWT, Haar) achieved better approximations than the smoother methods (Fourier, 14 Approximation k=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar HGLET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 GHWT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 ux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 50 100 150 200 250 300 350 # of elementsApproximation k=1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 HGLET GHWT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 x X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 200 400 600 800 1000 1200 1400 # of elementsApproximation k=2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 HGLET GHWT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 500 1000 1500 2000 2500 3000 # of elementsApproximation k=3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar HGLET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 GHWT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 1000 2000 3000 4000 5000 # of elementsApproximation k=4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar HGLET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 GHWT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 1000 2000 3000 4000 5000 # of elementsApproximation k=5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar HGLET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='8 GHWT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 1000 2000 3000 4000 # of elementsFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 12: Top: Approximation of the Citation Complex for κ = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Bottom: Log of the error for up to 50% of the terms retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Basis Methods Dictionary Methods Delta Fourier Haar Walsh HGLET (BB) GHWT (BB C2F) GHWT (BB F2C) Joint Separate HGLET GHWT # of terms 661 661 661 661 661 661 661 5288 5288 9254 9254 MNIST 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='675 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='053 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='388 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='011 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='991 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='779 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='156 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='202 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='038 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='001 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='089 FMNIST 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='370 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='753 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='779 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='230 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='117 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='991 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='121 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='761 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='738 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='739 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='789 Table 2: Test Accuracy for SVMs trained on transforms of MNIST signals interpolated to a random triangulation HGLET, Joint, and Separate Hodgelets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This is likely due to the near-binary nature of im- ages in both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Unsupervised Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A natural setting for studying κ = 1 valued signals is the analysis of trajectories [5, 36, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Of particular interest is the case where the domain has nontrivial topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Such is the case of the Global Drifter Program dataset, which tracks the positions of 334 buoys dropped into the ocean at various points around the island of Madagascar [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We begin by dividing the dataset into three subsets, train (|Xtr| = 176), test (|Xte| = 83) and validation (|Xvl| = 84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We then use orthogonal matching pursuit [3] (OMP) to compute the m significant features of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Next, we extract these features for the test set and use them to compute the centroids {c j }d j=1 for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' To evaluate these clusters K -score (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' the standard k-means objective) on the transformed features of the validation set: K −score := 1 N N � i=1 min 1≤j≤d ∥f (xi)−c j ∥2, xi ∈ Xvl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' where f (·) represents the feature extraction prescribed by applying OMP to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' We repeat this experiment for m = 5,10,15,20,25 (number of features) and d = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=',7 (num- ber of clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Figure 13 summarizes the results of this test, while Table 3 shows the full 15 Approximationk=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 Haar HGLET GHWT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 (llux 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 )601 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 0 25 50 75 100 125 150 175 #ofelementsApproximation k=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 HGLET GHWT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 0 100 200 300 400 500 600 700 # of elementsApproximation k=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 HGLET GHWT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 250 500 750 1000 1250 1500 # of elementsApproximation k=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta Fourier Haar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 HGLET GHWT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 : xII)60| 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0 500 1000 1500 2000 2500 # of elementsApproximation k=4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 Fourier Haar HGLET 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 GHWT 0 500 1000 1500 2000 2500 # of elementsApproximation k=5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 Delta 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='5 Fourier Haar 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0 HGLET GHWT 0 500 1000 1500 2000 # of elementsFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 13: Extensive results for buoy cluster test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Leftmost figure shows which method pre- formed best, the second to the left shows the second best and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The x-axis in each subplot indicates the number of coefficients used and the y-axis is the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Full numerical results are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this experiment, the GHWT outperformed all other bases because the trajectories are roughly constant and locally supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' The orthogonal matching pur- suit scheme can select elements with the correct support size, and the piecewise constant nature of the GHWT atoms can capture the action of the trajectory with very few elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Conclusions and Future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' In this article, we have developed several general- izations of orthonormal bases and overcomplete transforms/dictionaries for signals de- fined on κ-simplices, and demonstrated their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co- authorship/citation dataset and an ocean current/flow dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' However, there are many more tools from harmonic analysis that we have not addressed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From a theoretical standpoint, future work may involve 1) defining additional families of mul- tiscale transforms such as the extended Generalized Haar-Walsh Transform(eGHWT) [45] and Natural Graph Wavelet Packets (NGWPs) [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2) exploring different best-basis selection criteria tailored for classification and regression problems such as the Local Discriminant Basis [37, 39] and the Local Regression Basis [38] on simplicial complexes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' and 3) inves- tigating nonlinear feature extraction techniques such as the Geometric Scattering Trans- form [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' From an application standpoint, we look forward to applying the techniques presented here to data science problems in computational chemistry, weather forecasting, and genetic analysis, all of which have elements that are naturally modeled with simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' This research was partially supported by the US National Science Foundation grants DMS-1418779, DMS-1912747, CCF-1934568;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' the US Office of Naval Re- search grant N00014-20-1-2381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 16 Validation set Best Validation set Second Validation set Third 6 6 5 5+ num_clusters 4 GHWT 3 - 3 HGLET 2 - 2 5 5 5 num coefs num coefs Haar num coefs Validation set Fourth Validation set Fifth Validation set Sixth Separate 6 Joint 4 Fourier 3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2 num coefs num coefs 9 num coefsREFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' BARBAROSSA AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SARDELLITTI, Topological signal processing over simplicial complexes, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 68 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2992–3007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' BRUNA, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' ZAREMBA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SZLAM, AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LECUN, Spectral networks and locally connected networks on graphs, arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='6203, (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' CAI AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' WANG, Orthogonal matching pursuit for sparse signal recovery with noise, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Theory, 57 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 4680–4688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' CARLSSON, Topology and data, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 46 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 255–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' CHEN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' MEIL ˘A, AND I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' KEVREKIDIS, Helmholtzian eigenmap: Topological feature discovery & edge flow learning from point cloud data, arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='07626, (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' CHUNG AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LU, Complex Graphs and Networks, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 107 in CBMS Regional Conference Series in Math- ematics, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', Providence, RI, 2006, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1090/cbms/107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' CLONINGER, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LI, AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, Natural graph wavelet packet dictionaries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Fourier Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 27 (2021), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 41, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1007/s00041-021-09832-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' a part of “Topical Collection: Harmonic Analysis on Combinatorial Graphs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' COIFMAN AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' WICKERHAUSER, Entropy-based algorithms for best basis selection, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Theory, 38 (1992), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 713–718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' CRANE, Discrete differential geometry: An applied introduction, Notices of the AMS, Communication, (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 1153–1159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' DONG, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' THANOU, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' TONI, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' BRONSTEIN, AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' FROSSARD, Graph signal processing for machine learning: A review and new perspectives, IEEE Signal Processing Magazine, 37 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 117–127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' EASLEY AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' KLEINBERG, Networks, Crowds, and Markets: Reasoning and a Highly Connected World, Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Press, New York, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' EBLI, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' DEFFERRARD, AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SPREEMANN, Simplicial neural networks, arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='03633, (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' GAO, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' WOLF, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' HIRN, Geometric scattering for graph data analysis, in International Conference on Machine Learning, PMLR, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2122–2131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' GIUSTI, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' GHRIST, AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' BASSETT, Two’s company, three (or more) is a simplex, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Neurosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 41 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' HAAR, Zur Theorie der orthogonalen Funktionensysteme, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 69 (1910), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 331–371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' HOLZRICHTER AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' OLIVEIRA, A graph based method for generating the fiedler vector of irregular prob- lems, in International Parallel Processing Symposium, Springer, 1999, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 978–985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' IRION AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, The generalized Haar-Walsh transform, in 2014 IEEE Workshop on Statistical Signal Processing (SSP), IEEE, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 472–475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' IRION AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, Hierarchical graph Laplacian eigen transforms, JSIAM Letters, 6 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 21–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' IRION AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, Applied and computational harmonic analysis on graphs and networks, in Wavelets and Sparsity XVI, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SPIE 9597, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Papadakis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Goyal, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Van De Ville, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 2015, https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2186921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Paper # 95971F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' IRION AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, Learning sparsity and structure of matrices with multiscale graph basis dictionaries, in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Uncini, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Diamantaras, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Palmieri, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Larsen, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 2016, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1109/MLSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='7738892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' IRION AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, Efficient approximation and denoising of graph signals using the multiscale basis dictionaries, IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Signal Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 3 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 607–616, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1109/ TSIPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2632039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' IRION, Multiscale Transforms for Signals on Graphs: Methods and Applications, PhD thesis, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' California, Davis, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2015, https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='com/JeffLIrion/MTSG_Toolbox/blob/ master/Publications/irion_dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' JAFFARD, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' MEYER, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' RYAN, Wavelets: Tools for Science & Technology, SIAM, Philadelphia, PA, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [24] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' JIANG, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LIM, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' YAO, AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' YE, Statistical ranking and combinatorial Hodge theory, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Pro- gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 127 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 203–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' KONDOR AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' TRIVEDI, On the generalization of equivariance and convolution in neural networks to the action of compact groups, in International Conference on Machine Learning, PMLR, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2747–2755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' KUNEGIS, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHMIDT, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LOMMATZSCH, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LERNER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LUCA, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' ALBAYRAK, Spectral analy- sis of signed graphs for clustering, prediction and visualization, in Proceedings of the 2010 SIAM In- ternational Conference on Data Mining (SDM), SIAM, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 559–570, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1137/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 9781611972801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='49, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/abs/https://epubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='siam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1137/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='9781611972801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LECUN AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' CORTES, MNIST handwritten digit database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' http://yann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='lecun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='com/exdb/mnist/, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 17 [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LIM, Hodge Laplacians on graphs, SIAM Review, 62 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 685–715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LOVÁSZ, Large Networks and Graph Limits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 60 of Colloquium Publications, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', Provi- dence, RI, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' MALLAT, A Wavelet Tour of Signal Processing, Academic Press, Burlington, MA, third ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' NEWMAN, Networks, Oxford Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Press, Oxford, UK, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' ORTEGA, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' FROSSARD, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' KOVA ˇCEVI ´C, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' MOURA, AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' VANDERGHEYNST, Graph signal processing: Overview, challenges, and applications, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' IEEE, 106 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 808–828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' PATANIA, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' PETRI, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' VACCARINO, The shape of collaborations, EPJ Data Science, 6 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [34] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' PENNEBAKER AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' MITCHELL, JPEG Still Image Data Compression Standard, Van Nostrand Rein- hold, New York, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' RODDENBERRY, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' FRANTZEN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHAUB, AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SEGARRA, Hodgelets: Localized spectral repre- sentations of flows on simplicial complexes, in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 5922–5926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [36] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' RODDENBERRY, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHAUB, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' HAJIJ, Signal processing on cell complexes, in ICASSP 2022- 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 8852–8856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [37] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' COIFMAN, Local discriminant bases and their applications, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Imaging Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 5 (1995), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 337–358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Invited paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [38] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' COIFMAN, Extraction of geological information from acoustic well-logging wave- forms using time-frequency wavelets, Geophysics, 62 (1997), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 1921–1930, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1190/ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1444292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [39] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' COIFMAN, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' GESHWIND, AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' WARNER, Discriminant feature extraction using empir- ical probability density estimation and a local basis library, Pattern Recognition, 35 (2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 2841– 2852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [40] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO AND Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SHAO, eGHWT: The Extended Generalized Haar–Walsh Transform, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Imaging Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 64 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 261–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [41] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAYOOD, Introduction to Data Compression, Morgan Kaufmann Publishers, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', San Francisco, CA, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHAUB, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' BENSON, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' HORN, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LIPPNER, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' JADBABAIE, Random walks on simplicial com- plexes and the normalized Hodge 1-laplacian, SIAM Review, 62 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 353–391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [43] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHONSHECK AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHONSHECK, Spherical coordinates from persistent cohomology, arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='02791, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SCHONSHECK, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' DONG, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' LAI, Parallel transport convolution: Deformable convolutional net- works on manifold-structured data, SIAM Journal on Imaging Sciences, 15 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 367–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [45] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SHAO AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SAITO, The extended generalized Haar-Walsh transform and applications, in Wavelets and Sparsity XVIII, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SPIE 11138, Paper #111380C, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [46] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SHUMAN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' NARANG, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' FROSSARD, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' ORTEGA, AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' VANDERGHEYNST, The emerging field of signal processing on graphs, IEEE Signal Processing Magazine, 30 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 83–98, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1109/MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='2235192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [47] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' SHUMAN, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' NARANG, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' FROSSARD, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' ORTEGA, AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' VANDERGHEYNST, The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains, IEEE Signal Processing Magazine, 30 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 83–98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [48] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' ŠUBELJ AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' BAJEC, Model of complex networks based on citation dynamics, in Proceedings of the 22nd international conference on World Wide Web, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 527–530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [49] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' THIELE AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' VILLEMOES, A fast algorithm for adapted time-frequency tilings, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Harmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', 3 (1996), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 91–99, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1006/acha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' WICKERHAUSER, Adapted Wavelet Analysis from Theory to Software, A K Peters, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=', Wellesley, MA, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' [51] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' XIAO, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' RASUL, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' VOLLGRAF, Fashion-MNIST: a novel image dataset for benchmarking ma- chine learning algorithms, arXiv preprint arXiv:/1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='07747, (2017), https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='org/abs/cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='LG/1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 07747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' 18 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Full Results for Buoy Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Clusters # Feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content=' Fourier Joint Separate Haar HGLET GHWT 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='024 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='023 2 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='021 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='023 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='096 03073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='024 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='110 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='0115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='026 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='023 3 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='021 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='019 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='021 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='023 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='019 4 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='018 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='021 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='022 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='014 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='018 5 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='017 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='020 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='019 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='014 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='018 6 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='017 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='020 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='019 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='014 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='013 7 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='013 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='014 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} +page_content='013 Table 3: K -score for buoys tests, smaller is better 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dA0T4oBgHgl3EQfMv9X/content/2301.02136v1.pdf'} diff --git a/3tAzT4oBgHgl3EQfffwg/content/tmp_files/2301.01452v1.pdf.txt b/3tAzT4oBgHgl3EQfffwg/content/tmp_files/2301.01452v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..83a6895e5ca814974c2ee1c3699928e2ccedea04 --- /dev/null +++ b/3tAzT4oBgHgl3EQfffwg/content/tmp_files/2301.01452v1.pdf.txt @@ -0,0 +1,1010 @@ +THE PREDICTIVE FORWARD-FORWARD ALGORITHM +Alexander Ororbia +Rochester Institute of Technology +ago@cs.rit.edu +Ankur Mali +University of South Florida +ankurarjunmali@usf.edu +ABSTRACT +In this work, we propose a generalization of the forward-forward (FF) algorithm that we call the +predictive forward-forward (PFF) algorithm. Specifically, we design a dynamic, recurrent neural +system that learns a directed generative circuit jointly and simultaneously with a representation +circuit, combining elements of predictive coding, an emerging and viable neurobiological process +theory of cortical function, with the forward-forward adaptation scheme. Furthermore, PFF +efficiently learns to propagate learning signals and updates synapses with forward passes only, +eliminating some of the key structural and computational constraints imposed by a backprop- +based scheme. Besides computational advantages, the PFF process could be further useful for +understanding the learning mechanisms behind biological neurons that make use of local (and +global) signals despite missing feedback connections [11]. We run several experiments on image +data and demonstrate that the PFF procedure works as well as backprop, offering a promising +brain-inspired algorithm for classifying, reconstructing, and synthesizing data patterns. As a +result, our approach presents further evidence of the promise afforded by backprop-alternative +credit assignment algorithms within the context of brain-inspired computing. +Keywords Brain-inspired computing · Self-supervised learning · Neuromorphic · Forward learning +1 +Introduction +The algorithm known as backpropagation of errors [38], or “backprop” for short, has long faced criticism concerning +its neurobiological plausibility [8, 27, 12]. Despite having powered the tremendous progress and success behind +deep learning and its every-growing myriad of promising applications [44, 9], it is improbable that backprop is +a good, viable model of learning in the brain, such as in cortical regions. Notably, there are both practical and +biophysical issues [12, 27], and among these issues are the following: +• there is a lack of evidence that neural activities are explicitly stored to be used later for synaptic adjustment, +• error derivatives are backpropagated along a global feedback pathway to generate teaching signals, +• the error signals move back along the same neural pathways used to forward propagate information, and, +• inference and learning are locked to be largely sequential (instead of massively/easily parallel). +Furthermore, in processing temporal data, it is certainly not the case that the neural circuitry of the brain is unfolded +backwards through time in order to calculate and adjust synapses [33] (as it is for backprop through time). +Recently, there has been a growing interest in the research domain of brain-inspired computing, which focuses +on developing algorithms and computational models that attempt to circumvent or resolve the critical issues +such as those highlighted above. Among the most powerful and promising ones is predictive coding (PC) +[15, 37, 10, 3, 40, 32], and among the most recent ones is the forward-forward (FF) algorithm [16]. These +alternatives offer powerful, different means of conducting credit assignments that have shown similar performance +as backprop, but to the contrary, are more likely consistent with and similar to real biological neuron learning +(see Figure 1 for some representative credit assignment depictions). This paper will propose a novel model and +learning/inference process, the predictive forward-forward (PFF) process, that generalizes and combines FF and +PC into a robust (stochastic) neural system that simultaneously learns a representation and generative model in a +biologically-plausible fashion. Like the FF algorithm, the PFF procedure offers a promising, potentially helpful +model of biological neural circuits, a potential candidate system for low-power analog hardware and neuromorphic +circuits, and a potential backprop-alternative worthy of future investigation and study. +arXiv:2301.01452v1 [cs.LG] 4 Jan 2023 + +Preprint +h3 +h2 +h1 +W1 +W2 +W3 +WT +3 +WT +2 +WT +1 +h3 +h2 +h1 +W1 +W2 +W3 +B3 +B2 +B1 +L, Y +L, Y +h3 +h2 +h1 +W1 +W2 +W3 +WT +3 +WT +2 +WT +1 +L3, Y +L2 +L, +h3 +h2 +h1 +W1 +W2 +W3 +E3 +E2 +E1 +L3, Y +L2 +L1 +X +X +X +BP +h3 +h2 +h1 +W1 +W2 +W3 +L3, Ypos +L2 +L1 +X_pos +hNg +3 +hNg +2 +hNg +1 +X_neg +FF +h3 +h2 +h1 +W1 +W2 +W3 +L3, Ypos +L2 +L1 +X_pos +hNg +3 +hNg +2 +hNg +1 +X_neg +zG +3 +zG +2 +zG +1 +GM +L = Global loss +LN = Local loss +WN = Forward weights +BN = Fixed backward weights (random +weights) +EN = Learnable recurrent error weights +WT +N = Transpose of forward activity +GN = Generative weights +GM = Generative model +HN = Hidden States +HNG +N = Hidden States obtained by doing +2nd forward pass on negative data +ZN = Error Corrected State +ZG +N = Generative Model State +X = Input +Y = Output +X_pos = Positive input data +X_neg = Negative input data +Yneg +Yneg +G3 +G2 +G1 +FA +PC +LRA +h3 +h2 +h1 +W1 +W2 +W3 +E3 +E2 +E1 +L3, Y +L2 +L1 +[X, Y] +z3 +z2 +z1 +G3 +G2 +G1 +NGC +PFF +X +Figure 1: Comparison of learning algorithms that relax constraints imposed by backpropagation of errors (BP). +Algorithms visually depicted include feedback alignment (FA) [26], predictive coding (PC) [37, 41], local repre- +sentation alignment (LRA) [35], neural generative coding (NGC) [34, 32], the forward-forward procedure (FF) +[16], and predictive forward-forward algorithm (PFF). +2 +The Predictive Forward-Forward Learning Process +The brain-inspired neural process that we will design and study is called the predictive forward-forward (PFF) +algorithm, which builds on and generalizes aspects of the FF algorithm [16]. At a high level, the PFF process +consists of two neural structures or circuits, i.e., a representation circuit (parameterized by Θr) that focuses on +acquiring distributed representations of data samples and a top-down generative circuit (parameterized by Θg) that +focuses on learning to synthesize data given the activity values of the representation circuit. Thus, the PFF process +can be characterized as a complementary system with the aim of jointly learning a classifier and generative model. +We will first define the notation used throughout this paper, then proceed to describe the inference and learning +mechanics of the representation circuit followed by those of the generative circuit. +Notation: +We use ⊙ to indicate a Hadamard product and · to denote a matrix/vector multiplication. (v)T is the +transpose of v. Matrices/vectors are depicted in bold font, e.g., matrix M or vector v (scalars shown in italic). zj +will refer to extracting jth scalar from vector z. Finally, ||v||2 denotes the Euclidean norm of vector v. The sensory +input has shape x ∈ RJ0×1 (J0 is the number of input features, e.g., pixels), label has shape y ∈ RC×1 (where C +is the number of classes), and any neural layer has shape zℓ ∈ RJℓ×1 (Jℓ is the number of neurons in layer ℓ). +2.1 +The Forward-Forward Learning Rule +The PPF process, like the FF algorithm when it is applied to a recurrent network, involves adjusting the synaptic +efficacies of a group of neurons by measuring their “goodness”, or, in other words, the probability that their activity +indicates that an incoming signal comes from the target training data distribution (or the “positive class”). Formally, +for any single layer ℓ in an L-layered neural system, we calculate the goodness as the sum of the squared activities +for a given neural activity vector zℓ and compare it to particular threshold value θz in one of two ways: +p(c = 1)ℓ = +1 +1 + exp +� +− (�Jℓ +j (zℓ +j)2 − θz) +�, or, p(c = 1)ℓ = +1 +1 + exp +� +− (θz − �Jℓ +j (zℓ +j)2) +� +(1) +where p(c = 1)ℓ indicates the probability that the data comes from the data distribution (i.e., positive data, where +the positive class is labeled c = 1) while the probability that the data does not come from the training data +distribution is p(c = 0)ℓ = 1 − p(c = 1)ℓ. Note that p(c)ℓ indicates the probability that is assigned by a layer ℓ of +neurons in a system/network. This means the cost function that any layer is trying to solve/optimize is akin to a +binary class logistic regression problem formulated as follows: +L(Θℓ) = − 1 +N +N +� +i=1 +ci log p(ci = 1)ℓ + (1 − ci) log p(ci = 0)ℓ +(2) +2 + +Preprint +where the binary label ci (the label for the ith datapoint xi) can be generated correctly and automatically if one +formulates a generative process for producing negative data samples. Data patterns sampled from the training set +xj ∼ Dtrain can be labeled as cj = 1 and patterns sampled outside of Dtrain (from the negative data generating +process) can be automatically labeled as cj = 0. Crucial to the success of the FF procedure is the design of a useful +negative data distribution, much as is the case for noise contrastive estimation [13]. +It is important to notice that the FF learning rule is local in nature – this means that the synapses of any particular +layer of neurons can be adjusted independently of the others. The rule’s form is furthermore different from a +classical Hebbian update [14] (which produces a weight change by a product of incoming and outgoing neural +activities), given that this synaptic adjustment requires knowledge across a group of neurons (goodness depends on +the sum of squares of the activities of a group rather than an individual unit) and integrates contrastive learning +into the dynamics. Synaptic updates are specifically calculated by taking the gradient of Equation 2, i.e., ∂L(Θℓ) +∂Θℓ . +In effect, a neural layer optimizes Equation 2 by either maximizing the squared activities of a layer (to be above +threshold θz) (left form of the probability presented in Equation 1) or, alternatively, minimizing the squared +activities (right form of the probability presented in Equation 1). +2.2 +The Representation Circuit +In order to take advantage of the above FF learning rule (and to model contextual prediction via top-down and +bottom-up influences), a recurrent network was proposed in [16], where, at each layer, a set of top-down and +bottom-up forces are combined to compute the activity of any layer ℓ, much akin to the inference process of a deep +Boltzmann machine [39]. The core parameters of this model are housed in the construct Θr = {W1, W2, ..., WL} +(later referred to as the representation parameters). Note that no additional classification-specific parameters are +included in our model (in contrast to the model of [16]), although incorporating these is straightforward.1 Note that +the representation circuit of the the PFF process will take the form of a recurrent network. +To compute any layer’s activity within the representation circuit, top-down and bottom-up messages are combined +with an interpolation of the layer’s activity at the previous time step. Specifically, in PFF, this is done as follows: +zℓ(t) = β +� +φℓ� +Wℓ · LN(zℓ−1(t − 1)) + Vℓ · LN(zℓ+1(t − 1)) +� ++ ϵℓ +r +� ++ (1 − β)zℓ(t − 1) +(3) +where ϵℓ +r ∼ N(0, σ) is injected, centered Gaussian noise and z0(t − 1) = x. As in [16], we set the activation +function φℓ() for each layer ℓ to be the linear rectifier, i.e., φℓ(v) = max(0, v). Notice the introduction of an +interpolation coefficient β, which allows integration of the state zℓ over time (the new activity state at time t is a +convex combination of the newly proposed state and the previous value of the state at t − 1). Furthermore, notice +that this interpolation is similar to that of the “regression” factor introduced into the recirculation algorithm [19], a +classical local learning algorithm that made use of carefully crafted autoencoders to generate the signals needed +for computing synaptic adjustments. LN(z) is a layer normalization function applied to the activity vector, i.e., +LN(zℓ) = zℓ/(||zℓ||2 + ϵ) (ϵ is a small numerical stability factor for preventing division by zero). Note that the +topmost layer of the representation circuit is clamped to a context vector y (which could be provided by another +neural circuit or be set to be a data point’s label/target vector), i.e., zL+1 = y2, while the bottom layer is clamped +to sensory input, i.e., z0(t) = x(t) (where x(t) could be the frame of video or a repeated copy of a static image x). +Equation 3 depicts a synchronous update of all layer-wise activities, but, as noted in [16], the recurrent model could +alternatively be implemented by cycling between even and odd-number layers, i.e., first updating all even-numbered +layers given the activities of the odd-numbered layers followed by updating the values of the odd-numbered layers +given the new values of the even layers, much like the generative stochastic networks of [5]. +To create the negative data needed to train this system, we disregard the current class indicated by the label y of +the positive data xp and create an incorrect “negative label” yn by randomly (uniformly) sampling an incorrect +class index, excluding the correct one.3 A final mini-batch of samples is dynamically created by concatenating +positive and negative samples, i.e., x =< x, x > and y =< y, yn > (notice that positive image pixels are reused +1If classification-specific parameters are desired, one could include an additional set of synaptic weights Θd = {W, b} +that take in as input the top-most (normalized) activity LN(zL) of the recurrent representation circuit in order to make a rough +prediction of the label distribution over y, i.e, p(y = i|LN(zL)) = exp(W · LN(zL) + b)i/ +� � +c exp(W · LN(zL) + b)c +� +. +This would make the recurrent model of this work much more similar to that of [16]. Softmax parameters W and b would then +be adjusted by taking the relevant gradients of the objective Ly(W, b) = − log p(y = i|LN(zL)). +2It is important to scale the label/context vector by a factor of about 5, i.e., the topmost layer activity would be zL+1 = y ∗ 5 +(Geoffrey Hinton, personal communication, Dec 12, 2022). +3This deviates from how the negative label was made in [16], which chose an incorrect class index in proportion to the +probabilities produced by a forward pass of the classification-specific parameters. This was not needed for the PFF algorithm. +3 + +Preprint +and paired with the negative labels in order to create the negative samples). The PFF process then involves running +the combined mini-batch through the neural system and calculating the relevant synaptic updates. +Equation 3 is typically run several times (8 to 10 times as in this study and [16]), similar to the stimulus processing +window that is simulated for predictive coding systems [37, 32]. Each time Equation 3 is run, the (bottom-up and +top-down) synapses for layer ℓ are adjusted according to the following local update: +∆Wℓ = +� +2 +∂L(Θℓ) +∂ �Jℓ +j (zℓ +j)2 ⊙ zℓ� +· +� +LN(zℓ−1) +�T , and, ∆Vℓ = +� +2 +∂L(Θℓ) +∂ �Jℓ +j (zℓ +j)2 ⊙ zℓ� +· +� +LN(zℓ+1) +�T +(4) +which can then be applied to the relevant parameters, i.e., Wℓ and Vℓ, via methods such as stochastic gradient +descent (SGD) with momentum or Adam [22]. In principle, the neural layers of the representation circuit are +globally optimizing the objective L(Θr) = �L +ℓ=1 Lℓ(Θℓ = Wℓ) (the summation of local goodness functions). +On Classifying Sensory Patterns: +One might observe that our representation circuit does not include discrimi- +natory parameters that classify inputs directly. Nevertheless, given that the supervised target y is used as context to +mediate the top-most latent representations of the recurrent circuit above, the representation system should (positive +data samples) acquire distributed representations that implicitly encode label information. To take advantage of +the discriminative information encoded in PFF’s representations, as was also done in the FF algorithm, we may +still classify by executing an inference process similar to that of early hybrid Boltzmann machine models [23, 36]. +Specifically, to classify an input x, we iterate over all possible (one-hot) values that y could be, starting with the +first class index. Specifically, for any chosen y (such as the one-hot encoding of class index i), we run Equation +3 for the representation circuit for T steps and then record the goodness across the layers in the middle three +iterations (from T/2 − 1 to T/2 + 1), i.e., Gy=i = 1 +3 +�T/2+1 +T/2−1 +1 +L +�L +ℓ=1 θz − �Jℓ +j (zℓ +j +2). This goodness calculation +is made for all class indices y = 1, 2, ..., C, resulting in {Gy=1, Gy=2, ..., Gy=C} over which the argmax is applied +in order to obtain the index of the class with the highest average goodness value. Note that, as mentioned in [16], if +classification-specific parameters are included in PFF’s representation circuit, then a single feedforward pass could +be used to obtain initial class probabilities. Then the above search could instead be simplified by conducting it +over only the top M highest probabilities (and thus avoid an expensive search over a massive number of classes). +To estimate the label probability distribution under the representation circuit, as we do in this work, we run the +goodness (logits) through the softmax, i.e., p(y = i|x) ∼ exp(Gi)/(� +c exp(Gc)). +2.3 +The Generative Circuit +As mentioned before, the PFF algorithm incorporates the joint adaptation of a top-down directed generative model. +This aspect of the PFF process is motivated by the generative nature of predictive processing (PP) models [37, 10], +particularly those that focus on learning a top-down generative model as in the framework of neural generative +coding [32]. Crucially, we remark that jointly learning (in a biologically-plausible fashion) a generative feedback +system could favorably provide a means of inspecting the content of the representations acquired by an FF-centric +process as well as provide a plausible, alternative means for(internally) synthesizing negative data. +The generative circuit, which is comprised of the set of synaptic parameters Θg = {G0, G1, ..., GL}, attempts to +learn how to predict, at each layer, a local region of neural activity, which, as we will see by design, facilitates +simple error Hebbian updates (much like those calculated in a PP system). Formally, the objective that this +generative circuit will attempt to optimize (for a single data point) is: +L(Θg) = +L +� +ℓ=0 +Lℓ +g(Gℓ) = +L +� +ℓ=0 +Jℓ +� +j=1 +(¯zℓ +j − zℓ +j(t))2 +(5) +where z0 = x (the bottom layer target is clamped to the data point being processed). Each layer of the generative +circuit conducts the following computation: +¯zℓ = gℓ(Gℓ · LN(�zℓ+1)), where, �zℓ+1 = φℓ+1(zℓ+1(t) + ϵℓ+1 +z +) and, eℓ = ¯zℓ − zℓ(t) +(6) +¯zL = gL(GL · LN(zs)), where, zs ← zs − γ ∂LL +g (Gℓ) +∂zs +// Topmost latent layer activity zs +(7) +where ϵℓ +z ∼ N(0, σz) is controlled (additive) activity noise injected into layer ℓ (with a small scale, such as +σz = 0.025). gℓ() is the elementwise activation function applied to any generative layer’s prediction and, in this +work, we set the activation functions for layers ℓ >= 1 to be the linear rectifier while the bottom one is specifically +set to be the clipped identity, i.e., g0(v) = HardClip(v, 0, 1). At each step of the inference process that in Section +4 + +Preprint +y +x +Representation +Circuit +Generative +Circuit +z1 +z2 +z3 +e1 +e2 +e3 +e0 +𝛍1 +𝛍2 +𝛍3 +𝛍0 +zs +Figure 2: The PFF algorithmic process depicted over three-time steps for a three hidden layer network system +coupled to a four-layer generative system (topmost layer is the sampled latent variable zs). Solid arrows represent +synaptic weights, dashed blue arrows depict interpolation between left and right states, and dash-dotted arrows +depict state carry-over/direct copying. The dashed diamond curve represents a feedback pathway, gray circles +represent neural units, and red diamonds represent error neurons. Note that since all elements of the system are +adjusted dynamically, the generative circuit is run/updated each time the representation circuit is run/updated. +2.2, the synaptic weights of the generative model (at each layer) are adjusted via the following Hebbian rule: +∆Gℓ = eℓ · +� +LN(zℓ+1(t)) +�T , and, ∆GL = eℓ · +� +LN(zs) +�T . +(8) +Notice that the topmost layer of the generative circuit (i.e., layer L + 1) is treated a bit differently from the rest, i.e., +the highest latent generative layer zs predicts the topmost neural activity of the representation circuit zL and is +then adjusted by an iterative inference feedback scheme, much akin to that of sparse/predictive coding [31, 37, 32]. +Once trained, synthesizing data from the generative circuit can be done using ancestral sampling: +¯zL+1 = zs ∼ P(zs) +(9) +¯zℓ = gℓ(Gℓ · LN(¯zℓ+1)), ℓ = L, (L − 1), ..., 0 +(10) +where we choose the prior P(zs) to be a Gaussian mixture model (GMM) with 10 components, which, in this +study, was retro-fit to samples of the trained system’s topmost activity values (acquired by running the training +dataset Dtrain through the model), as was done for the top-down directed generative PP models of [32]. Note +that for all circuits in PFF (both the representation and generative circuits), we treat the derivative of the linear +rectifier activation function as a vector of ones with the same shape as the layer activity zℓ (as was done in [16]). +The learning process of the PFF procedure is shown in Algorithm 1 and its neural circuits are depicted in Figure 2. +Relationship to Contrastive Hebbian Learning: +When designing a network much as we do above, one might +notice that the inference process is quite similar to that of a neural system learned under contrastive Hebbian +learning (CHL) [28], although there are several significant differences. Layer activities in a CHL-based system are +updated as follows: +zℓ(t) = zℓ(t − 1) + β +� +− zℓ(t − 1) + φℓ� +Wℓ · zℓ−1(t − 1) + (Wℓ+1)T · zℓ+1(t − 1) +�� +(11) +where we notice that dynamics do not involve any normalization and the values for any layer ℓ are integrated a bit +differently than in Equation 3, i.e., neural values change as a function of a form of a leaky Euler integration, where +the top-down and bottom-up transmissions are combined to produce a perturbation to the layer rather than propose +a new value of the state itself. +Like CHL, FF and PFF require two phases (or modes of computation) where the signals propagated through the +neural system will be used in contrast with one another. Given data sample (x, y), CHL specifically entails running +the neural system first in an un-clamped phase (negative phase), where only the input image x is clamped to the +sensory input/bottom layer, followed by a clamped phase, where both x and its target y are clamped, i.e., y is +clamped to the output layer (positive phase). At the end of each phase (or inference cycle), the layer-wise activities +are recorded and then used in a subtractive/contrastive Hebbian rule to calculate the updates for each matrix of +5 + +Preprint +Algorithm 1 The predictive forward-forward (PFF) credit assignment algorithm. red denotes representation circuit +computation and blue denotes generative circuit computation. +1: Input: sample (yi, xi), data label ci (binary label: 1 = “positive”, 0 = “negative”), PFF parameters Θr and Θg +2: Hyperparameters: State interpolation β, SGD step size η, noise scales σr and σz, stimulus time T +3: // Note that LN(zℓ) = zℓ/(||zℓ||2 + 1e−8) +4: function SIMULATE((yi, xi, ci), Θr, Θg) +5: +// Run forward pass to get initial activities +6: +z0 = xi, +zℓ = φℓ(Wℓ · zℓ−1), for ℓ = 1, 2, ..., L, +zL+1 = yi, �zL+1 = 0 (same as zs) +7: +for t = 1 to T do +8: +// Run representation circuit +9: +for ℓ = 1 to L do +▷ Compute representation activities with layer-wise parameters Θℓ +r = {Wℓ, Vℓ} +10: +Θℓ +r = Θr[ℓ], +Wℓ, Vℓ ← Θℓ +r +▷ Extract relevant parameters +11: +ϵℓ +r ∼ N(0, σr), +zℓ(t) = β +� +φℓ� +Wℓ · LN(zℓ−1(t − 1)) + Vℓ · LN(zℓ+1(t − 1))� ++ ϵℓ +r +� ++ (1 − β)zℓ(t − 1) +12: +Calculate local goodness loss L(Θℓ +r) (Equation 1 using data label ci) +13: +∆Wℓ = +� +2 +∂L(Θℓ +r) +∂ �Jℓ +j +(zℓ +j)2 ⊙ zℓ� +· �LN(zℓ−1)�T , +∆Vℓ = +� +2 +∂L(Θℓ +r) +∂ �Jℓ +j +(zℓ +j)2 ⊙ zℓ� +· �LN(zℓ+1)�T +14: +Wℓ ← Wℓ − η∆Wℓ, +Vℓ ← Vℓ − η∆Vℓ +▷ SGD update with step size η shown (could use Adam [22] instead) +15: +// Run generative circuit +16: +for ℓ = L to 1 do +▷ Compute generative predictions with layer-wise parameters Θℓ +g = {Gℓ} +17: +Θℓ +g = Θg[ℓ], +Gℓ ← Θℓ +r +▷ Extract relevant parameters +18: +ϵℓ ∼ N(0, σz), �zℓ+1 = φℓ+1(zℓ+1 + ϵℓ+1), ¯zℓ = φℓ(Gℓ · LN(�zℓ+1)) +19: +Calculate local generative loss Lℓ +g(Gℓ) = 1 +2 +� +j(¯zℓ +j − zℓ +j(t))2 +20: +eℓ = ¯zℓ − zℓ, +∆Gℓ = eℓ · �LN(zℓ+1(t))�T +▷ Note that eℓ = +∂Lℓ +g(Gℓ) +∂¯zℓ +21: +Gℓ ← Gℓ − η∆Gℓ +22: +zL+1 ← zL+1 − γ +∂LL +g (GL) +∂zL+1 +▷ Update latent variable zs (one step of iterative inference) +23: +Return Θg, Θr +▷ Output newly updated PFF parameters +synapses. Note that the positive phase of CHL depends on first running the negative phase. FF and PFF, in contrast, +essentially amount to running the positive and negative phases in parallel (with each phase conditioned on different +data), resulting in an overall faster pattern processing time (instead of one inference cycle being conditioned on the +statistics of another, the same cycles are now run on either positive or negative data with opposite objectives [16]). +Relationship to Predictive Coding: +The PFF algorithm integrates the local hypothesis generation component +of predictive coding (PC) into the inference process by leveraging the representations acquired within the recurrent +representation network’s iterative processing window. Specifically, each layer of the representation circuit, at each +time step, becomes the prediction target for each layer of the generative circuit. In contrast, PC generative models +must leverage a set of feedback synapses to progressively modify their layerwise neural activities before finally +adjusting synaptic values. Furthermore, PFF iteratively/dynamically modifies the synapses within each processing +time step, whereas; typically, most PC circuits implement a form of expectation-maximization that, as a result, +generally requires longer stimulus processing windows in order to learn effective generative models [32] given +that Euler integration is being simulated (in this work, the PFF generative circuit learns a good-quality generative +model in only 8 steps whereas the models of [32] required at least 50 steps). +Relationship to Local Learning: +It has been strongly argued that the synapses in the brain are likely to be +adjusted according to a local scheme, i.e., only information closest spatially and in time to a target synapse is +involved in computing its change in efficacy. Methods that adhere to this biological constraint/setup are referred to +as local learning procedures [35, 25, 29, 30, 4, 21], offering a potential replacement for backprop for training deep +neural networks, relaxing one or more of its core constraints (see Figure 3 for details related to some of the key +ones). Desirably, it has even been shown that, empirically, updates from a local scheme can result in improved +model generalization [25, 35]. There have been many efforts in designing biologically-plausible local learning +algorithms, such as contrastive Hebbian learning (mentioned above) [28], contrastive divergence for learning +harmoniums (or restricted Boltzmann machines) [17], the wake-sleep algorithm for learning Helmholtz machines +[18], and algorithms such as equilibrium propagation [43]. Other efforts that directly integrate local learning into +the deep learning pipeline include kickback [1] and decoupled neural interfaces [20]. It is worth pointing out that +PFF does bear some similarity to the wake-sleep algorithm, which itself entails learning a generative model jointly +with an inference (recognition) model. However, the wake-sleep algorithm suffers from instability, given that the +recognition network could be damaged by random fantasies produced by the generative network and the generative +network could itself be hampered by the low-quality representation capability of the inference network (motivating +6 + +Preprint +Learning +Algorithms +BP +FA +PC +LRA +NGC +FF +PFF +Fwd locked +Global +Global +Local +Local +Local +None +None +Fwd error + ✅ + ✅ +Fwd target + ✅ + ✅ +Bwd locked +Global +Global +None +None +None +None +None +Bwd error + ✅ + ✅ + ✅ + +Bwd target + ✅ + ✅ + ✅ +Local loss + ✅ + ✅ + ✅ + ✅ + ✅ +Error Synapses +Fixed +Learned +Learned +Global signal + ✅ + ✅ + + ✅ + + + +Local Signal + ✅ + ✅ + ✅ + ✅ + ✅ +Generative +capabilities + ✅ + ✅ +Generative +Weights + ✅ + ✅ +Figure 3: Properties of different learning algorithms, i.e., backprop (BP), feedback alignment (FA), predictive +coding (PC), local representation alignment (LRA), neural generative coding (NGC), the forward-forward algorithm +(FF), and the predictive forward-forward process (PFF). +the design of improvements, such as reweighted wake-sleep [6]). PFF, in contrast, aims to learn the generative +model given the representation circuit, using the locally-adapted distributed neural activities as a guide for the +synthesization process rather than randomly sampling the generative model to generate teaching signals for the +recognition network (potentially distracting its optimization with nonsensical noisy signals). +3 +Experiments +This section describes the simulations/experiments that were run to test the proposed PFF procedure. We leverage +several benchmark image datasets to quantitatively evaluate PFF’s classification ability (in terms of test-set error) +and qualitatively evaluate its generative capability (in terms of visual inspection of sample reconstruction and +pattern synthesization). The PFF process (PFF-RNN) is compared with the FF algorithm (FF) as well as several +baselines, including the K-nearest neighbors algorithm (with K = 4, or 4-KNN), the recurrent network trained +with the original FF algorithm [16], and two backprop-based models, i.e., a feedforward network that uses backprop +to adjust all of its internal synapses (BP-FNN) and the same network but one that only adjusts the top-most +softmax/output layer parameters and fixes the hidden layer synaptic parameters (Rnd-FNN). Both backprop-based +networks are trained to minimize the categorical cross-entropy of each dataset’s provided labels. The partially- +trained model, i.e., the Rnd-FNN, serves as a sort of lower bound on the generalization ability of a neural system, +given that it is possible to obtain respectable classification performance with only random hidden feature detectors +(a neural credit assignment algorithm should not perform worse than this). +Datasets: +In this study, we experiment with two (gray-scale) image collections, i.e., the MNIST and the Kuzushiji- +MNIST databases. The MNIST dataset [24] specifically contains 28 × 28 images containing handwritten digits +across 10 different categories. Kuzushiji-MNIST (KMNIST) is a challenging drop-in replacement for MNIST, +containing 28 × 28 images depicting hand-drawn Japanese Kanji characters [7] (each class corresponding to the +character’s modern hiragana counterpart, with 10 classes in total). +7 + +Preprint +Table 1: Classification generalization results for neural systems trained under different learning algorithms (except +for 4-KNN, which is a non-parametric learning baseline model). Measurements of mean and standard deviation are +made across five experimental trial runs. +MNIST +K-MNIST +Model +Test Error (%) +Test Error (%) +4-KNN +2.860 ± 0.000 +7.900 ± 0.000 +Rnd-FNN +3.070 ± 0.018 +14.070 ± 0.189 +BP-FNN +1.300 ± 0.023 +6.340 ± 0.202 +FF-RNN [16] +1.320 ± 0.100 +6.590 ± 0.420 +PFF-RNN +1.360 ± 0.030 +6.460 ± 0.120 +(a) MNIST recon. +(b) MNIST synthesis. +(c) K-MNIST recon. +(d) K-MNIST synthesis.. +(e) MNIST rep. fields. +(f) MNIST gen. fields. +(g) K-MNIST rep. fields. +(h) K-MNIST gen. fields. +Figure 4: Model reconstruction (Left) and generated (Right) samples for MNIST and K-MNIST. In the bottom row, +the receptive fields of the bottom-most layer of the representation (rep.) circuit (Left) and those of the generative +(gen.) circuit (Right) are displayed. +Simulation Setup: +All models simulated in this study were constrained to use similar architectures in order to +ensure a more fair comparison. All networks for all neural-based learning algorithms contained two hidden layers +of 2000 neurons (which was also done for the FF models in [16]), with initial synaptic weight values selected +according to the random orthogonal initialization scheme [42] (using singular value decomposition). Once any +given learning algorithm calculated adjustment values for the synapses, parameters were adjusted, using the Adam +adaptive learning rate [22] with mini-batches containing 500 samples. Both FF and PFF were set to use a threshold +value of θz = 10.0 and PFF was set to use 20 latent variables (i.e., zs ∈ R20×1), representation noise ϵℓ = 0.05, +and generative noise ϵz = 0.025. +3.1 +Discussion +Observe in Table 1 that the PFF procedure performs well in the context of the models simulated in this study, +reaching a top/good-quality classification error of about 1.36% on MNIST, nearly reaching that of the well-tuned +backprop-based classifier BP-FNN. Notably, the PFF-RNN model outperforms BP-FNN slightly on K-MNIST, +arguably a more difficult benchmark. Both FF and PFF outperform the lower-bound baselines, i.e., 4-KNN and +Rnd-FNN, indicating that they acquire hidden feature detectors that facilitate good discriminative performance. +Qualitatively, in Figure 4 (Top Row), observe that PFF learns a good-quality reconstruction model and generative +model of the image inputs. The reconstructed digits and Kanji characters are excellent and the image samples +for both cases exhibit variety/diversity across the categories (albeit a bit blurry). Note that to sample from the +PFF’s directed generative model, as mentioned earlier in Section 2.3, we retro-fit a GMM to samples of its latent +8 + +Acquired Filters3 +7455131 +0 +2 +72417172 +8 +2/5 +012 +Sb +7375 +231 +02 +994 +038 +9b6b +7 +3 +32.3 +22Z +14 +3 +44 +4 +hhbth +hb +B0000000 +55555660 +5 +6 +5 +7777小Acquired FiltersAcquired FiltersPreprint +variable zs, specifically optimizing a GMM via expectation-maximization with 10 components. In addition, as +shown in the bottom row of Figure 4, the receptive fields (of the synapses of the layer closest to the sensory input +layer) acquired by the fully-connected representation circuits of both the representation and generative circuits +appear to extract useful/interesting structure related to digit or Kanji character strokes, often, as is expected for +fully-connected neural structures, acquiring representative full object templates (if one desired each receptive field +to acquire only single strokes/component features specifically, then an additional prior would need to be imposed, +such as convolution or the locally-connected receptive field structure employed in [2, 16]). +4 +Conclusion +In this work, we proposed the predictive forward-forward (PFF) process for dynamically adjusting the synaptic +efficacies of a recurrent neural system that jointly learns how to classify, reconstruct, and synthesize data samples +without backpropagation of errors. Our model and credit assignment procedure integrates elements of the forward- +forward algorithm, such as its local synaptic adaptation rule based on goodness and contrastive learning, with +aspects of predictive coding, such as its local error Hebbian manner of adjusting generative synaptic weights, +resulting in a promising brain-inspired, forward-only and backprop-free form of machine learning. +References +[1] BALDUZZI, D., VANCHINATHAN, H., AND BUHMANN, J. M. Kickback cuts backprop’s red-tape: Biologi- +cally plausible credit assignment in neural networks. In AAAI (2015), pp. 485–491. +[2] BARTUNOV, S., SANTORO, A., RICHARDS, B., MARRIS, L., HINTON, G. E., AND LILLICRAP, T. +Assessing the scalability of biologically-motivated deep learning algorithms and architectures. In Advances +in Neural Information Processing Systems (2018), pp. 9390–9400. +[3] BASTOS, A. M., USREY, W. M., ADAMS, R. A., MANGUN, G. R., FRIES, P., AND FRISTON, K. J. +Canonical microcircuits for predictive coding. Neuron 76, 4 (2012), 695–711. +[4] BELILOVSKY, E., EICKENBERG, M., AND OYALLON, E. Decoupled greedy learning of cnns. In Interna- +tional Conference on Machine Learning (2020), PMLR, pp. 736–745. +[5] BENGIO, Y., LAUFER, E., ALAIN, G., AND YOSINSKI, J. Deep generative stochastic networks trainable by +backprop. In International Conference on Machine Learning (2014), PMLR, pp. 226–234. +[6] BORNSCHEIN, J., AND BENGIO, Y. Reweighted wake-sleep. arXiv preprint arXiv:1406.2751 (2014). +[7] CLANUWAT, T., BOBER-IRIZAR, M., KITAMOTO, A., LAMB, A., YAMAMOTO, K., AND HA, D. Deep +learning for classical japanese literature, 2018. +[8] CRICK, F. The recent excitement about neural networks. Nature 337, 6203 (1989), 129–132. +[9] FLORIDI, L., AND CHIRIATTI, M. Gpt-3: Its nature, scope, limits, and consequences. Minds and Machines +30, 4 (2020), 681–694. +[10] FRISTON, K. The free-energy principle: a unified brain theory? Nature reviews neuroscience 11, 2 (2010), +127–138. +[11] GOLDMAN, M. S. Memory without feedback in a neural network. Neuron 61, 4 (2009), 621–634. +[12] GROSSBERG, S. Competitive learning: From interactive activation to adaptive resonance. Cognitive science +11, 1 (1987), 23–63. +[13] GUTMANN, M., AND HYVÄRINEN, A. Noise-contrastive estimation: A new estimation principle for +unnormalized statistical models. In Proceedings of the thirteenth international conference on artificial +intelligence and statistics (2010), JMLR Workshop and Conference Proceedings, pp. 297–304. +[14] HEBB, D. O., ET AL. The organization of behavior, 1949. +[15] HELMHOLTZ, H. V. Treatise on physiological optics, 3 vols. +[16] HINTON, +G. +The +forward-forward +algorithm: +Some +preliminary +investigations. +https://www.cs.toronto.edu/ hinton/FFA13.pdf (2022). +[17] HINTON, G. E. Training products of experts by minimizing contrastive divergence. Neural computation 14, +8 (2002), 1771–1800. +[18] HINTON, G. E., DAYAN, P., FREY, B. J., AND NEAL, R. M. The" wake-sleep" algorithm for unsupervised +neural networks. Science 268, 5214 (1995), 1158–1161. +9 + +Preprint +[19] HINTON, G. E., AND MCCLELLAND, J. L. Learning representations by recirculation. In Neural information +processing systems (1988), pp. 358–366. +[20] JADERBERG, M., CZARNECKI, W. M., OSINDERO, S., VINYALS, O., GRAVES, A., AND KAVUKCUOGLU, +K. Decoupled neural interfaces using synthetic gradients. arXiv preprint arXiv:1608.05343 (2016). +[21] KAISER, J., MOSTAFA, H., AND NEFTCI, E. Synaptic plasticity dynamics for deep continuous local learning +(decolle). Frontiers in Neuroscience 14 (2020), 424. +[22] KINGMA, D., AND BA, J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 +(2014). +[23] LAROCHELLE, H., AND BENGIO, Y. Classification using discriminative restricted boltzmann machines. In +Proceedings of the 25th international conference on Machine learning (2008), pp. 536–543. +[24] LECUN, Y. The mnist database of handwritten digits. http://yann. lecun. com/exdb/mnist/ (1998). +[25] LEE, D.-H., ZHANG, S., FISCHER, A., AND BENGIO, Y. Difference target propagation. In Joint European +Conference on Machine Learning and Knowledge Discovery in Databases (2015), Springer, pp. 498–515. +[26] LILLICRAP, T. P., COWNDEN, D., TWEED, D. B., AND AKERMAN, C. J. Random feedback weights +support learning in deep neural networks. arXiv preprint arXiv:1411.0247 (2014). +[27] MARBLESTONE, A. H., WAYNE, G., AND KORDING, K. P. Toward an integration of deep learning and +neuroscience. Frontiers in computational neuroscience (2016), 94. +[28] MOVELLAN, J. R. Contrastive hebbian learning in the continuous hopfield model. In Connectionist models. +Elsevier, 1991, pp. 10–17. +[29] NØKLAND, A. Direct feedback alignment provides learning in deep neural networks. In Advances in Neural +Information Processing Systems (2016), pp. 1037–1045. +[30] NØKLAND, A., AND EIDNES, L. H. Training neural networks with local error signals. In Proceedings of the +36th International Conference on Machine Learning (09–15 Jun 2019), K. Chaudhuri and R. Salakhutdinov, +Eds., vol. 97 of Proceedings of Machine Learning Research, PMLR, pp. 4839–4850. +[31] OLSHAUSEN, B. A., AND FIELD, D. J. Sparse coding with an overcomplete basis set: A strategy employed +by v1? Vision research 37, 23 (1997), 3311–3325. +[32] ORORBIA, A., AND KIFER, D. The neural coding framework for learning generative models. Nature +communications 13, 1 (2022), 1–14. +[33] ORORBIA, A., MALI, A., GILES, C. L., AND KIFER, D. Continual learning of recurrent neural architectures +by locally aligning distributed representations. arXiv preprint arXiv:1810.07411 (2018). +[34] ORORBIA, A., MALI, A., KIFER, D., AND GILES, C. L. Lifelong neural predictive coding: Sparsity yields +less forgetting when learning cumulatively. arXiv preprint arXiv:1905.10696 (2019). +[35] ORORBIA, A. G., AND MALI, A. Biologically motivated algorithms for propagating local target representa- +tions. In Proceedings of the AAAI Conference on Artificial Intelligence (2019), vol. 33, pp. 4651–4658. +[36] ORORBIA, A. G., REITTER, D., WU, J., AND GILES, C. L. Online learning of deep hybrid architectures +for semi-supervised categorization. In Joint European Conference on Machine Learning and Knowledge +Discovery in Databases (2015), Springer, pp. 516–532. +[37] RAO, R. P., AND BALLARD, D. H. Predictive coding in the visual cortex: a functional interpretation of some +extra-classical receptive-field effects. Nature neuroscience 2, 1 (1999). +[38] RUMELHART, D. E., HINTON, G. E., AND WILLIAMS, R. J. Learning representations by back-propagating +errors. nature 323, 6088 (1986), 533–536. +[39] SALAKHUTDINOV, R., AND LAROCHELLE, H. Efficient learning of deep boltzmann machines. In Proceed- +ings of the thirteenth international conference on artificial intelligence and statistics (2010), JMLR Workshop +and Conference Proceedings, pp. 693–700. +[40] SALVATORI, T., SONG, Y., HONG, Y., SHA, L., FRIEDER, S., XU, Z., BOGACZ, R., AND LUKASIEWICZ, +T. Associative memories via predictive coding. Advances in Neural Information Processing Systems 34 +(2021), 3874–3886. +[41] SALVATORI, T., SONG, Y., XU, Z., LUKASIEWICZ, T., BOGACZ, R., LIN, H., FAN, Y., ZHANG, J., +BAI, B., XU, Z., ET AL. Reverse differentiation via predictive coding. In Proceedings of the 36th AAAI +Conference on Artificial Intelligence ‚AAAI 2022 ‚Vancouver, BC, Canada, February 22–March 1 ‚2022 +(2022), vol. 10177, AAAI Press, pp. 507–524. +10 + +Preprint +[42] SAXE, A. M., MCCLELLAND, J. L., AND GANGULI, S. Exact solutions to the nonlinear dynamics of +learning in deep linear neural networks. arXiv preprint arXiv:1312.6120 (2013). +[43] SCELLIER, B., AND BENGIO, Y. Equilibrium propagation: Bridging the gap between energy-based models +and backpropagation. Frontiers in computational neuroscience 11 (2017), 24. +[44] SILVER, D., HUANG, A., MADDISON, C. J., GUEZ, A., SIFRE, L., VAN DEN DRIESSCHE, G., SCHRIT- +TWIESER, J., ANTONOGLOU, I., PANNEERSHELVAM, V., LANCTOT, M., ET AL. Mastering the game of go +with deep neural networks and tree search. nature 529, 7587 (2016), 484–489. +11 + +Preprint +Appendix +Visualized Samples (Expanded) +This appendix section presents the reconstruction and synthesized samples from the PFF models in the main paper +at a larger image scale/size. +(a) PFF reconstructed images. +(b) PFF sampled images. +(c) PFF representation receptive fields. +(d) PFF generative receptive fields. +Figure 5: MNIST model reconstruction (Left) and generated (Right) samples. In the bottom row, the receptive +fields of the bottom-most layer of the representation circuit (Left) and those of the generative circuit (Right). +12 + +3 +7455131 +0 +2 +72417172 +8 +2/5 +012 +Sb +7375 +231 +02 +994 +038 +9b6b +7 +3 +32.3 +22Z +14 +3 +44 +4 +hhbth +hb +B0000000 +55555660 +5 +6 +5 +7777Acquired FiltersAcquired FiltersPreprint +(a) PFF reconstructed images. +(b) PFF sampled images. +(c) PFF representation receptive fields. +(d) PFF generative receptive fields. +Figure 6: In the top row, Kuzushiji-MNIST model reconstruction (Left) and generated (Right) samples. In the +bottom row, the receptive fields of the bottom-most layer of the representation circuit (Left) and those of the +generative circuit (Right). +13 + +Acquired Filters小Acquired Filters \ No newline at end of file diff --git a/3tAzT4oBgHgl3EQfffwg/content/tmp_files/load_file.txt b/3tAzT4oBgHgl3EQfffwg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3905991b4e6c11d42414809184adfa45ee24c24 --- /dev/null +++ b/3tAzT4oBgHgl3EQfffwg/content/tmp_files/load_file.txt @@ -0,0 +1,737 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf,len=736 +page_content='THE PREDICTIVE FORWARD-FORWARD ALGORITHM Alexander Ororbia Rochester Institute of Technology ago@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='rit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='edu Ankur Mali University of South Florida ankurarjunmali@usf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='edu ABSTRACT In this work, we propose a generalization of the forward-forward (FF) algorithm that we call the predictive forward-forward (PFF) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Specifically, we design a dynamic, recurrent neural system that learns a directed generative circuit jointly and simultaneously with a representation circuit, combining elements of predictive coding, an emerging and viable neurobiological process theory of cortical function, with the forward-forward adaptation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Furthermore, PFF efficiently learns to propagate learning signals and updates synapses with forward passes only, eliminating some of the key structural and computational constraints imposed by a backprop- based scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Besides computational advantages, the PFF process could be further useful for understanding the learning mechanisms behind biological neurons that make use of local (and global) signals despite missing feedback connections [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' We run several experiments on image data and demonstrate that the PFF procedure works as well as backprop, offering a promising brain-inspired algorithm for classifying, reconstructing, and synthesizing data patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' As a result, our approach presents further evidence of the promise afforded by backprop-alternative credit assignment algorithms within the context of brain-inspired computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Keywords Brain-inspired computing · Self-supervised learning · Neuromorphic · Forward learning 1 Introduction The algorithm known as backpropagation of errors [38], or “backprop” for short, has long faced criticism concerning its neurobiological plausibility [8, 27, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Despite having powered the tremendous progress and success behind deep learning and its every-growing myriad of promising applications [44, 9], it is improbable that backprop is a good, viable model of learning in the brain, such as in cortical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Notably,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' there are both practical and biophysical issues [12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' and among these issues are the following: there is a lack of evidence that neural activities are explicitly stored to be used later for synaptic adjustment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' error derivatives are backpropagated along a global feedback pathway to generate teaching signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' the error signals move back along the same neural pathways used to forward propagate information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' inference and learning are locked to be largely sequential (instead of massively/easily parallel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Furthermore, in processing temporal data, it is certainly not the case that the neural circuitry of the brain is unfolded backwards through time in order to calculate and adjust synapses [33] (as it is for backprop through time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Recently, there has been a growing interest in the research domain of brain-inspired computing, which focuses on developing algorithms and computational models that attempt to circumvent or resolve the critical issues such as those highlighted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Among the most powerful and promising ones is predictive coding (PC) [15, 37, 10, 3, 40, 32], and among the most recent ones is the forward-forward (FF) algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' These alternatives offer powerful, different means of conducting credit assignments that have shown similar performance as backprop, but to the contrary, are more likely consistent with and similar to real biological neuron learning (see Figure 1 for some representative credit assignment depictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' This paper will propose a novel model and learning/inference process, the predictive forward-forward (PFF) process, that generalizes and combines FF and PC into a robust (stochastic) neural system that simultaneously learns a representation and generative model in a biologically-plausible fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Like the FF algorithm, the PFF procedure offers a promising, potentially helpful model of biological neural circuits, a potential candidate system for low-power analog hardware and neuromorphic circuits, and a potential backprop-alternative worthy of future investigation and study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='01452v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='LG] 4 Jan 2023 Preprint h3 h2 h1 W1 W2 W3 WT 3 WT 2 WT 1 h3 h2 h1 W1 W2 W3 B3 B2 B1 L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Y L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Y h3 h2 h1 W1 W2 W3 WT 3 WT 2 WT 1 L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Y L2 L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' h3 h2 h1 W1 W2 W3 E3 E2 E1 L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Y L2 L1 X X X BP h3 h2 h1 W1 W2 W3 L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Ypos L2 L1 X_pos hNg 3 hNg 2 hNg 1 X_neg FF h3 h2 h1 W1 W2 W3 L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Ypos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='L1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='X_pos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='hNg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='hNg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='hNg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='X_neg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='zG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='zG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='zG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='GM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='L = Global loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='LN = Local loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='WN = Forward weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='BN = Fixed backward weights (random ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='weights) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='EN = Learnable recurrent error weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='WT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='N = Transpose of forward activity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='GN = Generative weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='GM = Generative model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='HN = Hidden States ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='HNG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='N = Hidden States obtained by doing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='2nd forward pass on negative data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='ZN = Error Corrected State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='ZG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='N = Generative Model State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='X = Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Y = Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='X_pos = Positive input data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='X_neg = Negative input data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Yneg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Yneg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='G3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='G2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='G1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='FA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='PC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='LRA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='h3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='h1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='W1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='W2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='W3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='E3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='E1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='L3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Y L2 L1 [X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Y] z3 z2 z1 G3 G2 G1 NGC PFF X Figure 1: Comparison of learning algorithms that relax constraints imposed by backpropagation of errors (BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Algorithms visually depicted include feedback alignment (FA) [26], predictive coding (PC) [37, 41], local repre- sentation alignment (LRA) [35], neural generative coding (NGC) [34, 32], the forward-forward procedure (FF) [16], and predictive forward-forward algorithm (PFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 2 The Predictive Forward-Forward Learning Process The brain-inspired neural process that we will design and study is called the predictive forward-forward (PFF) algorithm, which builds on and generalizes aspects of the FF algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' At a high level, the PFF process consists of two neural structures or circuits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', a representation circuit (parameterized by Θr) that focuses on acquiring distributed representations of data samples and a top-down generative circuit (parameterized by Θg) that focuses on learning to synthesize data given the activity values of the representation circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Thus, the PFF process can be characterized as a complementary system with the aim of jointly learning a classifier and generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' We will first define the notation used throughout this paper, then proceed to describe the inference and learning mechanics of the representation circuit followed by those of the generative circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Notation: We use ⊙ to indicate a Hadamard product and · to denote a matrix/vector multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (v)T is the transpose of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Matrices/vectors are depicted in bold font, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', matrix M or vector v (scalars shown in italic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' zj will refer to extracting jth scalar from vector z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Finally, ||v||2 denotes the Euclidean norm of vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The sensory input has shape x ∈ RJ0×1 (J0 is the number of input features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', pixels), label has shape y ∈ RC×1 (where C is the number of classes), and any neural layer has shape zℓ ∈ RJℓ×1 (Jℓ is the number of neurons in layer ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='1 The Forward-Forward Learning Rule The PPF process, like the FF algorithm when it is applied to a recurrent network, involves adjusting the synaptic efficacies of a group of neurons by measuring their “goodness”, or, in other words, the probability that their activity indicates that an incoming signal comes from the target training data distribution (or the “positive class”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Formally, for any single layer ℓ in an L-layered neural system, we calculate the goodness as the sum of the squared activities for a given neural activity vector zℓ and compare it to particular threshold value θz in one of two ways: p(c = 1)ℓ = 1 1 + exp � − (�Jℓ j (zℓ j)2 − θz) �, or, p(c = 1)ℓ = 1 1 + exp � − (θz − �Jℓ j (zℓ j)2) � (1) where p(c = 1)ℓ indicates the probability that the data comes from the data distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', positive data, where the positive class is labeled c = 1) while the probability that the data does not come from the training data distribution is p(c = 0)ℓ = 1 − p(c = 1)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that p(c)ℓ indicates the probability that is assigned by a layer ℓ of neurons in a system/network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' This means the cost function that any layer is trying to solve/optimize is akin to a binary class logistic regression problem formulated as follows: L(Θℓ) = − 1 N N � i=1 ci log p(ci = 1)ℓ + (1 − ci) log p(ci = 0)ℓ (2) 2 Preprint where the binary label ci (the label for the ith datapoint xi) can be generated correctly and automatically if one formulates a generative process for producing negative data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Data patterns sampled from the training set xj ∼ Dtrain can be labeled as cj = 1 and patterns sampled outside of Dtrain (from the negative data generating process) can be automatically labeled as cj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Crucial to the success of the FF procedure is the design of a useful negative data distribution, much as is the case for noise contrastive estimation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' It is important to notice that the FF learning rule is local in nature – this means that the synapses of any particular layer of neurons can be adjusted independently of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The rule’s form is furthermore different from a classical Hebbian update [14] (which produces a weight change by a product of incoming and outgoing neural activities), given that this synaptic adjustment requires knowledge across a group of neurons (goodness depends on the sum of squares of the activities of a group rather than an individual unit) and integrates contrastive learning into the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Synaptic updates are specifically calculated by taking the gradient of Equation 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ∂L(Θℓ) ∂Θℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In effect, a neural layer optimizes Equation 2 by either maximizing the squared activities of a layer (to be above threshold θz) (left form of the probability presented in Equation 1) or, alternatively, minimizing the squared activities (right form of the probability presented in Equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='2 The Representation Circuit In order to take advantage of the above FF learning rule (and to model contextual prediction via top-down and bottom-up influences), a recurrent network was proposed in [16], where, at each layer, a set of top-down and bottom-up forces are combined to compute the activity of any layer ℓ, much akin to the inference process of a deep Boltzmann machine [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The core parameters of this model are housed in the construct Θr = {W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', WL} (later referred to as the representation parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that no additional classification-specific parameters are included in our model (in contrast to the model of [16]), although incorporating these is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='1 Note that the representation circuit of the the PFF process will take the form of a recurrent network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' To compute any layer’s activity within the representation circuit, top-down and bottom-up messages are combined with an interpolation of the layer’s activity at the previous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Specifically, in PFF, this is done as follows: zℓ(t) = β � φℓ� Wℓ · LN(zℓ−1(t − 1)) + Vℓ · LN(zℓ+1(t − 1)) � + ϵℓ r � + (1 − β)zℓ(t − 1) (3) where ϵℓ r ∼ N(0, σ) is injected, centered Gaussian noise and z0(t − 1) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' As in [16], we set the activation function φℓ() for each layer ℓ to be the linear rectifier, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', φℓ(v) = max(0, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Notice the introduction of an interpolation coefficient β, which allows integration of the state zℓ over time (the new activity state at time t is a convex combination of the newly proposed state and the previous value of the state at t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Furthermore, notice that this interpolation is similar to that of the “regression” factor introduced into the recirculation algorithm [19], a classical local learning algorithm that made use of carefully crafted autoencoders to generate the signals needed for computing synaptic adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' LN(z) is a layer normalization function applied to the activity vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', LN(zℓ) = zℓ/(||zℓ||2 + ϵ) (ϵ is a small numerical stability factor for preventing division by zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that the topmost layer of the representation circuit is clamped to a context vector y (which could be provided by another neural circuit or be set to be a data point’s label/target vector), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', zL+1 = y2, while the bottom layer is clamped to sensory input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', z0(t) = x(t) (where x(t) could be the frame of video or a repeated copy of a static image x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Equation 3 depicts a synchronous update of all layer-wise activities, but, as noted in [16], the recurrent model could alternatively be implemented by cycling between even and odd-number layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', first updating all even-numbered layers given the activities of the odd-numbered layers followed by updating the values of the odd-numbered layers given the new values of the even layers, much like the generative stochastic networks of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' To create the negative data needed to train this system, we disregard the current class indicated by the label y of the positive data xp and create an incorrect “negative label” yn by randomly (uniformly) sampling an incorrect class index, excluding the correct one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='3 A final mini-batch of samples is dynamically created by concatenating positive and negative samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', x =< x, x > and y =< y, yn > (notice that positive image pixels are reused 1If classification-specific parameters are desired, one could include an additional set of synaptic weights Θd = {W, b} that take in as input the top-most (normalized) activity LN(zL) of the recurrent representation circuit in order to make a rough prediction of the label distribution over y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e, p(y = i|LN(zL)) = exp(W · LN(zL) + b)i/ � � c exp(W · LN(zL) + b)c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' This would make the recurrent model of this work much more similar to that of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Softmax parameters W and b would then be adjusted by taking the relevant gradients of the objective Ly(W, b) = − log p(y = i|LN(zL)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 2It is important to scale the label/context vector by a factor of about 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', the topmost layer activity would be zL+1 = y ∗ 5 (Geoffrey Hinton, personal communication, Dec 12, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 3This deviates from how the negative label was made in [16], which chose an incorrect class index in proportion to the probabilities produced by a forward pass of the classification-specific parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' This was not needed for the PFF algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 3 Preprint and paired with the negative labels in order to create the negative samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The PFF process then involves running the combined mini-batch through the neural system and calculating the relevant synaptic updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Equation 3 is typically run several times (8 to 10 times as in this study and [16]), similar to the stimulus processing window that is simulated for predictive coding systems [37, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Each time Equation 3 is run, the (bottom-up and top-down) synapses for layer ℓ are adjusted according to the following local update: ∆Wℓ = � 2 ∂L(Θℓ) ∂ �Jℓ j (zℓ j)2 ⊙ zℓ� � LN(zℓ−1) �T , and, ∆Vℓ = � 2 ∂L(Θℓ) ∂ �Jℓ j (zℓ j)2 ⊙ zℓ� � LN(zℓ+1) �T (4) which can then be applied to the relevant parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', Wℓ and Vℓ, via methods such as stochastic gradient descent (SGD) with momentum or Adam [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In principle, the neural layers of the representation circuit are globally optimizing the objective L(Θr) = �L ℓ=1 Lℓ(Θℓ = Wℓ) (the summation of local goodness functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' On Classifying Sensory Patterns: One might observe that our representation circuit does not include discrimi- natory parameters that classify inputs directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Nevertheless, given that the supervised target y is used as context to mediate the top-most latent representations of the recurrent circuit above, the representation system should (positive data samples) acquire distributed representations that implicitly encode label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' To take advantage of the discriminative information encoded in PFF’s representations, as was also done in the FF algorithm, we may still classify by executing an inference process similar to that of early hybrid Boltzmann machine models [23, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Specifically, to classify an input x, we iterate over all possible (one-hot) values that y could be, starting with the first class index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Specifically, for any chosen y (such as the one-hot encoding of class index i), we run Equation 3 for the representation circuit for T steps and then record the goodness across the layers in the middle three iterations (from T/2 − 1 to T/2 + 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', Gy=i = 1 3 �T/2+1 T/2−1 1 L �L ℓ=1 θz − �Jℓ j (zℓ j 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' This goodness calculation is made for all class indices y = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', C, resulting in {Gy=1, Gy=2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', Gy=C} over which the argmax is applied in order to obtain the index of the class with the highest average goodness value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that, as mentioned in [16], if classification-specific parameters are included in PFF’s representation circuit, then a single feedforward pass could be used to obtain initial class probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Then the above search could instead be simplified by conducting it over only the top M highest probabilities (and thus avoid an expensive search over a massive number of classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' To estimate the label probability distribution under the representation circuit, as we do in this work, we run the goodness (logits) through the softmax, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', p(y = i|x) ∼ exp(Gi)/(� c exp(Gc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='3 The Generative Circuit As mentioned before, the PFF algorithm incorporates the joint adaptation of a top-down directed generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' This aspect of the PFF process is motivated by the generative nature of predictive processing (PP) models [37, 10], particularly those that focus on learning a top-down generative model as in the framework of neural generative coding [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Crucially, we remark that jointly learning (in a biologically-plausible fashion) a generative feedback system could favorably provide a means of inspecting the content of the representations acquired by an FF-centric process as well as provide a plausible, alternative means for(internally) synthesizing negative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The generative circuit, which is comprised of the set of synaptic parameters Θg = {G0, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', GL}, attempts to learn how to predict, at each layer, a local region of neural activity, which, as we will see by design, facilitates simple error Hebbian updates (much like those calculated in a PP system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Formally, the objective that this generative circuit will attempt to optimize (for a single data point) is: L(Θg) = L � ℓ=0 Lℓ g(Gℓ) = L � ℓ=0 Jℓ � j=1 (¯zℓ j − zℓ j(t))2 (5) where z0 = x (the bottom layer target is clamped to the data point being processed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Each layer of the generative circuit conducts the following computation: ¯zℓ = gℓ(Gℓ · LN(�zℓ+1)), where, �zℓ+1 = φℓ+1(zℓ+1(t) + ϵℓ+1 z ) and, eℓ = ¯zℓ − zℓ(t) (6) ¯zL = gL(GL · LN(zs)), where, zs ← zs − γ ∂LL g (Gℓ) ∂zs // Topmost latent layer activity zs (7) where ϵℓ z ∼ N(0, σz) is controlled (additive) activity noise injected into layer ℓ (with a small scale, such as σz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' gℓ() is the elementwise activation function applied to any generative layer’s prediction and, in this work, we set the activation functions for layers ℓ >= 1 to be the linear rectifier while the bottom one is specifically set to be the clipped identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', g0(v) = HardClip(v, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' At each step of the inference process that in Section 4 Preprint y x Representation Circuit Generative Circuit z1 z2 z3 e1 e2 e3 e0 𝛍1 𝛍2 𝛍3 𝛍0 zs Figure 2: The PFF algorithmic process depicted over three-time steps for a three hidden layer network system coupled to a four-layer generative system (topmost layer is the sampled latent variable zs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Solid arrows represent synaptic weights, dashed blue arrows depict interpolation between left and right states, and dash-dotted arrows depict state carry-over/direct copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The dashed diamond curve represents a feedback pathway, gray circles represent neural units, and red diamonds represent error neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that since all elements of the system are adjusted dynamically, the generative circuit is run/updated each time the representation circuit is run/updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='2, the synaptic weights of the generative model (at each layer) are adjusted via the following Hebbian rule: ∆Gℓ = eℓ · � LN(zℓ+1(t)) �T , and, ∆GL = eℓ · � LN(zs) �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (8) Notice that the topmost layer of the generative circuit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', layer L + 1) is treated a bit differently from the rest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', the highest latent generative layer zs predicts the topmost neural activity of the representation circuit zL and is then adjusted by an iterative inference feedback scheme, much akin to that of sparse/predictive coding [31, 37, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Once trained, synthesizing data from the generative circuit can be done using ancestral sampling: ¯zL+1 = zs ∼ P(zs) (9) ¯zℓ = gℓ(Gℓ · LN(¯zℓ+1)), ℓ = L, (L − 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', 0 (10) where we choose the prior P(zs) to be a Gaussian mixture model (GMM) with 10 components, which, in this study, was retro-fit to samples of the trained system’s topmost activity values (acquired by running the training dataset Dtrain through the model), as was done for the top-down directed generative PP models of [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that for all circuits in PFF (both the representation and generative circuits), we treat the derivative of the linear rectifier activation function as a vector of ones with the same shape as the layer activity zℓ (as was done in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The learning process of the PFF procedure is shown in Algorithm 1 and its neural circuits are depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Relationship to Contrastive Hebbian Learning: When designing a network much as we do above, one might notice that the inference process is quite similar to that of a neural system learned under contrastive Hebbian learning (CHL) [28], although there are several significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Layer activities in a CHL-based system are updated as follows: zℓ(t) = zℓ(t − 1) + β � − zℓ(t − 1) + φℓ� Wℓ · zℓ−1(t − 1) + (Wℓ+1)T · zℓ+1(t − 1) �� (11) where we notice that dynamics do not involve any normalization and the values for any layer ℓ are integrated a bit differently than in Equation 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', neural values change as a function of a form of a leaky Euler integration, where the top-down and bottom-up transmissions are combined to produce a perturbation to the layer rather than propose a new value of the state itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Like CHL, FF and PFF require two phases (or modes of computation) where the signals propagated through the neural system will be used in contrast with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Given data sample (x, y), CHL specifically entails running the neural system first in an un-clamped phase (negative phase), where only the input image x is clamped to the sensory input/bottom layer, followed by a clamped phase, where both x and its target y are clamped, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', y is clamped to the output layer (positive phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' At the end of each phase (or inference cycle), the layer-wise activities are recorded and then used in a subtractive/contrastive Hebbian rule to calculate the updates for each matrix of 5 Preprint Algorithm 1 The predictive forward-forward (PFF) credit assignment algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' red denotes representation circuit computation and blue denotes generative circuit computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 1: Input: sample (yi, xi), data label ci (binary label: 1 = “positive”, 0 = “negative”), PFF parameters Θr and Θg 2: Hyperparameters: State interpolation β, SGD step size η, noise scales σr and σz, stimulus time T 3: // Note that LN(zℓ) = zℓ/(||zℓ||2 + 1e−8) 4: function SIMULATE((yi, xi, ci), Θr, Θg) 5: // Run forward pass to get initial activities 6: z0 = xi, zℓ = φℓ(Wℓ · zℓ−1), for ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' zL+1 = yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' �zL+1 = 0 (same as zs) 7: for t = 1 to T do 8: // Run representation circuit 9: for ℓ = 1 to L do ▷ Compute representation activities with layer-wise parameters Θℓ r = {Wℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Vℓ} 10: Θℓ r = Θr[ℓ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Wℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Vℓ ← Θℓ r ▷ Extract relevant parameters 11: ϵℓ r ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' σr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' zℓ(t) = β � φℓ� Wℓ · LN(zℓ−1(t − 1)) + Vℓ · LN(zℓ+1(t − 1))� + ϵℓ r � + (1 − β)zℓ(t − 1) 12: Calculate local goodness loss L(Θℓ r) (Equation 1 using data label ci) 13: ∆Wℓ = � 2 ∂L(Θℓ r) ∂ �Jℓ j (zℓ j)2 ⊙ zℓ� �LN(zℓ−1)�T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' ∆Vℓ = � 2 ∂L(Θℓ r) ∂ �Jℓ j (zℓ j)2 ⊙ zℓ� �LN(zℓ+1)�T 14: Wℓ ← Wℓ − η∆Wℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Vℓ ← Vℓ − η∆Vℓ ▷ SGD update with step size η shown (could use Adam [22] instead) 15: // Run generative circuit 16: for ℓ = L to 1 do ▷ Compute generative predictions with layer-wise parameters Θℓ g = {Gℓ} 17: Θℓ g = Θg[ℓ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Gℓ ← Θℓ r ▷ Extract relevant parameters 18: ϵℓ ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' σz),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' �zℓ+1 = φℓ+1(zℓ+1 + ϵℓ+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' ¯zℓ = φℓ(Gℓ · LN(�zℓ+1)) 19: Calculate local generative loss Lℓ g(Gℓ) = 1 2 � j(¯zℓ j − zℓ j(t))2 20: eℓ = ¯zℓ − zℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' ∆Gℓ = eℓ · �LN(zℓ+1(t))�T ▷ Note that eℓ = ∂Lℓ g(Gℓ) ∂¯zℓ 21: Gℓ ← Gℓ − η∆Gℓ 22: zL+1 ← zL+1 − γ ∂LL g (GL) ∂zL+1 ▷ Update latent variable zs (one step of iterative inference) 23: Return Θg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Θr ▷ Output newly updated PFF parameters synapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that the positive phase of CHL depends on first running the negative phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' FF and PFF, in contrast, essentially amount to running the positive and negative phases in parallel (with each phase conditioned on different data), resulting in an overall faster pattern processing time (instead of one inference cycle being conditioned on the statistics of another, the same cycles are now run on either positive or negative data with opposite objectives [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Relationship to Predictive Coding: The PFF algorithm integrates the local hypothesis generation component of predictive coding (PC) into the inference process by leveraging the representations acquired within the recurrent representation network’s iterative processing window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Specifically, each layer of the representation circuit, at each time step, becomes the prediction target for each layer of the generative circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In contrast, PC generative models must leverage a set of feedback synapses to progressively modify their layerwise neural activities before finally adjusting synaptic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Furthermore, PFF iteratively/dynamically modifies the synapses within each processing time step, whereas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' typically, most PC circuits implement a form of expectation-maximization that, as a result, generally requires longer stimulus processing windows in order to learn effective generative models [32] given that Euler integration is being simulated (in this work, the PFF generative circuit learns a good-quality generative model in only 8 steps whereas the models of [32] required at least 50 steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Relationship to Local Learning: It has been strongly argued that the synapses in the brain are likely to be adjusted according to a local scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', only information closest spatially and in time to a target synapse is involved in computing its change in efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Methods that adhere to this biological constraint/setup are referred to as local learning procedures [35, 25, 29, 30, 4, 21], offering a potential replacement for backprop for training deep neural networks, relaxing one or more of its core constraints (see Figure 3 for details related to some of the key ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Desirably, it has even been shown that, empirically, updates from a local scheme can result in improved model generalization [25, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' There have been many efforts in designing biologically-plausible local learning algorithms, such as contrastive Hebbian learning (mentioned above) [28], contrastive divergence for learning harmoniums (or restricted Boltzmann machines) [17], the wake-sleep algorithm for learning Helmholtz machines [18], and algorithms such as equilibrium propagation [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Other efforts that directly integrate local learning into the deep learning pipeline include kickback [1] and decoupled neural interfaces [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' It is worth pointing out that PFF does bear some similarity to the wake-sleep algorithm, which itself entails learning a generative model jointly with an inference (recognition) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' the wake-sleep algorithm suffers from instability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' given that the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='recognition network could be damaged by random fantasies produced by the generative network and the generative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='network could itself be hampered by the low-quality representation capability of the inference network (motivating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Preprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Algorithms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='BP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='FA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='PC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='LRA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='NGC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='FF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='PFF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Fwd locked ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Global ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Global ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Fwd error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Fwd target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Bwd locked ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Global ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Global ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='None ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Bwd error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Bwd target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Local loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Error Synapses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Fixed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Learned ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Learned ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Global signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Local Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Generative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='capabilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Generative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='✅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='Figure 3: Properties of different learning algorithms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', backprop (BP), feedback alignment (FA), predictive coding (PC), local representation alignment (LRA), neural generative coding (NGC), the forward-forward algorithm (FF), and the predictive forward-forward process (PFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' the design of improvements, such as reweighted wake-sleep [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' PFF, in contrast, aims to learn the generative model given the representation circuit, using the locally-adapted distributed neural activities as a guide for the synthesization process rather than randomly sampling the generative model to generate teaching signals for the recognition network (potentially distracting its optimization with nonsensical noisy signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 3 Experiments This section describes the simulations/experiments that were run to test the proposed PFF procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' We leverage several benchmark image datasets to quantitatively evaluate PFF’s classification ability (in terms of test-set error) and qualitatively evaluate its generative capability (in terms of visual inspection of sample reconstruction and pattern synthesization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The PFF process (PFF-RNN) is compared with the FF algorithm (FF) as well as several baselines, including the K-nearest neighbors algorithm (with K = 4, or 4-KNN), the recurrent network trained with the original FF algorithm [16], and two backprop-based models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', a feedforward network that uses backprop to adjust all of its internal synapses (BP-FNN) and the same network but one that only adjusts the top-most softmax/output layer parameters and fixes the hidden layer synaptic parameters (Rnd-FNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Both backprop-based networks are trained to minimize the categorical cross-entropy of each dataset’s provided labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The partially- trained model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', the Rnd-FNN, serves as a sort of lower bound on the generalization ability of a neural system, given that it is possible to obtain respectable classification performance with only random hidden feature detectors (a neural credit assignment algorithm should not perform worse than this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Datasets: In this study, we experiment with two (gray-scale) image collections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', the MNIST and the Kuzushiji- MNIST databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The MNIST dataset [24] specifically contains 28 × 28 images containing handwritten digits across 10 different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Kuzushiji-MNIST (KMNIST) is a challenging drop-in replacement for MNIST, containing 28 × 28 images depicting hand-drawn Japanese Kanji characters [7] (each class corresponding to the character’s modern hiragana counterpart, with 10 classes in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 7 Preprint Table 1: Classification generalization results for neural systems trained under different learning algorithms (except for 4-KNN, which is a non-parametric learning baseline model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Measurements of mean and standard deviation are made across five experimental trial runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' MNIST K-MNIST Model Test Error (%) Test Error (%) 4-KNN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='860 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='000 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='900 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='000 Rnd-FNN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='070 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='018 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='070 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='189 BP-FNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='300 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='023 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='340 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='202 FF-RNN [16] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='320 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='100 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='590 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='420 PFF-RNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='360 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='030 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='460 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='120 (a) MNIST recon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (b) MNIST synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (c) K-MNIST recon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (d) K-MNIST synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='. (e) MNIST rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (f) MNIST gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (g) K-MNIST rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (h) K-MNIST gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Figure 4: Model reconstruction (Left) and generated (Right) samples for MNIST and K-MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In the bottom row, the receptive fields of the bottom-most layer of the representation (rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=') circuit (Left) and those of the generative (gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=') circuit (Right) are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Simulation Setup: All models simulated in this study were constrained to use similar architectures in order to ensure a more fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' All networks for all neural-based learning algorithms contained two hidden layers of 2000 neurons (which was also done for the FF models in [16]), with initial synaptic weight values selected according to the random orthogonal initialization scheme [42] (using singular value decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Once any given learning algorithm calculated adjustment values for the synapses, parameters were adjusted, using the Adam adaptive learning rate [22] with mini-batches containing 500 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Both FF and PFF were set to use a threshold value of θz = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='0 and PFF was set to use 20 latent variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', zs ∈ R20×1), representation noise ϵℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='05, and generative noise ϵz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='1 Discussion Observe in Table 1 that the PFF procedure performs well in the context of the models simulated in this study, reaching a top/good-quality classification error of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='36% on MNIST, nearly reaching that of the well-tuned backprop-based classifier BP-FNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Notably, the PFF-RNN model outperforms BP-FNN slightly on K-MNIST, arguably a more difficult benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Both FF and PFF outperform the lower-bound baselines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', 4-KNN and Rnd-FNN, indicating that they acquire hidden feature detectors that facilitate good discriminative performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Qualitatively, in Figure 4 (Top Row), observe that PFF learns a good-quality reconstruction model and generative model of the image inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The reconstructed digits and Kanji characters are excellent and the image samples for both cases exhibit variety/diversity across the categories (albeit a bit blurry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Note that to sample from the PFF’s directed generative model, as mentioned earlier in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='3, we retro-fit a GMM to samples of its latent 8 Acquired Filters3 7455131 0 2 72417172 8 2/5 012 Sb 7375 231 02 994 038 9b6b 7 3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='3 22Z 14 3 44 4 hhbth hb B0000000 55555660 5 6 5 7777小Acquired FiltersAcquired FiltersPreprint variable zs, specifically optimizing a GMM via expectation-maximization with 10 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' as shown in the bottom row of Figure 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' the receptive fields (of the synapses of the layer closest to the sensory input layer) acquired by the fully-connected representation circuits of both the representation and generative circuits appear to extract useful/interesting structure related to digit or Kanji character strokes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' often,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' as is expected for fully-connected neural structures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' acquiring representative full object templates (if one desired each receptive field to acquire only single strokes/component features specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' then an additional prior would need to be imposed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' such as convolution or the locally-connected receptive field structure employed in [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 4 Conclusion In this work, we proposed the predictive forward-forward (PFF) process for dynamically adjusting the synaptic efficacies of a recurrent neural system that jointly learns how to classify, reconstruct, and synthesize data samples without backpropagation of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Our model and credit assignment procedure integrates elements of the forward- forward algorithm, such as its local synaptic adaptation rule based on goodness and contrastive learning, with aspects of predictive coding, such as its local error Hebbian manner of adjusting generative synaptic weights, resulting in a promising brain-inspired, forward-only and backprop-free form of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' References [1] BALDUZZI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', VANCHINATHAN, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND BUHMANN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Kickback cuts backprop’s red-tape: Biologi- cally plausible credit assignment in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In AAAI (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 485–491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [2] BARTUNOV, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', SANTORO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', RICHARDS, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', MARRIS, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', HINTON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND LILLICRAP, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Assessing the scalability of biologically-motivated deep learning algorithms and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 9390–9400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [3] BASTOS, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', USREY, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ADAMS, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', MANGUN, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', FRIES, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND FRISTON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Canonical microcircuits for predictive coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Neuron 76, 4 (2012), 695–711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [4] BELILOVSKY, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', EICKENBERG, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND OYALLON, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Decoupled greedy learning of cnns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Interna- tional Conference on Machine Learning (2020), PMLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 736–745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [5] BENGIO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', LAUFER, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ALAIN, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND YOSINSKI, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Deep generative stochastic networks trainable by backprop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In International Conference on Machine Learning (2014), PMLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 226–234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [6] BORNSCHEIN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND BENGIO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Reweighted wake-sleep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv preprint arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='2751 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [7] CLANUWAT, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', BOBER-IRIZAR, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', KITAMOTO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', LAMB, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', YAMAMOTO, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND HA, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Deep learning for classical japanese literature, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [8] CRICK, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The recent excitement about neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Nature 337, 6203 (1989), 129–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [9] FLORIDI, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND CHIRIATTI, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Gpt-3: Its nature, scope, limits, and consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Minds and Machines 30, 4 (2020), 681–694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [10] FRISTON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The free-energy principle: a unified brain theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Nature reviews neuroscience 11, 2 (2010), 127–138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [11] GOLDMAN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Memory without feedback in a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Neuron 61, 4 (2009), 621–634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [12] GROSSBERG, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Competitive learning: From interactive activation to adaptive resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Cognitive science 11, 1 (1987), 23–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [13] GUTMANN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND HYVÄRINEN, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Noise-contrastive estimation: A new estimation principle for unnormalized statistical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Proceedings of the thirteenth international conference on artificial intelligence and statistics (2010), JMLR Workshop and Conference Proceedings, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 297–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [14] HEBB, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The organization of behavior, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [15] HELMHOLTZ, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Treatise on physiological optics, 3 vols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [16] HINTON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The forward-forward algorithm: Some preliminary investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='edu/ hinton/FFA13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='pdf (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [17] HINTON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Training products of experts by minimizing contrastive divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Neural computation 14, 8 (2002), 1771–1800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [18] HINTON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', DAYAN, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', FREY, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND NEAL, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The" wake-sleep" algorithm for unsupervised neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Science 268, 5214 (1995), 1158–1161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 9 Preprint [19] HINTON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND MCCLELLAND, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Learning representations by recirculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Neural information processing systems (1988), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 358–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [20] JADERBERG, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', CZARNECKI, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', OSINDERO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', VINYALS, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', GRAVES, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND KAVUKCUOGLU, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Decoupled neural interfaces using synthetic gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv preprint arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='05343 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [21] KAISER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', MOSTAFA, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND NEFTCI, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Synaptic plasticity dynamics for deep continuous local learning (decolle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Frontiers in Neuroscience 14 (2020), 424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [22] KINGMA, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND BA, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='6980 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [23] LAROCHELLE, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND BENGIO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Classification using discriminative restricted boltzmann machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Proceedings of the 25th international conference on Machine learning (2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 536–543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [24] LECUN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The mnist database of handwritten digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' http://yann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' lecun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' com/exdb/mnist/ (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [25] LEE, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ZHANG, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', FISCHER, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND BENGIO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Difference target propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Joint European Conference on Machine Learning and Knowledge Discovery in Databases (2015), Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 498–515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [26] LILLICRAP, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', COWNDEN, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', TWEED, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND AKERMAN, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Random feedback weights support learning in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv preprint arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='0247 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [27] MARBLESTONE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', WAYNE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND KORDING, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Toward an integration of deep learning and neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Frontiers in computational neuroscience (2016), 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [28] MOVELLAN, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Contrastive hebbian learning in the continuous hopfield model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Connectionist models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Elsevier, 1991, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 10–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [29] NØKLAND, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Direct feedback alignment provides learning in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 1037–1045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [30] NØKLAND, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND EIDNES, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Training neural networks with local error signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning (09–15 Jun 2019), K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Chaudhuri and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Salakhutdinov, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 97 of Proceedings of Machine Learning Research, PMLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 4839–4850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [31] OLSHAUSEN, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND FIELD, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Sparse coding with an overcomplete basis set: A strategy employed by v1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Vision research 37, 23 (1997), 3311–3325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [32] ORORBIA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND KIFER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' The neural coding framework for learning generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Nature communications 13, 1 (2022), 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [33] ORORBIA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', MALI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', GILES, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND KIFER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Continual learning of recurrent neural architectures by locally aligning distributed representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='07411 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [34] ORORBIA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', MALI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', KIFER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND GILES, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Lifelong neural predictive coding: Sparsity yields less forgetting when learning cumulatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='10696 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [35] ORORBIA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND MALI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Biologically motivated algorithms for propagating local target representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence (2019), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 4651–4658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [36] ORORBIA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', REITTER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', WU, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND GILES, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Online learning of deep hybrid architectures for semi-supervised categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Joint European Conference on Machine Learning and Knowledge Discovery in Databases (2015), Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 516–532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [37] RAO, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND BALLARD, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Nature neuroscience 2, 1 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [38] RUMELHART, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', HINTON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND WILLIAMS, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Learning representations by back-propagating errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' nature 323, 6088 (1986), 533–536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [39] SALAKHUTDINOV, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND LAROCHELLE, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Efficient learning of deep boltzmann machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Proceed- ings of the thirteenth international conference on artificial intelligence and statistics (2010), JMLR Workshop and Conference Proceedings, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 693–700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [40] SALVATORI, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', SONG, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', HONG, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', SHA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', FRIEDER, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', XU, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', BOGACZ, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND LUKASIEWICZ, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Associative memories via predictive coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 34 (2021), 3874–3886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [41] SALVATORI, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', SONG, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', XU, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', LUKASIEWICZ, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', BOGACZ, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', LIN, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', FAN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ZHANG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', BAI, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', XU, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Reverse differentiation via predictive coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In Proceedings of the 36th AAAI Conference on Artificial Intelligence ‚AAAI 2022 ‚Vancouver, BC, Canada, February 22–March 1 ‚2022 (2022), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 10177, AAAI Press, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 507–524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 10 Preprint [42] SAXE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', MCCLELLAND, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND GANGULI, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Exact solutions to the nonlinear dynamics of learning in deep linear neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='6120 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [43] SCELLIER, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', AND BENGIO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Equilibrium propagation: Bridging the gap between energy-based models and backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Frontiers in computational neuroscience 11 (2017), 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' [44] SILVER, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', HUANG, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', MADDISON, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', GUEZ, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', SIFRE, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', VAN DEN DRIESSCHE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', SCHRIT- TWIESER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ANTONOGLOU, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', PANNEERSHELVAM, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', LANCTOT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=', ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Mastering the game of go with deep neural networks and tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' nature 529, 7587 (2016), 484–489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 11 Preprint Appendix Visualized Samples (Expanded) This appendix section presents the reconstruction and synthesized samples from the PFF models in the main paper at a larger image scale/size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (a) PFF reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (b) PFF sampled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (c) PFF representation receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (d) PFF generative receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Figure 5: MNIST model reconstruction (Left) and generated (Right) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In the bottom row, the receptive fields of the bottom-most layer of the representation circuit (Left) and those of the generative circuit (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 12 3 7455131 0 2 72417172 8 2/5 012 Sb 7375 231 02 994 038 9b6b 7 3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content='3 22Z 14 3 44 4 hhbth hb B0000000 55555660 5 6 5 7777Acquired FiltersAcquired FiltersPreprint (a) PFF reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (b) PFF sampled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (c) PFF representation receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' (d) PFF generative receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' Figure 6: In the top row, Kuzushiji-MNIST model reconstruction (Left) and generated (Right) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' In the bottom row, the receptive fields of the bottom-most layer of the representation circuit (Left) and those of the generative circuit (Right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} +page_content=' 13 Acquired Filters小Acquired Filters' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfffwg/content/2301.01452v1.pdf'} diff --git a/6tE2T4oBgHgl3EQf7Qi3/content/tmp_files/2301.04208v1.pdf.txt b/6tE2T4oBgHgl3EQf7Qi3/content/tmp_files/2301.04208v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2279e5a8084cad007afb6ce5f5d711560bbb2ba --- /dev/null +++ b/6tE2T4oBgHgl3EQf7Qi3/content/tmp_files/2301.04208v1.pdf.txt @@ -0,0 +1,1277 @@ +Sequential Structure and Control Co-Design of +Lightweight Precision Stages with Active Control of +Flexible Modes +Jingjie Wu +Walker Department of Mechanical Engineering +The University of Texas at Austin +Austin, TX, 78712 +wujingjie@utexas.edu +Lei Zhou +Walker Department of Mechanical Engineering +The University of Texas at Austin +Austin, TX, 78712 +lzhou@utexas.edu +Abstract—Precision motion stages are playing a prominent role +in various manufacturing equipment. The drastically increasing +demand for higher throughput in integrated circuit (IC) man- +ufacturing and inspection calls for the next-generation preci- +sion stages that have light weight and high control bandwidth +simultaneously. In today’s design techniques, the stage’s first +flexible mode is limiting its achievable control bandwidth, which +enforces a trade-off between the stage’s acceleration and closed- +loop stiffness and thus limits the system’s overall performance. +To overcome this challenge, this paper proposes a new hardware +design and control framework for lightweight precision motion +stages with the stage’s low-frequency flexible modes actively +controlled. Our method proposes to minimize the resonance +frequency of the controlled mode to reduce the stage’s weight, and +to maximize that of the uncontrolled mode to enable high control +bandwidth. In addition, the proposed framework determines +the placement of the actuators and sensors to maximize the +controllability/observability of the stage’s controlled flexible mode +while minimizing that of the uncontrolled mode, which effectively +simplifies the controller designs. Two case studies are used to +evaluate the effectiveness of the proposed framework. Simulation +results show that the stage designed using the proposed method +has a weight reduction of more than 55% compared to a +baseline stage design. Improvement in control bandwidth was +also achieved. These results demonstrate the effectiveness of the +proposed method in achieving lightweight precision positioning +stages with high acceleration, bandwidth, and precision. +Index Terms—Precision positioning systems, control co-design, +structure control +I. INTRODUCTION +High-precision positioning stages are playing a critical role +in a wide range of manufacturing and inspection tools such as +photolithography scanners [2] and MEMS inspection systems +[1]. The drastically growing demand for higher throughput in +semiconductor manufacturing necessitates the next-generation +precision motion stages with higher acceleration capability +while maintaining excellent positioning accuracy and high +control bandwidth [9]. Creating new lightweight precision +positioning stages is critical to achieve this goal. However, as +the stage’s weight reduces, its structural resonance frequencies +will decrease to near or even within the control bandwidth +(Fig. 1), which limits the stage’s control bandwidth and +Control +Bandwidth +Frequency +Loop gain +Flexible dynamics +- Well above bandwidth +- no stability challenges +Control +Bandwidth +Frequency +Loop gain +Flexible dynamics +- Close or within bandwidth +- Cause stability challenges +How to design? +Conventional Stages +Lightweight Stages +Fig. 1. Design challenge of lightweight precision positioning stages. +Stage Acceleration +✓ High Acceleration, +✓ Low power consumption +X Low control bandwidth +X Deformation during acceleration +Closed-loop Stiffness/Control Bandwidth +Conventional Rigid Stage +✓ Rigid structure, high precision +✓ High control bandwidth +X Low acceleration +X High power consumption +Today’s +stages +Feasible Range +New Feasible Range +Lightweight stage +w/ Flexible Mode +Control +Proposed Approach: +Lightweight Stage w/o +Flexible Mode Control +Acceleration +Fig. 2. Illustration of acceleration and bandwidth trade-off in today’s precision +positioning systems and motivation for the proposed lightweight stage with +flexible mode control. +positioning accuracy, and can even cause stability challenges +[8]. +In the past decade, a number of research and engineering +efforts have studied the design and control for lightweight +precision positioning stages. For example, Laro et al. [7] +presented an over-actuation approach to place actuators/sensors +at the stage’s nodal locations to prevent the flexible dynamics +from being excited by the feedback loops. Oomen et al. [9] +proposed a system identification and robust control framework +for wafer stages, which provides a systematic approach to +create controller designs for stages exhibiting low-frequency +flexible dynamics. Although effective, these studies mostly +investigate the motion control for flexible stages, and the +arXiv:2301.04208v1 [eess.SY] 10 Jan 2023 + +Control +Bandwidth +Frequency +Loop gain +Flexible dynamics +- Well above bandwidth +- no stability challenges +Control +Bandwidth +Frequency +Loop gain +Flexible dynamics +- Close or within bandwidth +- Cause stability challenges +How to design? +Conventional Stages +Lightweight Stages +Control +Bandwidth +Frequency +Loop gain +Uncontrolled +flexible dynamics +- well above bandwidth +- no stability challenges +Actively controlled +flexible dynamics +- highly compliant +- well within bandwidth +baseline +Design challenge +proposed +Fig. 3. Illustration of the proposed lightweight stage design with active control +for flexible modes. +synergy between the structure design and controller design +is not fully exploited. In recent years, the hardware-control +co-design, or control co-design (CCD) [6], has been studied for +the lightweight precision positioning stages, aiming at enabling +a synergistic structure-control design method for precision +positioning stages. For example, Van der Veen et al. [13] +studied the integrated topology and controller optimization +for a simple 2D motion stage structure. Delissen et al. [3] +presented a topology-optimized wafer stage fabricated via +metal additive manufacturing. In a recent study, Wu et al. +[15] presented a nested CCD formulation of for lightweight +precision stages with controller design constraints explicitly +considered. Despite these advances, we make a key observation +that in these prior lightweight precision stages designs, the +first resonance frequency of the stage structure sets an upper +limit for the achievable control bandwidth. This fact enforces +a fundamental trade-off between the stage’s bandwidth and +acceleration as illustrated in Fig. 2. Fundamental advances in +the stage’s mechatronic design must be made to break this trade- +off and thus enable stages with improved overall performance. +Aiming at overcoming the aforementioned trade-off and thus +creating new lightweight stages that can simultaneously have +high acceleration and high closed-loop stiffness, this paper +presents a sequential structure and control design framework +where the low-frequency flexible modes of the stage are +under active control. This approach has been explored in +van Herpen et al. [14] where additional actuators and sensors +are introduced for a lightweight stage to enhance the control +bandwidth. However, in [14], the control for flexible dynamics +is not considered in the stage’s structural design phase, which +limits the achievable performance. In our work, to facilitate +the controller design, we propose to minimize the resonance +frequency of the stage’s mode being controlled and to maximize +the resonance frequency of the uncontrolled mode. The target +control bandwidth of the stage is in between the resonance fre- +quencies, as shown in Fig. 3. We envision that this formulation +will remove material in the stage’s structure to allow compliance +in the actively-controlled modes thereby breaking the trade-off +in lightweight stages, as shown in Fig. 2. With the stage’s +structure designed, we further propose to use an optimization +method to compute the best actuator/sensor placement. Our +hypothesis is that maximizing the controllability/observability +of the actively-controlled flexible modes while minimizing that +of the uncontrolled modes will deliver the best positioning +performance with reasonable control signal magnitude. Two +case studies are simulated to evaluate the effectiveness of the +proposed approach, where a stage weight reduction of > 55% +is demonstrated compared to a baseline case. These results +demonstrate the potential of the proposed lightweight precision +stage design framework. +The rest of the paper is organized as follows. Section II de- +scribes the problem statement. Section III presents the proposed +design framework for the lightweight precision positioning +stage. Section IV shows the simulation evaluations with two +case studies. Conclusion and future work are summarized in +Section V. +II. PROBLEM STATEMENT +The dynamics of a precision positioning stage considering +its flexible structural behaviors can be described by +M(θp)¨x + D(θp) ˙x + K(θp)x = B(θp, θa)u, +y = C(θp, θs)x, +(1) +where x is a vector of state variables of both rigid-body displace- +ments and flexible displacements in the modal coordinate, M, +D, K are the mass, damping and stiffness matrices, respectively, +u is the vector of control signals, y is a vector of measurement +signals, B is the input matrix which maps the control input u to +corresponding states, C is the output matrix which maps state +variables to measurements, θp is a vector of stage’s geometric +design parameters, and θa, θs are the vectors of actuator and +sensor locations, respectively. +The design optimization problem for a lightweight precision +stage described by (1) aims at finding a set of hardware design +parameters θp, θa, and θs and a controller design that can +minimize the stage’s weight while maximizing the control +bandwidth, meanwhile satisfying certain robustness criteria. +III. SEQUENTIAL HARDWARE AND CONTROL +OPTIMIZATION FRAMEWORK +This section presents a sequential framework of designing the +hardware and controller for lightweight stages with their low- +frequency flexible modes actively controlled. In the first step, +an optimization problem that determines the stage’s geometric +parameter is formulated to facilitate the active control for the +stage’s low-frequency flexible modes. In the second step, an +optimization is performed to determine the location of actuators +and sensors. Finally, feedback controllers are synthesized for +the designed stage to control the stage’s motion as well as the +low-frequency flexible modes. The three steps are introduced +in detail in the following sections. +A. Stage Geometry Design Optimization +In a lightweight precision stage with active control for +low-frequency flexible modes, the stage’s geometry design + +ALoop-gain +ANoop-gain +How to design? +Flexible +vnamic +Frequency +Frequency +Control +Flexible +Control +bandwidth +dynamics +bandwidthoptimization is formulated as +min +θp +Jm(θp), +s.t. +ωi ≤ ωlow, +i = 1, ..., n +ωj ≥ ωhigh, +j = n + 1, ..., m +θp,min ≤ θp ≤ θp,max. +(2) +Here, the objective function Jm represents the stage’s weight, +θp is a vector for the stage’s geometric parameters, ωi is the i-th +modal frequency with its corresponding vibration mode actively +controlled, and ωj is the j-th resonance frequency where +the corresponding mode shape is not controlled. ωlow is the +upper bound for the actively-controlled resonance frequencies, +and ωhigh is the lower bound for the uncontrolled resonance +frequencies. θp,min and θp,max are the lower and upper bounds +for the stage’s geometric parameter, respectively. +With the stage structure design optimization formulation (2), +the stage’s flexible modes under active control are having +resonance frequencies below ωlow, and that of the uncontrolled +modes are beyond ωhigh. Such an optimization process can +enforce material removal in the stage’s structure to allow for +compliance in the actively-controlled flexible modes, and add +material to stiffen the uncontrolled modes. +Remark 3.1: The selection of ωlow and ωhigh are highly +important and determine the system’s dynamic behavior. The +system’s target control bandwidth must be between ωlow and +ωhigh, and ωhigh sets the new upper bound for the achievable +control bandwidth for the lightweight precision stage with +actively controlled flexible modes, as illustrated in Fig. 2. +To facilitate controller design while maintaining design +feasibility, the values of ωlow and ωhigh need to be se- +lected according to the target control bandwidth, for example +ωlow ∼ +1 +2 × ωbw and ωhigh ∼ 5 × ωbw, where ωbw is the +target bandwidth. This method, although robust, may lead to +a relatively conservative stage design. To fully evaluate the +feasible design range in Fig. 2, the value of ωhigh needs to be +swept while considering the actuator/sensor positioning, which +will be introduced in Section IV-B. +B. Actuator and Sensor Placement +The actuator and sensor placement optimization problem +for the proposed lightweight stage with active flexible mode +controlled can be formulated as +max +θa∈Da Ja(θa) = +� +i=1,...,n +Wpi(θa) − γ +� +i=n+1,...,m +Wpi(θa), +(3) +max +θs∈Ds Jo(θs) = +� +i=1,...,n +Woi(θs) − γ +� +i=n+1,...,m +Woi(θs), +(4) +where θa and θs are vectors of actuator and sensor placement +parameters, respectively; Da and Ds are the design domains +for actuator/sensor locations, and γ is a positive user-defined +weighting constant. Wpi and Woi are the controllability and ++ ++ +𝐶𝜃𝑥 +𝐶𝜃𝑦 +u𝜃𝑦 +u𝜃𝑥 +𝐶𝑧 +𝐶𝑚1 +Control Diagram +Lightweight +Stage +Dynamics +Actuation +Recoupling +Transformation +𝑢𝑧 +𝑢𝑚1 +Measurement +Decoupling +Transformation +𝑢1 +𝑢2 +𝑢3 +𝑢4 +𝑦1 +𝑦2 +𝑦3 +𝑦4 +𝑟𝑧 +𝑟𝜃𝑥 +𝑟𝜃𝑦 +𝑟𝑞1 +𝑥𝑧 +𝜃𝑥 +𝜃𝑦 +𝑞1 ++ +− + +− +− +− +Fig. 4. Control block diagram for the lightweight precision positioning stage +with model decoupling. +TABLE I +CONTROLLER PARAMETERS [2]. +Parameter +Description +Typical +Value +ωbw +Desired bandwidth [rad/s] +– +α +PID frequency ratio +0.3 +Kp +Proportional gain +– +ωi +Integrator frequency +ωbw/α2 +ωd +Differentiator frequency +ωbw/α +ωlp +Lowpass filter frequency +αωbw +zlp +Lowpass filter damping ratio +0.7 +observability grammians of i-th flexible mode, respectively, +which can be calculated as +Wpi = ∥φi(θa)⊤Ba(θa)∥2 +2 +4ζiωi +, Woi = ∥Cs(θs)⊤φi(θs)∥2 +2 +4ζiωi +, (5) +where φi is the mass-normalized mode shape of i-th flexible +mode, Ba and Cs are the force and measurement assembling +matrices, ζi is the modal damping ratio, and ωi is the i-th +resonance natural frequency. The controllability/observability +grammians Wpi and Woi quantitatively evaluate the control- +lability/observability of the corresponding flexible mode in +the control system, which will reflect on the peak resonance +magnitude in the system’s frequency response. +With actuator/sensor placement optimization formulation +in (3) and (4), our goal is to maximize the controllabil- +ity/observability for the actively-controlled modes to reduce +the required controller gain, and to minimize those of the +uncontrolled modes to reduce their coupling with the control +systems. The value of γ provides a trade-off between the two +design goals: a low value in γ emphasizes reducing the needed +controller gain for actively-controlled modes, and a high value +in γ emphasizes reducing the cross-talk between uncontrolled +modes and controlled modes. +C. Feedback Control Design +With the stage’s structure and actuator/sensor locations +determined, the plant dynamics of the stage can be found. +Feedback controllers can be designed for each degree of +freedom (DOF) to enable precision positioning and disturbance +rejection. Figure 4 shows a block diagram for the control loop +for a lightweight stage with three rigid-body DOFs and one +flexible mode under active control. Here, the lightweight stage +plant dynamics P : u → y can be obtained from solving (2), +(3), and (4). The sensor measurements y are transformed to +individual DOFs via a measurement decoupling transformation. +Four single-input, single-output (SISO) feedback controllers +can then be designed for four decoupled channels assuming the + +Rib width: +4 mm +Rib distance: +30 mm +𝑥 +𝑦 +𝑧 +Rib height: +25 mm +Base height: +3 mm +Rib width 2: +𝜃𝑝2 +Rib width 1: +𝜃𝑝1 +Rib distance: +𝜃𝑝3 +Rib Height: 𝜃𝑝5 +Base Height: 𝜃𝑝4 +𝑥 +𝑦 +𝑧 +𝑎1 +𝑎2 +𝑎3 +𝑎4 +𝑥𝑎 +𝑦𝑎 +𝑥𝑠 +𝑦𝑠 +𝑠1 +𝑠2 +𝑠3 +𝑠4 +𝑎1 +𝑠1 +𝑎2 +𝑠3 +𝑎3 +𝑠2 +1st: 38 Hz +2nd: 500 Hz +3rd: 500 Hz +4th: 553 Hz +Proposed: Lightweight stage w/ 1st flexible mode controlled +Baseline: Precision stage w/o flexible mode control +1st: 250 Hz +3rd: 1394 Hz +4th: 1415 Hz +Thought maybe colorful one is better for proposed case. Rainbow. +Flexible Modes: +2nd: 1260 Hz +Flexible Modes: +Fig. 5. Case study #1: proposed and baseline stage parameter definition and resultant dynamics. +cross-coupling between different DOFs is negligible. For each +DOF, a fixed-structure SISO controller is selected following +reference [5] as +Ck(s) = Kp +�s + ωi +s +�� s +ωd ++ 1 +�� +ω2 +lp +s2 + 2zlpωlps + ω2 +lp +� +, +(6) +where the controller parameters are described in Table I. This +controller design follows reference [2], [4] where all the con- +troller parameters except the controller gain can be determined +by a target control bandwidth ωbw. This approach effectively +simplifies the parameter tuning process. The proportional gain +Kp and the target bandwidth are determined such that the +control bandwidth is maximized while satisfying a robustness +criteria[10] of +∥Sk(s)∥∞ ≤ 2, k = 1, ..., n, +(7) +where Sk(s) is the closed-loop sensitivity function of the k- +th channel as Sk = (I − GkCk)−1. With the control effort +signals uk for each channel computed, an actuation recoupling +transformation is used to map the control signals to individual +actuators. +IV. SIMULATION EVALUATION +Two case studies are simulated to evaluate the potential and +effectiveness of the proposed lightweight precision stage design +method. Case study #1 considers a simple rib-enhanced stage +structure with arbitrary sensor/actuator placements, aiming at +demonstrating the impact of the selection of the weighting +variable γ on controller design. Case study #2 implements +the proposed framework for a practical lightweight planar +motor stage with the actuator’s weight and location constraints +considered. The performance of both case studies compared to +that of a baseline stage design without flexible mode control +for evaluation. +A. Case study #1 +Figure 5 shows the diagrams of the stage structure being +considered, which shows a rib-reinforced structure made of +6061-T6 aluminum alloy of 300 mm × 300 mm in size. The +coordinate system being used is also shown in Fig. 5. Herein, +the rigid-body motion of the stage in three DOFs, including +vertical translation (z), roll (θx), and pitch (θy) are actively +controlled. In addition, the proposed stage also actively controls +its first vibration mode, and the baseline stage has no control +for flexible modes. Therefore, three actuators and three sensors +are used for the baseline stages for exact constraint, while +the proposed case uses four actuators and four sensors. The +geometric parameters θp ∈ R5 and the actuator/sensor location +parameters θa = [xa, ya]⊤ and θs = [xs, ys]⊤ are also shown +in Fig. 5. +Due to the geometric complexity of the ribbed stage structure, +analytical models are not sufficient to capture its structural dy- +namics accurately. In this work, finite element (FE) simulation +(with COMSOL Multiphysics) is used to simulate the stage’s +spatial-temporal behavior. In the stage geometry optimization +problem (2) formulation for the proposed stage in Fig. 5, to +facilitate controller design with a target control bandwidth +of ∼ 100 Hz, the values of ωlow and ωhigh are selected as +50 Hz and 500 Hz, respectively. In addition, the rib width and +base height are constrained to be larger than 1 mm for the +sake of manufactuability. With the stage geometry optimization +problem (2) fully formulated, the Optimization Module in +COMSOL Multiphysics is selected to solve the problem, where +an iterative method for derivative-free constrained optimization +COBYLA [11] is employed. The resultant stage resonance +frequencies and mode shapes are illustrated in Fig. 5. +The actuator/sensor placement optimization problems (3)-(4) +are then solved for the optimized structure. In case study #1, the +actuator/sensor location range is over the entire top surface of +the stage, i.e., Da = Ds = {(x, y, z) +�� ∥x∥, ∥y∥ ≤ 0.15 m, z = +0}. The normalized mode shapes over all mesh nodes φi(x, y, z) +for the stage and their corresponding natural frequencies ωi can +be obtained from the FE simulations. For each node location +within placement domain, let θa or θs = (x, y, z) ∈ Da or +Ds and thus the actuation/sensing matrices Ba(θa) or Cs(θs) +can be found. A modal damping of ζ = 0.01 is assumed +for all modes, and the grammians (5) for each mode can be +computed. A direct search algorithm is utilized to find the +optimal actuator/sensor locations. +Remark 4.1: When γ and the placement domain of actuators +and sensors being identical, i.e., Da = Ds, the optimal solution +for both (3) and (4) will be identical too. Therefore, the optimal +configuration is a “collocated” case with the actuator and +sensors configured at the same location [12]. In addition, the + +stage structure being considered is symmetrical about the x +and y axes. Therefore, the optimal actuator/sensor location +will be also symmetrical as shown in Fig. 5. These two facts +significantly simplify the numerical computation required for +the actuator/sensor placement optimization problems. +With the stage’s geometric design and the placement of +actuator/sensor decided, we are able to extract the state- +space models for the proposed lightweight stage from the FE +simulations. The system’s undamped dynamics can be written +as +MF E ¨xF E + KF ExF E = BF Eu, +y = CF ExF E, +(8) +where xF E ∈ RnF E is the vector of displacement of all nodes +in the FE simulation, nF E is the number of nodes from mesh +setting, MF E, KF E ∈ RnF E×nF E are the mass and stiffness +matrices, respectively, and BF E and CF E are the input and +output matrices determined by the actuator and sensor locations. +Note that the dimension of the FE-computed system dynamics +(8) is typically very large (nF E ∼ 104) especially when a fine +mesh is used in the simulation. To overcome this problem, the +system dynamics (8) is transformed into the modal coordinate +as +¨q + Kq = B(θa)u, +y = C(θs)q, +(9) +where q = Φ−1xF E is the decoupled modal state vector, +Φ = [φ1, · · · , φn] is an n × n matrix where φi represents +the vector of corresponding i-th mode shape with mass matrix +normalized, i.e. Φ⊤MF EΦ = I, K = Φ⊤KF EΦ is the diago- +nal stiffness matrix, and B(θa) and C(θs) are decoupled input +and output matrix, respectively. In this decoupled coordinate, +we can reduce the model order by truncating high-frequency +vibration modes. We keep only the 3 rigid-body modes and +first 10 flexible modes in this paper. Such model is able to +capture the system dynamics accurately up to 1200 Hz, which +is sufficient for controller design. Then, a modal damping term +is introduced into the (9), and a reduced-order model in the +form of (1) can be derived. Finally, the actuation signals u +and measurement signals y are transformed into the decouple +DOFs as shown in Fig. 4. +As is stated in Section III-B, the weighting parameter γ +in (3)-(4) provides a balance between the need to have small +control gains and the need to decouple controlled modes and +uncontrolled modes. In this case study, (3)-(4) are solved for +the stage structure with a varying value of γ, and the resultant +actuator/sensor locations are shown in Fig. 6. Here in Fig. 6, +the red crosses represent the optimal actuator/sensor locations +with different γ values (note that the actuators and sensors +are collocated), and the blue lines represent the nodal lines +of the stage’s second to fourth vibration modes. Figure 7 +shows the decoupled plant frequency responses of the proposed +lightweight stage with actuator/sensor location optimized under +different values of γ. Several selections of the value γ are +discussed as below. +Gamma Gamma +𝛾 = 0 +𝛾 = 5 +𝛾 = 5.5 +𝛾 = 6 +𝛾 = 50 +𝛾 = 10 +𝑥 +𝑦 +𝛾 +𝑥 +𝑦 +0 +5 +5.5 +6 +10 +50 +Fig. 6. +Optimal actuator/sensor placements under varying γ. Blue: nodal +points of uncontrolled modes. +Phase [deg] +Phase [deg] +Magnitude +[𝐦/𝐍] +Phase [deg] +Magnitude +[𝐫𝐚𝐝/(𝐍 ∙ 𝐦)] +Phase [deg] +Magnitude +[𝐫𝐚𝐝/(𝐍 ∙ 𝐦)] +Magnitude +[𝐦/𝐍] +Frequency [Hz] +Frequency [Hz] +Fig. 7. Open-loop plant with different γ. +(a): γ = 0: With γ = 0, the optimal actuator/sensor locations +are at the corners of the stage (Fig. 6), where the first vibration +mode’s modal displacement is maximized. This is because +with γ = 0 we are only considering the need to maximize the +controllability/observability of the actively-controlled modes, +and not considering the effects of high-frequency uncontrolled +modes. This is confirmed by the plant frequency response +shown in Fig. 7 with γ = 0 (blue dashed line), where the +last channel of the plant dynamics (the stage’s first flexible +mode) is having high magnitude. However, this design results in +strong coupling between the stage’s rigid body motion and the +uncontrolled flexible modes (e.g. the second mode at 500 Hz). +(b): γ = 50. As γ increases, the actuator/sensor locations move +towards the the nodal location of the stage’s uncontrolled +flexible modes, as shown in Fig. 6. This is also confirmed by +the plant frequency responses shown in Fig. 7: as γ increases, +the peak of uncontrolled flexible modes decreases, while the +magnitude of the last channel in the plant dynamics (the stage’s +first flexible mode) reduces as well. +From the discussions above, it can be concluded that a large +value in γ is beneficial for obtaining high control bandwidth +at the cost of needing a higher controller gain in the flexible +mode control. Therefore, the value of γ should be selected +as its maximum allowed value to produce an acceptable plant +magnitude in the actively controlled flexible mode. In this +case study, γ = 50 (i.e. the plant as red solid lines in Fig. 7) +is selected to enable a high control bandwidth. The resultant + +×0.2 +0.15 +0.1 +XX +X +0.05 +0 +0.05 +X X +-0.1 ++ +-0.15 +-0.2 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2G1:vertical translation +10~3 +!! ! ! +10~5 +10-7 +101 +102 +103 +0 +7=0 +=5 +-90 +=6 +=50 +-180 +101 +102 +103G2 : tip +10~3 +10~5 +10-7 +107 +102 +103 +0 +=0 +/=5 +-90 +=6 +=50 +-180 +101 +102 +103G3 : tilt +10~3 +10~5 +10~7 +107 +102 +103 +0 +2=0 +1=5 +-90 +=6 +=50 +-180 +101 +102 +103=0 +=5010~1 +104 +10-7 +107 +102 +103 +0 +06- +180 +101 +102 +103𝑎 +𝑏 +𝝎𝒃 = 𝟏𝟎𝟎 𝐇𝐳 +Frequency [Hz] +Frequency [Hz] +𝝓𝒎 = 𝟑𝟕° +𝝎𝒃 = 𝟏𝟎𝟎 𝑯𝒛 +𝝎𝒃 = 𝟐𝟔 𝑯𝒛 +𝝓𝒎 = 𝟑𝟖° +𝝓𝒎 = 𝟑𝟕° +𝟓𝟎𝟎 𝐇𝐳 +Magnitude [abs] +𝟏𝟐𝟔𝟎 𝐇𝐳 +𝟐𝟓𝟎 𝐇𝐳 +𝟓𝟓𝟑 𝐇𝐳 +Phase [deg] +Fig. 8. Case study #1: comparison of loop gains of the proposed stage design (red solid) and baseline stage design (blue dashed). (a) z-DOF (translation in the +vertical direction). (b) θx-DOF (pitch). +TABLE II +CASE STUDY #1 PERFORMANCE COMPARISON. +Baseline Design +Proposed Design +Stage weight +2.31 kg +0.34 kg +1st res. freq. +250 Hz +38 Hz +2nd res. freq. +1260 Hz +500 Hz +z bandwidth +100 Hz +100 Hz +θx/θy bandwidth +26 Hz +100 Hz +Max sensitivity +1.89 +1.84 +optimal actuator/sensor locations are close to the nodal positions +of the uncontrolled flexible modes, see Fig. 6. Finally, four +SISO controllers in the form of (6) are designed for each +actively-controlled DOFs, with a target control bandwidth of +ωbw = 100 Hz. +To evaluate the effectiveness of our proposed design method, +a baseline lightweight precision stage as illustrated in Fig. 5 is +used for comparison. This baseline stage lightweight stage does +not have active control for its flexible modes, and only has the +rigid body motions under feedback control. Three actuators and +three sensors are used to achieve exact constraint in the stage +actuation and control. In such a design, the first resonance +frequency of the stage structure places an upper limit to the +achievable control bandwidth. With a target control bandwidth +of 50 Hz, the geometric parameters of the baseline stage are +designed such that the first resonance frequency of the stage +structure is above 250 Hz (i.e. 5× of the target bandwidth). +Similarly, SISO controllers in the form of (6) are designed +for all decoupled DOFs under active control such that the +robustness criteria 7 is satisfied. +Table. II summarizes the performance of the proposed +lightweight stage in case study #1 and that of the baseline stage, +and Fig. 8 shows the loop gains of both proposed and baseline +designs in the z-DOF (translation in the vertical direction) and +the θx-DOF (pitch direction). Comparing the loop frequency +responses shown in Fig. 8a, it can be observed both stages +can reach a high control bandwidth of 100 Hz with sufficient +stability margins in the z-DOF, and the 250 Hz resonance +in the baseline stage is not shown in its z-DOF frequency +response. This is because the baseline’s first flexible mode is +not controllable or not excitable by the z-axis control loop, +and thus this resonance does not limit the stage’s control +bandwidth in this axis. However, the 250 Hz resonance of +the baseline stage can couple in the stage’s z-axis dynamics +under imperfect actuator or position placement, and stability +issue can arise in the control under such situations. In addition, +the lightly-damped resonance at 250 Hz in the baseline stage +is not actively controlled and thus can be easily excited by +disturbances, which can impair the stage’s positioning accuracy. +Comparing the loop frequency responses shown in Fig. 8b, +it can be observed that the bandwidth of the baseline stage is +only 26 Hz. This is primarily due to the 250 Hz resonance peak +in the stage dynamics is coupled into the stage’s control in +the θx direction with the current actuator/sensor configuration, +and thus limits the achievable control bandwidth. In contrast, +the proposed design can robustly achieve a control bandwidth +of 100 Hz since the stage’s first resonance mode at 50Hz is +actively controlled. +Finally, comparing the performance shown in Table II, it can +be seen that the weight of the proposed stage design is reduced +by 85% compared to baseline design. To our understanding, +this significant gain in weight reduction is due to the proposed +stage is allowing compliance in the first flexible mode, which +effectively removes material in the stage structure needed to +reinforce the stage. This result shows the tremendous potential +of the proposed approach in stage acceleration improvement and +the power consumption reduction. In addition, comparing the +closed-loop damping performance of the stage’s first resonance +mode, it can be seen that the baseline stage’s resonance at +250 Hz is only having a low damping ratio of 0.01, which can +be excited by external disturbances. In contrast, the first flexible +mode of the proposed stage is under closed-loop control, which +has a bandwidth of 100 Hz and has a closed-loop damping ratio +of 0.37. This improvement in the structural damping shows +the potential of the proposed approach to improve the stage’s +positioning accuracy under external disturbances. +B. Case study #2 +Case study #2 considers a magnetically-levitated planar +motion stage as illustrated in Fig. 9, where four neodymium + +100 +360 +Proposed +180 +Baseline +0 +180 +360 +10 +102 +10310° +06- +-180 +-270 +-360 +Proposed +Baseline +450 +10 +102 +103permanent magnet arrays of 60mm × 60 mm × 6 mm are +arranged at the corner of the stage to provide both thrust forces +for planar motion and the levitation forces. The inclusion of the +actuator magnets enhances the practical relevance of the case +study for wafer positioning application. The vertical-directional +levitation forces are assumed to be located at the center of the +permanent magnet arrays. All other stage geometry parameters +are defined in the same way with case study #1. +As stated in Remark 3.1, the value of ωhigh sets an upper +bound for the achievable control bandwidth for the proposed +positioning stage. However, using a high value of ωhigh can +enforce the stage design to increase materials to stiffen the +corresponding resonance mode, and thus increase the stage’s +weight. Therefore, to fully explore the feasible designs set +as illustrated in Fig. 2 and thus to remove possible design +conservatism, the value of ωhigh needs to be swept. It is +worth pointing out that the stage geometry design (2) and +the actuator/sensor placement design (3)-(4) collaboratively +determine the plant dynamics of the positioning stage. When +conducting a parameter sweep for ωhigh, the actuator/sensor +placement problems must also be solved for each stage +geometry design for effective design optimization. +To reduce possible design conservatism and thus fully exploit +the advantages brought by the flexible mode control, the feasible +stage design set for case study #2 is explored as follows: +First, a target control bandwidth is selected to be 120 Hz for +the positioning stage. Next, the stage geometry optimization +problem (2) is solved with ωhigh = 600 Hz, i.e. 5× of the +target bandwidth. Then, the sensor positioning optimization +problem (4) is solved with γ = 50. Note that the actuator’s +locations are fixed due to the inclusion of magnet arrays. With +one feasible stage and sensor positioning design provided by +the previous steps, we then decrease the value of ωhigh by a +constant step δω = 10 Hz and resolve (2) and (4). Assuming δω +is sufficiently small, the change in optimal geometric parameters +can be assumed continuous, which allows us to use the optimal +solution from the previous run as the initial parameters when +resolving (2). This method effectively reduces the required +computation time. The previous steps are repeated until ωhigh +is sufficiently low such that it may be excited by external +disturbances. In this case study, the lowest value of ωhigh is +selected to be at 300 Hz. +In the stage geometry optimization problem, the optimal +solutions always have the stage’s second resonance frequency +match ωhigh. Fig. 11 shows the stage geometric parameters +and the resultant stage weight and actuator/sensor placement +objectives under varying ωhigh. It can be observed that the +stage’s weight is reducing as the value of ωhigh decreases, +and the value of Jp + Jo (i.e. sum of objectives of (3)-(4)) +is also decreasing along with the reduction of ωhigh. These +observations reveal new trade-off between the stage’s achievable +control bandwidth and acceleration (assuming constant thrust +force generation), which is illustrated by the orange line in +Fig 2. +The stage hardware design can be manually made among +the optimal designs based on the results shown in Fig. 11. +TABLE III +CASE #2 OPTIMAL PARAMETERS +Baseline Design +Proposed Design +Stage weight +2.67 kg +1.20 kg +First res. freq. +251 Hz +50 Hz +2nd res. freq. +1080 Hz +540 Hz +z motion bandwidth +25 Hz +120 Hz +θx/θy bandwidth +120 Hz +120 Hz +Max sensitivity +1.80 +1.94 +In this case study, ωhigh = 540 Hz is selected to provide +sufficiently high Jp + Jo values while reducing the stage’s +weight. Compared to the initial stage design using ωhigh = +600 Hz, the stage’s weight is reduced by 4.5%. Although the +improvement is not significant, it is worth pointing out that the +geometry optimization of the stage is relatively limited in the +current formulation with only five parameters that can be varied. +A more significant improvement in the stage’s performance +may be expected given increased design flexibility is allowed +in the stage structure. The resultant stage’s flexible modes are +illustrated in the bottom left in Fig. 9. The state-space dynamic +model of the stage can be derived for this stage in the same +way as discussed in case study #1, and controllers are designed +for the decoupled motions. +To evaluate the effectiveness of our proposed framework con- +sidering actuator weight and constraints, a baseline lightweight +stage with same magnet array is simulated for comparison. +In the baseline stage, only the rigid-body motions are under +active control, and all flexible modes are uncontrolled. With a +target bandwidth of 50 Hz, the stage’s geometric parameters +are designed to constrain the first resonance frequency above +250 Hz. Fig. 9 show the baseline stage design parameters +and actuator/sensor location. Three SISO controllers as (6) are +designed for all decoupled DOFs in the same way with case +study #1. +Table. III summarizes the performance and comparison of +the proposed and baseline stage design in case #2, Fig. 10 +illustrates the loop gains of both proposed and baseline designs +in z- and θx-DOFs. Comparing the loop frequency responses in +Fig. 10a, it can be observed that the bandwidth of the baseline +design is limited to 25 Hz due to the 251 Hz resonance peak. +In contrast, the proposed design can reach a bandwidth of +120 Hz with sufficient stability margin. Fig. 10b shows that +both designs can reach a bandwidth of 120 Hz in the θx-DOF. +This is because the 251 Hz resonance peak in the baseline +stage is not excitable by the θx feedback loop. However, similar +to the z-DOF in case study #1, stability issue can be caused +if the actuator/sensor placement is imperfect. Moreover, the +lightly-damped 251 Hz resonance mode can be easily excited +by external disturbance and thus impair the stage’s positioning +precision. +Finally, Table III shows that the weight of the proposed +stage design is reduced by 55% compared to baseline design. +The significant improvement for a stage considering the weight +of magnet array shows the effectiveness and generality of our +proposed approach. In addition, comparing the closed-loop +damping performance of stage’s first resonance mode, it can be + +𝑥 +𝑦 +𝑧 +60 mm +𝑎1 +𝑎4 +𝑎2 +𝑎3 +Rib width 1: +𝜃1 +Rib width 2: 𝜃2 +𝑥 +𝑦 +Rib distance: +𝜃3 +Rib Height: +𝜃5 +Base +Height: 𝜃4 +6 mm +𝑠1 +𝑠2 +𝑠3 +𝑠4 +60 mm +Rib distance: +30 mm +Rib width: 3 +mm +6 mm +𝑥 +𝑦 +𝑥 +𝑦 +𝑧 +𝑎1 +𝑎4 +𝑎2 +𝑎3 +Rib height: +25 mm +Base height: +3 mm +𝑠1 +𝑠2 +𝑠3 +𝑠4 +Proposed: Practical lightweight stage w/ 1st flexible mode controlled +Baseline: Practical precision stage w/o flexible mode control +1st: 50 Hz +2nd: 540 Hz +3rd: 540 Hz +4th: 547 Hz +1st: 251Hz +2nd: 1080 Hz +3rd: 1183 Hz +4th: 1241 Hz +Flexible Modes: +Flexible Modes: +Fig. 9. Case study #2 proposed and baseline stages. Both stages consider a permanent magnet array with 60 mm × 60 mm × 6 mm for planar motor force +generation. +𝑎 +𝑏 +Frequency [Hz] +Frequency [Hz] +Magnitude [abs] +Phase [deg] +𝟐𝟓𝟏 𝐇𝐳 +𝟏𝟐𝟒𝟎 𝐇𝐳 +𝟓𝟒𝟕 𝐇𝐳 +𝟓𝟒𝟎 𝐇𝐳 +𝝓𝒎 = 𝟑𝟕° +𝝎𝒃 = 𝟏𝟐𝟎 𝑯𝒛 +𝝎𝒃 = 𝟐𝟓 𝑯𝒛 +𝝎𝒃 = 𝟏𝟐𝟎 𝑯𝒛 +𝝓𝒎 = 𝟑𝟕° +Fig. 10. Case study #2: comparison of loop gains of the proposed stage design (red solid) and baseline stage design (blue dashed). (a) z-DOF (translation in +the vertical direction). (b) θx-DOF (pitch). +Rib Distance +Rib Height +Base Height, Rib Width 1&2 +2nd Resonance Frequency [Hz] +Length [mm] +2nd Resonance Frequency [Hz] +Mass [kg] +Jp+Jo [𝐚𝐛𝐬] +𝑎 +𝑏 +2nd Resonance Frequency [Hz] +Fig. 11. +(a) Geometric parameter history. (b) Stage weight and grammian +history. +stated that the proposed design is more robust against external +disturbances with the first lightly-damped mode at 547 Hz, +while that of the baseline stage is at 251 Hz. The comparison +indicates the huge potential of our framework to improve both +the stage’s acceleration capability and positioning accuracy +simultaneously. +V. CONCLUSION AND FUTURE WORK +In this work, we proposed and evaluated a sequential +hardware and controller co-design framework for lightweight +precision stages, aiming at enabling designs that can achieve +high control bandwidth and high acceleration simultaneously. +The algorithm of the framework is presented, and the effec- +tiveness of the proposed method is demonstrated by numerical +simulations using two case studies. The significant weight +reduction (>55%) and improvement in control bandwidth +show the potential. Future work will consider the experimental +evaluations for the proposed method. A fully integrated +controller and hardware co-optimization that can better exploit +the synergy between hardware and control designs will also +be studied. +REFERENCES +[1] J. Albero, S. Bargiel, N. Passilly, P. Dannberg, M. Stumpf, U. Zeitner, +C. Rousselot, K. Gastinger, and C. Gorecki, “Micromachined array- +type mirau interferometer for parallel inspection of mems,” Journal of +Micromechanics and Microengineering, vol. 21, no. 6, p. 065005, 2011. +[2] H. Butler, “Position control in lithographic equipment [applications of +control],” IEEE Control Sys. Mag., vol. 31, no. 5, pp. 28–47, 2011. +[3] A. Delissen, D. Laro, H. Kleijnen, F. van Keulen, and M. Langelaar, +“High-precision motion system design by topology optimization consider- +ing additive manufacturing,” in 20th Int. Conf. of the European Society +for Precision Eng. and Nanotech., EUSPEN 2020. +EUSPEN, 2020, pp. +257–258. +[4] R. Ding, C. Ding, Y. Xu, W. Liu, and X. Yang, “An optimal actuator +placement method for direct-drive stages to maximize control bandwidth,” +in IECON 2020 The 46th Annual Conference of the IEEE Industrial +Electronics Society. +IEEE, 2020, pp. 556–561. +[5] G. F. Franklin, J. D. Powell, A. Emami-Naeini, and J. D. Powell, Feedback +control of dynamic systems. +Prentice hall Upper Saddle River, 2002, +vol. 4. +[6] M. Garcia-Sanz, “Control co-design: an engineering game changer,” +Advanced Control for Appl.: Eng. and Ind. Sys., vol. 1, no. 1, p. e18, +2019. +[7] D. A. Laro, R. Boshuisen, and J. van Eijk, “Design and control of +over-actuated lightweight 450 mm wafer chuck,” in 2010 ASPE Spring +Topical meeting, Cambridge, Massachusetts, USA. +ASPE, 2010, pp. +141–144. + +25 +20 +15 +600 +550 +500 +450 +400 +350 +300 +70 +60 +600 +550 +500 +450 +400 +350 +300 +1.05 +1 +0.95 +600 +550 +500 +450 +400 +350 +3001.3 +1.2 +1.1 +600 +550 +500 +450 +400 +350 +3000.4 +0.6 +0.8 +.1 +600 +550 +500 +450 +400 +350 +300100 +Proposed +Baseline +0 +-90 +-180 +270 +10 1 +102 +10 310° +Proposed +Baseline +10 +0 +06- +U +180 +270 +101 +102 +103[8] T. Oomen, “Advanced motion control for precision mechatronics: Control, +identification, and learning of complex systems,” IEEJ Journal of Ind. +Appl., vol. 7, no. 2, pp. 127–140, 2018. +[9] T. Oomen, R. van Herpen, S. Quist, M. van de Wal, O. Bosgra, and +M. Steinbuch, “Connecting system identification and robust control for +next-generation motion control of a wafer stage,” IEEE Trans. on Ctrl. +Sys. Tech., vol. 22, no. 1, pp. 102–118, 2013. +[10] M. Ortega and F. Rubio, “Systematic design of weighting matrices for +the h mixed sensitivity problem,” Journal of Process Control, vol. 14, +no. 1, pp. 89–98, 2004. +[11] M. J. Powell, “A direct search optimization method that models the +objective and constraint functions by linear interpolation,” in Adv. in opt. +and num. analysis. +Springer, 1994, pp. 51–67. +[12] A. M. Rankers, “Machine dynamics in mechatronic systems: An +engineering approach.” 1998. +[13] G. van der Veen, M. Langelaar, S. van der Meulen, D. Laro, W. Aangenent, +and F. van Keulen, “Integrating topology optimization in precision +motion system design for optimal closed-loop control performance,” +Mechatronics, vol. 47, pp. 1–13, 2017. +[14] R. van Herpen, T. Oomen, E. Kikken, M. van de Wal, W. Aangenent, +and M. Steinbuch, “Exploiting additional actuators and sensors for nano- +positioning robust motion control,” Mechatronics, vol. 24, no. 6, pp. +619–631, 2014. +[15] J. Wu and L. Zhou, “Control co-design of actively controlled lightweight +structures for high-acceleration precision motion systems,” in 2022 +American Control Conference (ACC), 2022, pp. 5320–5327. + diff --git a/6tE2T4oBgHgl3EQf7Qi3/content/tmp_files/load_file.txt b/6tE2T4oBgHgl3EQf7Qi3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c61881f571748d00c27b5db6e50958e7761eaed5 --- /dev/null +++ b/6tE2T4oBgHgl3EQf7Qi3/content/tmp_files/load_file.txt @@ -0,0 +1,648 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf,len=647 +page_content='Sequential Structure and Control Co-Design of Lightweight Precision Stages with Active Control of Flexible Modes Jingjie Wu Walker Department of Mechanical Engineering The University of Texas at Austin Austin, TX, 78712 wujingjie@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='edu Lei Zhou Walker Department of Mechanical Engineering The University of Texas at Austin Austin, TX, 78712 lzhou@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='edu Abstract—Precision motion stages are playing a prominent role in various manufacturing equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The drastically increasing demand for higher throughput in integrated circuit (IC) man- ufacturing and inspection calls for the next-generation preci- sion stages that have light weight and high control bandwidth simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In today’s design techniques, the stage’s first flexible mode is limiting its achievable control bandwidth, which enforces a trade-off between the stage’s acceleration and closed- loop stiffness and thus limits the system’s overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To overcome this challenge, this paper proposes a new hardware design and control framework for lightweight precision motion stages with the stage’s low-frequency flexible modes actively controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Our method proposes to minimize the resonance frequency of the controlled mode to reduce the stage’s weight, and to maximize that of the uncontrolled mode to enable high control bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In addition, the proposed framework determines the placement of the actuators and sensors to maximize the controllability/observability of the stage’s controlled flexible mode while minimizing that of the uncontrolled mode, which effectively simplifies the controller designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Two case studies are used to evaluate the effectiveness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Simulation results show that the stage designed using the proposed method has a weight reduction of more than 55% compared to a baseline stage design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Improvement in control bandwidth was also achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' These results demonstrate the effectiveness of the proposed method in achieving lightweight precision positioning stages with high acceleration, bandwidth, and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Index Terms—Precision positioning systems, control co-design, structure control I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' INTRODUCTION High-precision positioning stages are playing a critical role in a wide range of manufacturing and inspection tools such as photolithography scanners [2] and MEMS inspection systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The drastically growing demand for higher throughput in semiconductor manufacturing necessitates the next-generation precision motion stages with higher acceleration capability while maintaining excellent positioning accuracy and high control bandwidth [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Creating new lightweight precision positioning stages is critical to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' However, as the stage’s weight reduces, its structural resonance frequencies will decrease to near or even within the control bandwidth (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1), which limits the stage’s control bandwidth and Control Bandwidth Frequency Loop gain Flexible dynamics Well above bandwidth no stability challenges Control Bandwidth Frequency Loop gain Flexible dynamics Close or within bandwidth Cause stability challenges How to design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Conventional Stages Lightweight Stages Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Design challenge of lightweight precision positioning stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Stage Acceleration ✓ High Acceleration, ✓ Low power consumption X Low control bandwidth X Deformation during acceleration Closed-loop Stiffness/Control Bandwidth Conventional Rigid Stage ✓ Rigid structure, high precision ✓ High control bandwidth X Low acceleration X High power consumption Today’s stages Feasible Range New Feasible Range Lightweight stage w/ Flexible Mode Control Proposed Approach: Lightweight Stage w/o Flexible Mode Control Acceleration Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Illustration of acceleration and bandwidth trade-off in today’s precision positioning systems and motivation for the proposed lightweight stage with flexible mode control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' positioning accuracy, and can even cause stability challenges [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In the past decade, a number of research and engineering efforts have studied the design and control for lightweight precision positioning stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' For example, Laro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [7] presented an over-actuation approach to place actuators/sensors at the stage’s nodal locations to prevent the flexible dynamics from being excited by the feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Oomen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [9] proposed a system identification and robust control framework for wafer stages, which provides a systematic approach to create controller designs for stages exhibiting low-frequency flexible dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Although effective, these studies mostly investigate the motion control for flexible stages, and the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='04208v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='SY] 10 Jan 2023 Control Bandwidth Frequency Loop gain Flexible dynamics Well above bandwidth no stability challenges Control Bandwidth Frequency Loop gain Flexible dynamics Close or within bandwidth Cause stability challenges How to design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Conventional Stages Lightweight Stages Control Bandwidth Frequency Loop gain Uncontrolled flexible dynamics well above bandwidth no stability challenges Actively controlled flexible dynamics highly compliant well within bandwidth baseline Design challenge proposed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Illustration of the proposed lightweight stage design with active control for flexible modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' synergy between the structure design and controller design is not fully exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In recent years, the hardware-control co-design, or control co-design (CCD) [6], has been studied for the lightweight precision positioning stages, aiming at enabling a synergistic structure-control design method for precision positioning stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' For example, Van der Veen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [13] studied the integrated topology and controller optimization for a simple 2D motion stage structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Delissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [3] presented a topology-optimized wafer stage fabricated via metal additive manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In a recent study, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [15] presented a nested CCD formulation of for lightweight precision stages with controller design constraints explicitly considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Despite these advances, we make a key observation that in these prior lightweight precision stages designs, the first resonance frequency of the stage structure sets an upper limit for the achievable control bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This fact enforces a fundamental trade-off between the stage’s bandwidth and acceleration as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Fundamental advances in the stage’s mechatronic design must be made to break this trade- off and thus enable stages with improved overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Aiming at overcoming the aforementioned trade-off and thus creating new lightweight stages that can simultaneously have high acceleration and high closed-loop stiffness, this paper presents a sequential structure and control design framework where the low-frequency flexible modes of the stage are under active control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This approach has been explored in van Herpen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [14] where additional actuators and sensors are introduced for a lightweight stage to enhance the control bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' However, in [14], the control for flexible dynamics is not considered in the stage’s structural design phase, which limits the achievable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In our work, to facilitate the controller design, we propose to minimize the resonance frequency of the stage’s mode being controlled and to maximize the resonance frequency of the uncontrolled mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The target control bandwidth of the stage is in between the resonance fre- quencies, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' We envision that this formulation will remove material in the stage’s structure to allow compliance in the actively-controlled modes thereby breaking the trade-off in lightweight stages, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With the stage’s structure designed, we further propose to use an optimization method to compute the best actuator/sensor placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Our hypothesis is that maximizing the controllability/observability of the actively-controlled flexible modes while minimizing that of the uncontrolled modes will deliver the best positioning performance with reasonable control signal magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Two case studies are simulated to evaluate the effectiveness of the proposed approach, where a stage weight reduction of > 55% is demonstrated compared to a baseline case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' These results demonstrate the potential of the proposed lightweight precision stage design framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Section II de- scribes the problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Section III presents the proposed design framework for the lightweight precision positioning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Section IV shows the simulation evaluations with two case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Conclusion and future work are summarized in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' PROBLEM STATEMENT The dynamics of a precision positioning stage considering its flexible structural behaviors can be described by M(θp)¨x + D(θp) ˙x + K(θp)x = B(θp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' θa)u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' y = C(θp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' θs)x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (1) where x is a vector of state variables of both rigid-body displace- ments and flexible displacements in the modal coordinate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' K are the mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' damping and stiffness matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' u is the vector of control signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' y is a vector of measurement signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' B is the input matrix which maps the control input u to corresponding states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' C is the output matrix which maps state variables to measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' θp is a vector of stage’s geometric design parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' and θa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' θs are the vectors of actuator and sensor locations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The design optimization problem for a lightweight precision stage described by (1) aims at finding a set of hardware design parameters θp, θa, and θs and a controller design that can minimize the stage’s weight while maximizing the control bandwidth, meanwhile satisfying certain robustness criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' SEQUENTIAL HARDWARE AND CONTROL OPTIMIZATION FRAMEWORK This section presents a sequential framework of designing the hardware and controller for lightweight stages with their low- frequency flexible modes actively controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In the first step, an optimization problem that determines the stage’s geometric parameter is formulated to facilitate the active control for the stage’s low-frequency flexible modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In the second step, an optimization is performed to determine the location of actuators and sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Finally, feedback controllers are synthesized for the designed stage to control the stage’s motion as well as the low-frequency flexible modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The three steps are introduced in detail in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Stage Geometry Design Optimization In a lightweight precision stage with active control for low-frequency flexible modes, the stage’s geometry design ALoop-gain ANoop-gain How to design?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Flexible vnamic Frequency Frequency Control Flexible Control bandwidth dynamics bandwidthoptimization is formulated as min θp Jm(θp), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' ωi ≤ ωlow, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', n ωj ≥ ωhigh, j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', m θp,min ≤ θp ≤ θp,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (2) Here, the objective function Jm represents the stage’s weight, θp is a vector for the stage’s geometric parameters, ωi is the i-th modal frequency with its corresponding vibration mode actively controlled, and ωj is the j-th resonance frequency where the corresponding mode shape is not controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' ωlow is the upper bound for the actively-controlled resonance frequencies, and ωhigh is the lower bound for the uncontrolled resonance frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' θp,min and θp,max are the lower and upper bounds for the stage’s geometric parameter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With the stage structure design optimization formulation (2), the stage’s flexible modes under active control are having resonance frequencies below ωlow, and that of the uncontrolled modes are beyond ωhigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Such an optimization process can enforce material removal in the stage’s structure to allow for compliance in the actively-controlled flexible modes, and add material to stiffen the uncontrolled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1: The selection of ωlow and ωhigh are highly important and determine the system’s dynamic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The system’s target control bandwidth must be between ωlow and ωhigh, and ωhigh sets the new upper bound for the achievable control bandwidth for the lightweight precision stage with actively controlled flexible modes, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To facilitate controller design while maintaining design feasibility, the values of ωlow and ωhigh need to be se- lected according to the target control bandwidth, for example ωlow ∼ 1 2 × ωbw and ωhigh ∼ 5 × ωbw, where ωbw is the target bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This method, although robust, may lead to a relatively conservative stage design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To fully evaluate the feasible design range in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 2, the value of ωhigh needs to be swept while considering the actuator/sensor positioning, which will be introduced in Section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Actuator and Sensor Placement The actuator and sensor placement optimization problem for the proposed lightweight stage with active flexible mode controlled can be formulated as max θa∈Da Ja(θa) = � i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=',n Wpi(θa) − γ � i=n+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=',m Wpi(θa), (3) max θs∈Ds Jo(θs) = � i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=',n Woi(θs) − γ � i=n+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=',m Woi(θs), (4) where θa and θs are vectors of actuator and sensor placement parameters, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Da and Ds are the design domains for actuator/sensor locations, and γ is a positive user-defined weighting constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Wpi and Woi are the controllability and + + 𝐶𝜃𝑥 𝐶𝜃𝑦 u𝜃𝑦 u𝜃𝑥 𝐶𝑧 𝐶𝑚1 Control Diagram Lightweight Stage Dynamics Actuation Recoupling Transformation 𝑢𝑧 𝑢𝑚1 Measurement Decoupling Transformation 𝑢1 𝑢2 𝑢3 𝑢4 𝑦1 𝑦2 𝑦3 𝑦4 𝑟𝑧 𝑟𝜃𝑥 𝑟𝜃𝑦 𝑟𝑞1 𝑥𝑧 𝜃𝑥 𝜃𝑦 𝑞1 + − + − − − Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Control block diagram for the lightweight precision positioning stage with model decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' TABLE I CONTROLLER PARAMETERS [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Parameter Description Typical Value ωbw Desired bandwidth [rad/s] – α PID frequency ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3 Kp Proportional gain – ωi Integrator frequency ωbw/α2 ωd Differentiator frequency ωbw/α ωlp Lowpass filter frequency αωbw zlp Lowpass filter damping ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='7 observability grammians of i-th flexible mode, respectively, which can be calculated as Wpi = ∥φi(θa)⊤Ba(θa)∥2 2 4ζiωi , Woi = ∥Cs(θs)⊤φi(θs)∥2 2 4ζiωi , (5) where φi is the mass-normalized mode shape of i-th flexible mode, Ba and Cs are the force and measurement assembling matrices, ζi is the modal damping ratio, and ωi is the i-th resonance natural frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The controllability/observability grammians Wpi and Woi quantitatively evaluate the control- lability/observability of the corresponding flexible mode in the control system, which will reflect on the peak resonance magnitude in the system’s frequency response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With actuator/sensor placement optimization formulation in (3) and (4), our goal is to maximize the controllabil- ity/observability for the actively-controlled modes to reduce the required controller gain, and to minimize those of the uncontrolled modes to reduce their coupling with the control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The value of γ provides a trade-off between the two design goals: a low value in γ emphasizes reducing the needed controller gain for actively-controlled modes, and a high value in γ emphasizes reducing the cross-talk between uncontrolled modes and controlled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Feedback Control Design With the stage’s structure and actuator/sensor locations determined, the plant dynamics of the stage can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Feedback controllers can be designed for each degree of freedom (DOF) to enable precision positioning and disturbance rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Figure 4 shows a block diagram for the control loop for a lightweight stage with three rigid-body DOFs and one flexible mode under active control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Here, the lightweight stage plant dynamics P : u → y can be obtained from solving (2), (3), and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The sensor measurements y are transformed to individual DOFs via a measurement decoupling transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Four single-input,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' single-output (SISO) feedback controllers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='can then be designed for four decoupled channels assuming the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib width: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='4 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib distance: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='30 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑧 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib height: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='25 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Base height: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib width 2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝜃𝑝2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib width 1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝜃𝑝1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib distance: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝜃𝑝3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib Height: 𝜃𝑝5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Base Height: 𝜃𝑝4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑧 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1st: 38 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2nd: 500 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3rd: 500 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='4th: 553 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Proposed: Lightweight stage w/ 1st flexible mode controlled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Baseline: Precision stage w/o flexible mode control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1st: 250 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3rd: 1394 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='4th: 1415 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Thought maybe colorful one is better for proposed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Rainbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Flexible Modes: 2nd: 1260 Hz Flexible Modes: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #1: proposed and baseline stage parameter definition and resultant dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' cross-coupling between different DOFs is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' For each DOF, a fixed-structure SISO controller is selected following reference [5] as Ck(s) = Kp �s + ωi s �� s ωd + 1 �� ω2 lp s2 + 2zlpωlps + ω2 lp � , (6) where the controller parameters are described in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This controller design follows reference [2], [4] where all the con- troller parameters except the controller gain can be determined by a target control bandwidth ωbw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This approach effectively simplifies the parameter tuning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The proportional gain Kp and the target bandwidth are determined such that the control bandwidth is maximized while satisfying a robustness criteria[10] of ∥Sk(s)∥∞ ≤ 2, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', n, (7) where Sk(s) is the closed-loop sensitivity function of the k- th channel as Sk = (I − GkCk)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With the control effort signals uk for each channel computed, an actuation recoupling transformation is used to map the control signals to individual actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' SIMULATION EVALUATION Two case studies are simulated to evaluate the potential and effectiveness of the proposed lightweight precision stage design method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #1 considers a simple rib-enhanced stage structure with arbitrary sensor/actuator placements, aiming at demonstrating the impact of the selection of the weighting variable γ on controller design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #2 implements the proposed framework for a practical lightweight planar motor stage with the actuator’s weight and location constraints considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The performance of both case studies compared to that of a baseline stage design without flexible mode control for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #1 Figure 5 shows the diagrams of the stage structure being considered, which shows a rib-reinforced structure made of 6061-T6 aluminum alloy of 300 mm × 300 mm in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The coordinate system being used is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Herein, the rigid-body motion of the stage in three DOFs, including vertical translation (z), roll (θx), and pitch (θy) are actively controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In addition, the proposed stage also actively controls its first vibration mode, and the baseline stage has no control for flexible modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Therefore, three actuators and three sensors are used for the baseline stages for exact constraint, while the proposed case uses four actuators and four sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The geometric parameters θp ∈ R5 and the actuator/sensor location parameters θa = [xa, ya]⊤ and θs = [xs, ys]⊤ are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Due to the geometric complexity of the ribbed stage structure, analytical models are not sufficient to capture its structural dy- namics accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In this work, finite element (FE) simulation (with COMSOL Multiphysics) is used to simulate the stage’s spatial-temporal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In the stage geometry optimization problem (2) formulation for the proposed stage in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5, to facilitate controller design with a target control bandwidth of ∼ 100 Hz, the values of ωlow and ωhigh are selected as 50 Hz and 500 Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In addition, the rib width and base height are constrained to be larger than 1 mm for the sake of manufactuability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With the stage geometry optimization problem (2) fully formulated, the Optimization Module in COMSOL Multiphysics is selected to solve the problem, where an iterative method for derivative-free constrained optimization COBYLA [11] is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The resultant stage resonance frequencies and mode shapes are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The actuator/sensor placement optimization problems (3)-(4) are then solved for the optimized structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In case study #1, the actuator/sensor location range is over the entire top surface of the stage, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', Da = Ds = {(x, y, z) �� ∥x∥, ∥y∥ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='15 m, z = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The normalized mode shapes over all mesh nodes φi(x, y, z) for the stage and their corresponding natural frequencies ωi can be obtained from the FE simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' For each node location within placement domain, let θa or θs = (x, y, z) ∈ Da or Ds and thus the actuation/sensing matrices Ba(θa) or Cs(θs) can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' A modal damping of ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='01 is assumed for all modes, and the grammians (5) for each mode can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' A direct search algorithm is utilized to find the optimal actuator/sensor locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1: When γ and the placement domain of actuators and sensors being identical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', Da = Ds, the optimal solution for both (3) and (4) will be identical too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Therefore, the optimal configuration is a “collocated” case with the actuator and sensors configured at the same location [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In addition, the stage structure being considered is symmetrical about the x and y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Therefore, the optimal actuator/sensor location will be also symmetrical as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' These two facts significantly simplify the numerical computation required for the actuator/sensor placement optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With the stage’s geometric design and the placement of actuator/sensor decided, we are able to extract the state- space models for the proposed lightweight stage from the FE simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The system’s undamped dynamics can be written as MF E ¨xF E + KF ExF E = BF Eu, y = CF ExF E, (8) where xF E ∈ RnF E is the vector of displacement of all nodes in the FE simulation, nF E is the number of nodes from mesh setting, MF E, KF E ∈ RnF E×nF E are the mass and stiffness matrices, respectively, and BF E and CF E are the input and output matrices determined by the actuator and sensor locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Note that the dimension of the FE-computed system dynamics (8) is typically very large (nF E ∼ 104) especially when a fine mesh is used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To overcome this problem, the system dynamics (8) is transformed into the modal coordinate as ¨q + Kq = B(θa)u, y = C(θs)q, (9) where q = Φ−1xF E is the decoupled modal state vector, Φ = [φ1, · · · , φn] is an n × n matrix where φi represents the vector of corresponding i-th mode shape with mass matrix normalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Φ⊤MF EΦ = I, K = Φ⊤KF EΦ is the diago- nal stiffness matrix, and B(θa) and C(θs) are decoupled input and output matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In this decoupled coordinate, we can reduce the model order by truncating high-frequency vibration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' We keep only the 3 rigid-body modes and first 10 flexible modes in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Such model is able to capture the system dynamics accurately up to 1200 Hz, which is sufficient for controller design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Then, a modal damping term is introduced into the (9), and a reduced-order model in the form of (1) can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Finally, the actuation signals u and measurement signals y are transformed into the decouple DOFs as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' As is stated in Section III-B, the weighting parameter γ in (3)-(4) provides a balance between the need to have small control gains and the need to decouple controlled modes and uncontrolled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In this case study, (3)-(4) are solved for the stage structure with a varying value of γ, and the resultant actuator/sensor locations are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Here in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6, the red crosses represent the optimal actuator/sensor locations with different γ values (note that the actuators and sensors are collocated), and the blue lines represent the nodal lines of the stage’s second to fourth vibration modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Figure 7 shows the decoupled plant frequency responses of the proposed lightweight stage with actuator/sensor location optimized under different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Several selections of the value γ are discussed as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Gamma Gamma 𝛾 = 0 𝛾 = 5 𝛾 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='5 𝛾 = 6 𝛾 = 50 𝛾 = 10 𝑥 𝑦 𝛾 𝑥 𝑦 0 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='5 6 10 50 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Optimal actuator/sensor placements under varying γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Blue: nodal points of uncontrolled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Phase [deg] Phase [deg] Magnitude [𝐦/𝐍] Phase [deg] Magnitude [𝐫𝐚𝐝/(𝐍 ∙ 𝐦)] Phase [deg] Magnitude [𝐫𝐚𝐝/(𝐍 ∙ 𝐦)] Magnitude [𝐦/𝐍] Frequency [Hz] Frequency [Hz] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Open-loop plant with different γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (a): γ = 0: With γ = 0, the optimal actuator/sensor locations are at the corners of the stage (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6), where the first vibration mode’s modal displacement is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This is because with γ = 0 we are only considering the need to maximize the controllability/observability of the actively-controlled modes, and not considering the effects of high-frequency uncontrolled modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This is confirmed by the plant frequency response shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 7 with γ = 0 (blue dashed line), where the last channel of the plant dynamics (the stage’s first flexible mode) is having high magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' However, this design results in strong coupling between the stage’s rigid body motion and the uncontrolled flexible modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' the second mode at 500 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (b): γ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' As γ increases, the actuator/sensor locations move towards the the nodal location of the stage’s uncontrolled flexible modes, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This is also confirmed by the plant frequency responses shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 7: as γ increases, the peak of uncontrolled flexible modes decreases, while the magnitude of the last channel in the plant dynamics (the stage’s first flexible mode) reduces as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' From the discussions above, it can be concluded that a large value in γ is beneficial for obtaining high control bandwidth at the cost of needing a higher controller gain in the flexible mode control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Therefore, the value of γ should be selected as its maximum allowed value to produce an acceptable plant magnitude in the actively controlled flexible mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In this case study, γ = 50 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' the plant as red solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 7) is selected to enable a high control bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The resultant ×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1 XX X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='05 X X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2G1:vertical translation 10~3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 10~5 10-7 101 102 103 0 7=0 =5 90 =6 =50 180 101 102 103G2 : tip 10~3 10~5 10-7 107 102 103 0 =0 /=5 90 =6 =50 180 101 102 103G3 : tilt 10~3 10~5 10~7 107 102 103 0 2=0 1=5 90 =6 =50 180 101 102 103=0 =5010~1 104 10-7 107 102 103 0 06- 180 101 102 103𝑎 𝑏 𝝎𝒃 = 𝟏𝟎𝟎 𝐇𝐳 Frequency [Hz] Frequency [Hz] 𝝓𝒎 = 𝟑𝟕° 𝝎𝒃 = 𝟏𝟎𝟎 𝑯𝒛 𝝎𝒃 = 𝟐𝟔 𝑯𝒛 𝝓𝒎 = 𝟑𝟖° 𝝓𝒎 = 𝟑𝟕° 𝟓𝟎𝟎 𝐇𝐳 Magnitude [abs] 𝟏𝟐𝟔𝟎 𝐇𝐳 𝟐𝟓𝟎 𝐇𝐳 𝟓𝟓𝟑 𝐇𝐳 Phase [deg] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #1: comparison of loop gains of the proposed stage design (red solid) and baseline stage design (blue dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (a) z-DOF (translation in the vertical direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (b) θx-DOF (pitch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' TABLE II CASE STUDY #1 PERFORMANCE COMPARISON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Baseline Design Proposed Design Stage weight 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='31 kg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='34 kg 1st res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 250 Hz 38 Hz 2nd res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1260 Hz 500 Hz z bandwidth 100 Hz 100 Hz θx/θy bandwidth 26 Hz 100 Hz Max sensitivity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='84 optimal actuator/sensor locations are close to the nodal positions of the uncontrolled flexible modes, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Finally, four SISO controllers in the form of (6) are designed for each actively-controlled DOFs, with a target control bandwidth of ωbw = 100 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To evaluate the effectiveness of our proposed design method, a baseline lightweight precision stage as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5 is used for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This baseline stage lightweight stage does not have active control for its flexible modes, and only has the rigid body motions under feedback control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Three actuators and three sensors are used to achieve exact constraint in the stage actuation and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In such a design, the first resonance frequency of the stage structure places an upper limit to the achievable control bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With a target control bandwidth of 50 Hz, the geometric parameters of the baseline stage are designed such that the first resonance frequency of the stage structure is above 250 Hz (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5× of the target bandwidth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Similarly, SISO controllers in the form of (6) are designed for all decoupled DOFs under active control such that the robustness criteria 7 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' II summarizes the performance of the proposed lightweight stage in case study #1 and that of the baseline stage, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 8 shows the loop gains of both proposed and baseline designs in the z-DOF (translation in the vertical direction) and the θx-DOF (pitch direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Comparing the loop frequency responses shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 8a, it can be observed both stages can reach a high control bandwidth of 100 Hz with sufficient stability margins in the z-DOF, and the 250 Hz resonance in the baseline stage is not shown in its z-DOF frequency response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This is because the baseline’s first flexible mode is not controllable or not excitable by the z-axis control loop, and thus this resonance does not limit the stage’s control bandwidth in this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' However, the 250 Hz resonance of the baseline stage can couple in the stage’s z-axis dynamics under imperfect actuator or position placement, and stability issue can arise in the control under such situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In addition, the lightly-damped resonance at 250 Hz in the baseline stage is not actively controlled and thus can be easily excited by disturbances, which can impair the stage’s positioning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Comparing the loop frequency responses shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 8b, it can be observed that the bandwidth of the baseline stage is only 26 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This is primarily due to the 250 Hz resonance peak in the stage dynamics is coupled into the stage’s control in the θx direction with the current actuator/sensor configuration, and thus limits the achievable control bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In contrast, the proposed design can robustly achieve a control bandwidth of 100 Hz since the stage’s first resonance mode at 50Hz is actively controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Finally, comparing the performance shown in Table II, it can be seen that the weight of the proposed stage design is reduced by 85% compared to baseline design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To our understanding, this significant gain in weight reduction is due to the proposed stage is allowing compliance in the first flexible mode, which effectively removes material in the stage structure needed to reinforce the stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This result shows the tremendous potential of the proposed approach in stage acceleration improvement and the power consumption reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In addition, comparing the closed-loop damping performance of the stage’s first resonance mode, it can be seen that the baseline stage’s resonance at 250 Hz is only having a low damping ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='01, which can be excited by external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In contrast, the first flexible mode of the proposed stage is under closed-loop control, which has a bandwidth of 100 Hz and has a closed-loop damping ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This improvement in the structural damping shows the potential of the proposed approach to improve the stage’s positioning accuracy under external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #2 Case study #2 considers a magnetically-levitated planar motion stage as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 9, where four neodymium 100 360 Proposed 180 Baseline 0 180 360 10 102 10310° 06- 180 270 360 Proposed Baseline 450 10 102 103permanent magnet arrays of 60mm × 60 mm × 6 mm are arranged at the corner of the stage to provide both thrust forces for planar motion and the levitation forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The inclusion of the actuator magnets enhances the practical relevance of the case study for wafer positioning application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The vertical-directional levitation forces are assumed to be located at the center of the permanent magnet arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' All other stage geometry parameters are defined in the same way with case study #1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' As stated in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1, the value of ωhigh sets an upper bound for the achievable control bandwidth for the proposed positioning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' However, using a high value of ωhigh can enforce the stage design to increase materials to stiffen the corresponding resonance mode, and thus increase the stage’s weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Therefore, to fully explore the feasible designs set as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 2 and thus to remove possible design conservatism, the value of ωhigh needs to be swept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' It is worth pointing out that the stage geometry design (2) and the actuator/sensor placement design (3)-(4) collaboratively determine the plant dynamics of the positioning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' When conducting a parameter sweep for ωhigh, the actuator/sensor placement problems must also be solved for each stage geometry design for effective design optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To reduce possible design conservatism and thus fully exploit the advantages brought by the flexible mode control, the feasible stage design set for case study #2 is explored as follows: First, a target control bandwidth is selected to be 120 Hz for the positioning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Next, the stage geometry optimization problem (2) is solved with ωhigh = 600 Hz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5× of the target bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Then, the sensor positioning optimization problem (4) is solved with γ = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Note that the actuator’s locations are fixed due to the inclusion of magnet arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With one feasible stage and sensor positioning design provided by the previous steps, we then decrease the value of ωhigh by a constant step δω = 10 Hz and resolve (2) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Assuming δω is sufficiently small, the change in optimal geometric parameters can be assumed continuous, which allows us to use the optimal solution from the previous run as the initial parameters when resolving (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This method effectively reduces the required computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The previous steps are repeated until ωhigh is sufficiently low such that it may be excited by external disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In this case study, the lowest value of ωhigh is selected to be at 300 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In the stage geometry optimization problem, the optimal solutions always have the stage’s second resonance frequency match ωhigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 11 shows the stage geometric parameters and the resultant stage weight and actuator/sensor placement objectives under varying ωhigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' It can be observed that the stage’s weight is reducing as the value of ωhigh decreases, and the value of Jp + Jo (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' sum of objectives of (3)-(4)) is also decreasing along with the reduction of ωhigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' These observations reveal new trade-off between the stage’s achievable control bandwidth and acceleration (assuming constant thrust force generation), which is illustrated by the orange line in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The stage hardware design can be manually made among the optimal designs based on the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' TABLE III CASE #2 OPTIMAL PARAMETERS Baseline Design Proposed Design Stage weight 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='67 kg 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='20 kg First res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 251 Hz 50 Hz 2nd res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1080 Hz 540 Hz z motion bandwidth 25 Hz 120 Hz θx/θy bandwidth 120 Hz 120 Hz Max sensitivity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='94 In this case study, ωhigh = 540 Hz is selected to provide sufficiently high Jp + Jo values while reducing the stage’s weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Compared to the initial stage design using ωhigh = 600 Hz, the stage’s weight is reduced by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Although the improvement is not significant, it is worth pointing out that the geometry optimization of the stage is relatively limited in the current formulation with only five parameters that can be varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' A more significant improvement in the stage’s performance may be expected given increased design flexibility is allowed in the stage structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The resultant stage’s flexible modes are illustrated in the bottom left in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The state-space dynamic model of the stage can be derived for this stage in the same way as discussed in case study #1, and controllers are designed for the decoupled motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' To evaluate the effectiveness of our proposed framework con- sidering actuator weight and constraints, a baseline lightweight stage with same magnet array is simulated for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In the baseline stage, only the rigid-body motions are under active control, and all flexible modes are uncontrolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' With a target bandwidth of 50 Hz, the stage’s geometric parameters are designed to constrain the first resonance frequency above 250 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 9 show the baseline stage design parameters and actuator/sensor location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Three SISO controllers as (6) are designed for all decoupled DOFs in the same way with case study #1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' III summarizes the performance and comparison of the proposed and baseline stage design in case #2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 10 illustrates the loop gains of both proposed and baseline designs in z- and θx-DOFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Comparing the loop frequency responses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 10a, it can be observed that the bandwidth of the baseline design is limited to 25 Hz due to the 251 Hz resonance peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In contrast, the proposed design can reach a bandwidth of 120 Hz with sufficient stability margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 10b shows that both designs can reach a bandwidth of 120 Hz in the θx-DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' This is because the 251 Hz resonance peak in the baseline stage is not excitable by the θx feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' However, similar to the z-DOF in case study #1, stability issue can be caused if the actuator/sensor placement is imperfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Moreover, the lightly-damped 251 Hz resonance mode can be easily excited by external disturbance and thus impair the stage’s positioning precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Finally, Table III shows that the weight of the proposed stage design is reduced by 55% compared to baseline design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The significant improvement for a stage considering the weight of magnet array shows the effectiveness and generality of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' comparing the closed-loop damping performance of stage’s first resonance mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' it can be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑧 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='60 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib width 1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝜃1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib width 2: 𝜃2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib distance: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝜃3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib Height: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝜃5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Height: 𝜃4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='6 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='60 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib distance: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='30 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib width: 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='6 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑥 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑦 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑧 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑎3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Rib height: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='25 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Base height: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3 mm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='𝑠4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Proposed: Practical lightweight stage w/ 1st flexible mode controlled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Baseline: Practical precision stage w/o flexible mode control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1st: 50 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2nd: 540 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3rd: 540 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='4th: 547 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1st: 251Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2nd: 1080 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3rd: 1183 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='4th: 1241 Hz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Flexible Modes: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Flexible Modes: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #2 proposed and baseline stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Both stages consider a permanent magnet array with 60 mm × 60 mm × 6 mm for planar motor force generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 𝑎 𝑏 Frequency [Hz] Frequency [Hz] Magnitude [abs] Phase [deg] 𝟐𝟓𝟏 𝐇𝐳 𝟏𝟐𝟒𝟎 𝐇𝐳 𝟓𝟒𝟕 𝐇𝐳 𝟓𝟒𝟎 𝐇𝐳 𝝓𝒎 = 𝟑𝟕° 𝝎𝒃 = 𝟏𝟐𝟎 𝑯𝒛 𝝎𝒃 = 𝟐𝟓 𝑯𝒛 𝝎𝒃 = 𝟏𝟐𝟎 𝑯𝒛 𝝓𝒎 = 𝟑𝟕° Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Case study #2: comparison of loop gains of the proposed stage design (red solid) and baseline stage design (blue dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (a) z-DOF (translation in the vertical direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (b) θx-DOF (pitch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Rib Distance Rib Height Base Height, Rib Width 1&2 2nd Resonance Frequency [Hz] Length [mm] 2nd Resonance Frequency [Hz] Mass [kg] Jp+Jo [𝐚𝐛𝐬] 𝑎 𝑏 2nd Resonance Frequency [Hz] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (a) Geometric parameter history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' (b) Stage weight and grammian history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' stated that the proposed design is more robust against external disturbances with the first lightly-damped mode at 547 Hz, while that of the baseline stage is at 251 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The comparison indicates the huge potential of our framework to improve both the stage’s acceleration capability and positioning accuracy simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this work, we proposed and evaluated a sequential hardware and controller co-design framework for lightweight precision stages, aiming at enabling designs that can achieve high control bandwidth and high acceleration simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The algorithm of the framework is presented, and the effec- tiveness of the proposed method is demonstrated by numerical simulations using two case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' The significant weight reduction (>55%) and improvement in control bandwidth show the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Future work will consider the experimental evaluations for the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' A fully integrated controller and hardware co-optimization that can better exploit the synergy between hardware and control designs will also be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Albero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Bargiel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Passilly, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Dannberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Stumpf, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Zeitner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Rousselot, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Gastinger, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Gorecki, “Micromachined array- type mirau interferometer for parallel inspection of mems,” Journal of Micromechanics and Microengineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 065005, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Butler, “Position control in lithographic equipment [applications of control],” IEEE Control Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 28–47, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Delissen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Laro, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Kleijnen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van Keulen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Langelaar, “High-precision motion system design by topology optimization consider- ing additive manufacturing,” in 20th Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' of the European Society for Precision Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' and Nanotech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', EUSPEN 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' EUSPEN, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 257–258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Ding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Ding, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Liu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Yang, “An optimal actuator placement method for direct-drive stages to maximize control bandwidth,” in IECON 2020 The 46th Annual Conference of the IEEE Industrial Electronics Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 556–561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Franklin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Powell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Emami-Naeini, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Powell, Feedback control of dynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Prentice hall Upper Saddle River, 2002, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Garcia-Sanz, “Control co-design: an engineering game changer,” Advanced Control for Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' : Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' and Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' e18, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Laro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Boshuisen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van Eijk, “Design and control of over-actuated lightweight 450 mm wafer chuck,” in 2010 ASPE Spring Topical meeting, Cambridge, Massachusetts, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' ASPE, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 141–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 25 20 15 600 550 500 450 400 350 300 70 60 600 550 500 450 400 350 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='05 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='95 600 550 500 450 400 350 3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1 600 550 500 450 400 350 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content='1 600 550 500 450 400 350 300100 Proposed Baseline 0 90 180 270 10 1 102 10 310° Proposed Baseline 10 0 06- U 180 270 101 102 103[8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Oomen, “Advanced motion control for precision mechatronics: Control, identification, and learning of complex systems,” IEEJ Journal of Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 127–140, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Oomen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van Herpen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Quist, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van de Wal, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Bosgra, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Steinbuch, “Connecting system identification and robust control for next-generation motion control of a wafer stage,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' on Ctrl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 102–118, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Ortega and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Rubio, “Systematic design of weighting matrices for the h mixed sensitivity problem,” Journal of Process Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 89–98, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Powell, “A direct search optimization method that models the objective and constraint functions by linear interpolation,” in Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' in opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' and num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Springer, 1994, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 51–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Rankers, “Machine dynamics in mechatronic systems: An engineering approach.” 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van der Veen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Langelaar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van der Meulen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Laro, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Aangenent, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van Keulen, “Integrating topology optimization in precision motion system design for optimal closed-loop control performance,” Mechatronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 47, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 1–13, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van Herpen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Oomen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Kikken, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' van de Wal, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Aangenent, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Steinbuch, “Exploiting additional actuators and sensors for nano- positioning robust motion control,” Mechatronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 619–631, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Wu and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' Zhou, “Control co-design of actively controlled lightweight structures for high-acceleration precision motion systems,” in 2022 American Control Conference (ACC), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} +page_content=' 5320–5327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf'} diff --git a/7dAzT4oBgHgl3EQfgPzo/content/2301.01467v1.pdf b/7dAzT4oBgHgl3EQfgPzo/content/2301.01467v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..80d8a69224e15af99bd183baaa1a1e19d8fb5cba --- /dev/null +++ b/7dAzT4oBgHgl3EQfgPzo/content/2301.01467v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:22cbb578573b9f4866b4f2d4ec09b63a31e49c577fd1e000d365b6b3d24bc3cb +size 1100841 diff --git a/89E1T4oBgHgl3EQfUAPq/content/tmp_files/2301.03086v1.pdf.txt b/89E1T4oBgHgl3EQfUAPq/content/tmp_files/2301.03086v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..47d24253efcf6fe6b4720b140c72e8cf4b589830 --- /dev/null +++ b/89E1T4oBgHgl3EQfUAPq/content/tmp_files/2301.03086v1.pdf.txt @@ -0,0 +1,359 @@ +SYNERGY BETWEEN NP AND HEP RESEARCH GOALS +AND EFFORTS IN FUNDAMENTAL SYMMETRIES AND +INTERACTIONS +Tanmoy Bhattacharya and Rajan Gupta +Los Alamos National Laboratory, T-2, Los Alamos, NM 87545, USA +Kate Scholberg +Department of Physics, Duke University, Durham, NC, 27708, USA +(Dated: January 10, 2023) +The aim of this white paper is to highlight several areas for which the Department +of Energy’s Office of Nuclear Physics has primary stewardship or significant invest- +ment and expertise, and for which there is also significant interest and expertise +within the HEP community. These areas of overlap offer exciting opportunities for +collaboration. +The 2021 Snowmass process brought to the fore a remarkable collaboration between nu- +clear and high energy physicists to elucidate the potential for significant progress through +joint experimental and theoretical efforts in four areas of great interest to the “Fundamental +Symmetries” subprogram of the DOE Office of Science, Nuclear Physics. This collaboration +is evident from the joint authorship of four contributions [1–4], including the associated top- +ical group reports [5–7]. These four areas are: (i) neutrinoless double beta decay (0νββ), (ii) +the neutron electric dipole moment (nEDM), (iii) tests of CKM unitarity through precision +calculations for the extraction of the Vud matrix element, and (iv) lepton-nucleus scattering. +In addition, there are ongoing searches for novel scalar and tensor interactions at the TeV +scale, and NN oscillations for baryon number violation. Conclusive results in any of these +areas could merit the Nobel prize, and will open new directions in beyond-the-standard- +model (BSM) physics. In this short document we summarize the physics goals, the open +challenges and why collaborative efforts by multiple communities would greatly accelerate +progress.1 +Neutrinoless double beta decay [8]: A signal in experiments searching for 0νββ will +be a clear evidence of lepton-number-violation (LNV) and will demonstrate the Majorana +nature of neutrinos. An observation in the next-generation experiments will either identify +the neutrino mass ordering or, if oscillation experiments and advances in cosmology will +show that neutrinos are organized in the “normal ordering”, may provide decisive evidence +of BSM physics, shedding light on the mechanism of neutrino mass generation. There are +several experiments worldwide [9], with the US program stewarded by DOE NP pursuing a +multi-experiment international strategy. +1 We note that the areas highlighted here do not represent all possible opportunities for joint NP/HEP +collaboration. For example, instrumentation development challenges are shared between the communities +as well. +arXiv:2301.03086v1 [hep-ph] 8 Jan 2023 + +2 +0νββ experiments are sensitive to a variety of LNV mechanisms, from the “standard +mechanism” of light-Majorana-neutrino exchange, to contributions mediated by new parti- +cles at the TeV scale, or by weakly coupled light particles such as sterile neutrinos. Identi- +fying the microscopic mechanism behind a signal demands a rich theoretical program over +a wide range of energy scales [4]. At high energy, particle physics models with LNV need to +be further developed, and the complementarity between 0νββ experiments, cosmology, and +searches at present and future high-energy colliders needs to be further explored. +The 0νββ rates induced by light-Majorana exchange or less minimal LNV models can +be computed by using a tower of effective field theories (EFTs), systematically linking the +electroweak to the nuclear scale. Because of the lack of experimental data, the couplings +in the nuclear EFTs need to be determined directly from QCD. Lattice QCD is currently +the only way to systematically and reliably compute the necessary matrix elements. Signif- +icant progress has already been achieved in the calculation of LNV pion couplings [10–14]. +The determination of 0νββ transition operators requires, in addition, nucleon-nucleon LNV +couplings [15], even for light-Majorana-neutrino exchange [16]. Progress on this front will re- +quire further theoretical developments to relate lattice QCD results to physical two-nucleon +matrix elements [17, 18], coupled with computational advances to obtain precise two-nucleon +spectra and matrix elements. +The results from Lattice QCD will then serve as input for many-body calculations of nu- +clear matrix elements (NME) in experimentally relevant isotopes. Here ab initio methods are +starting to appear alongside more traditional phenomenological approaches. If accompanied +by more Lattice QCD and EFT work towards the construction of nuclear interactions and +transition operators at the same order and in the same regularization scheme, these meth- +ods will provide NMEs, and thus 0νββ rates, with a controlled estimate of the theoretical +uncertainties. +Neutron Electric Dipole Moment (nEDM) [3]: One of the profound mysteries of +nature is the lack of matter-antimatter symmetry in the universe, i.e., the almost total +absence of antibaryons. The symmetry between baryons and antibaryons is expected to +have been broken during the evolution of the universe post inflation [19], and requires CP +violation (�� +CP) [20]. If it is in the quark sector, then it has to be larger than that present +in the CKM quark mixing matrix [20]. In that case, weak-scale Baryogenesis is the favored +mechanism for creating the asymmetry [21]. If it is in the neutrino mixing matrix, then +it would be through Leptogenesis [22]. Any �� +CP interaction in the quark sector necessarily +contributes to the nEDM, and most popular BSM models have additional �� +CP that would +give a dn > 10−28 e-cm [23]. +The DOE NP Flagship SNS EDM experiment being built in the US at Oak Ridge is +designed to reach dn ∼ 3×10−28 e-cm [24], and there is a less ambitious effort at LANL [25] +using already proven technology. A successful measurement will give credence to electroweak +baryogenesis [26] as the mechanism for the baryon asymmetry. The value (or the lowering +of the bound in case of a null result) for dn will provide stringent constraints on possible +BSM theories, provided results for the matrix elements of low energy novel �� +CP operators +of dimension six or less can be calculated between the neutron ground state with O(20%) +accuracy. Lattice QCD [3], with effective field theory methods [27] providing the connection +between �� +CP couplings in BSM theories and the low-energy effective �� +CP operators [23, 28, 29], +is attempting to reach this precision over the next decade—there are currently multiple col- +laborations between nuclear and HEP physicists doing the lattice and the EFT calculations +to achieve this. This combined effort is designed to elucidate fundamental symmetries and + +3 +interactions at far beyond the TeV scale, often complementary to the searches at the LHC. +Lepton-Nucleus scattering [2]: The flagship of the HEP program in the US is the +DUNE experiment at Fermilab [30]. It is designed to quantify �� +CP in the neutrino sector. +Since there is �� +CP in the quark sector, it is important to quantify it in the neutrino sector. +Reaching the design precision requires accurate measurements of the ν-nucleus cross-section. +Essential, but the least constrained, ingredients for this are the nucleon axial vector form fac- +tors and transition matrix elements over the range of a few hundred MeV to a couple of GeV +incident neutrino energy, and corrections to these from nuclear effects [2]. This energy range +covers the difficult-to-model quasi-elastic and resonant regions, making the cross-section cal- +culations and Monte Carlo event generators challenging. The most promising approach to +reach the required precision is to use lattice QCD to calculate the axial form factors of the +nucleons and input them into nuclear many-body calculations of the cross-section. +At lower energies (few to few-hundred MeV), neutrino-nucleus interactions are relevant +for astrophysical neutrinos (e.g., solar, atmospheric and supernova neutrinos), and their +understanding is important both for the interpretation of detected signals and for processes +occurring in the sources. Thus, astrophysical signals provide information on both the sources +and the properties of neutrinos themselves. Neutrino cross-section measurements in this +regime are also relevant for the understanding of weak couplings and nuclear transitions, as +well as for searches for BSM physics [31, 32]. Experimental data in this energy regime are +sparse and theoretical understanding is also modest. Joint HEP-NP efforts for both theory +and experiment are underway, for example in the context of experiments at stopped-pion +sources [33–35]. +The planned electron-ion collider (EIC) is designed to provide a detailed 3D tomographic +map of the structure of nucleons in terms of quarks and gluons [36]. Experiments at the +EIC will significantly improve the measurements of electric and magnetic form factors that +also enter the analysis of ν-nucleus interactions. Similarly, improvements in the extraction +of parton distribution functions are of interest to both the NP and HEP communities [37]. +In all three areas, the ongoing collaborative efforts between HEP and NP physicists again +demonstrate that the relevant communities are already working together. +Test of CKM unitarity [38–41]: Understanding of nuclear β decays was instrumental +in the discovery of the Standard Model. Even in the era of the LHC, β decay experiments +can probe BSM physics at scales of ≳ 10 TeV, highly competitive with direct searches. +Tests of unitarity of the first row of the Cabibbo-Kobayashi-Maskawa mixing matrix +are particularly sensitive to these effects. Recently, a revaluation of the “inner radiative +correction” [39, 41–43] has led to a reduction of the uncertainty in the extraction of Vud from +superallowed 0+ → 0+ decays, while progress in lattice QCD resulted in permille accuracy +on the form factor f+(0) and on the ratio fK+/fπ+, needed to extract Vus and Vus/Vud from +kaon decays [44]. These advances revealed a ∼ 3σ tension with the SM [38–41]. Understand- +ing the tension is limited by theoretical errors, with an uncertainty currently dominated by +nuclear corrections in 0+ → 0+ decays [43]. In the near future, measurements of the neutron +lifetime τn with uncertainty ∆τn ∼ 0.1 s, and of ratio λ = gA/gV of the neutron axial and +vector coupling with uncertainty ∆λ/|λ| ∼ 0.03%, will allow for the extraction of Vud from +neutron decay with accuracy comparable to superallowed β decay. Such an extraction will +have the advantage of not being affected by nuclear corrections. Lattice QCD can play an +important role in validating and reducing the error on the radiative corrections to meson +and nucleon decays. The first calculations for pion and kaon decays have already appeared +[45–49], and work on nucleon decay is ongoing. In addition to CKM unitarity, decay spectra + +4 +and correlations also provide tests of new charged-current interactions at scales of about +10 TeV. Lattice QCD has provided precise calculations of the scalar and tensor charges +[44, 50–53], which are needed to convert bounds on the Fierz interference terms onto bounds +on quark-level operators (see below). Comparing experimental extractions and lattice QCD +calculations of the nucleon axial charge gA can provide strong bounds on right-handed +charged currents. With lattice QCD approaching the percent level precision [44, 50, 54, 55], +these comparisons are now limited by electromagnetic corrections [56]. +Novel Scalar and Tensor Interactions at the TeV scale [57]: The two commu- +nities are also working to search for novel scalar and tensor interactions at the TeV scale. +The low-energy approach requires precision measurements of the neutron or nuclear decay +distributions, the calculation of neutron matrix elements using lattice QCD [50, 58], and, +in the case of nuclear decays, the ab initio calculation of nuclear matrix elements. At the +moment, the best bounds on the neutron Fierz interference term, a probe of both scalar +and tensor currents, come from the UCNA and Perkeo III experiments [59, 60], while the +Nab experiment will provide bounds of a few per-mil [61]. The Fierz interference term in- +duced by scalar interactions is sensitively probed in 0+ → 0+ superallowed β decays [43], +while new experiments such as He6-CRES [62, 63] can investigate TeV-scale tensor cur- +rents. At high energy, scalar and tensor interactions affect the high transverse mass tail +of the charged-current Drell-Yan process at the LHC [64]. The latest high-transverse-mass +Drell-Yan dataset from the ATLAS and CMS collaborations [65, 66], which use the full lu- +minosity of the LHC Run II, provide constraints on scalar and tensor interactions that are +very competitive with present and future β decay experiments [67]. +∆B = 2 baryon number violation in NN oscillations: The current limit on the free +neutron oscillation time τNN ≳ 108 sec can be converted into new physics scales of 102 −103 +TeV, and upcoming experiments at the European Spallation Source will probe parameter +space relevant to low-scale baryogenesis scenarios in which the baryon asymmetry is induced +by the B violating decays of new particles that mediate NN oscillations [68, 69]. +Synergy in theoretical methods used: There is close synergy and often collaborations +between NP and HEP physicists exploiting two theoretical tools needed to achieve physics +goals: lattice QCD and effective field theory methods. +Lattice QCD [1, 44, 55]: Large-scale simulations of lattice QCD is the most promising +tool for many of the theoretical calculations of matrix elements needed in all physics drivers. +Explicit examples are connecting the 0νββ, nEDM, neutron decay distributions, and N ¯N +oscillation experiments to BSM physics, and obtaining crucial input in the the extraction +of Vud, Vus and axial vector form factors for lepton-nucleus scattering. The US lattice QCD +communities in both nuclear and high energy physics collaborate and work jointly, for exam- +ple, to procure resources that are then allocated by the umbrella USQCD collaboration [70]. +Many of the teams that receive these awards have members from both communities work- +ing collaboratively on the above six areas and have a history of producing state-of-the-art +results. These efforts would benefit from an increase in computing resources. +Effective Field Theory Methods [71] EFT is a systematic method to express interac- +tions and their couplings arising in BSM theories in terms of low-energy effective operators +composed of quark and gluon fields and organized by symmetries and dimension (roughly +translating into importance). The renomalization group and QCD perturbation theory are +used to run the associated couplings from the high scale to the hadronic scale of a few GeV, +and in the process integrating out the heavy degrees of freedom systematically. Lattice QCD + +5 +can then be used to calculate, incorporating full non-perturbative QCD dynamics, the ma- +trix elements of these effective operators between hadron states. These matrix elements then +provide the connection between low energy experiments and possible fundamental theories, +for example, between bound/value of neutron EDM and allowed values for �� +CP couplings in +BSM theories, i.e., constraining the space of possible theories. +Acknowledgments +T. Bhattacharya and R. Gupta, were partly supported by the U.S. DOE, Office of Sci- +ence, HEP under Contract No. DE-AC52-06NA25396 and the LANL LDRD program. K. +Scholberg is funded by the Department of Energy Office of Science, HEP and the National +Science Foundation. + +6 +[1] USQCD, A. S. Kronfeld et al., (2022), arXiv:2207.07641 [hep-lat]. +[2] L. A. Ruso et al., (2022), arXiv:2203.09030 [hep-ph]. +[3] R. Alarcon et al., Electric dipole moments and the search for new physics, in 2022 Snowmass +Summer Study, 2022, arXiv:2203.08103 [hep-ph]. +[4] V. Cirigliano et al., (2022), arXiv:2203.12169 [hep-ph]. +[5] Z. Davoudi et al., Report of the Snowmass 2021 Topical Group on Lattice Gauge Theory, in +2022 Snowmass Summer Study, 2022, arXiv:2209.10758 [hep-lat]. +[6] A. de Gouvˆea et al., (2022), arXiv:2209.07983 [hep-ph]. +[7] T. Blum et al., (2022), arXiv:2209.08041 [hep-ex]. +[8] B. J. P. Jones, The Physics of Neutrinoless Double Beta Decay: A Primer, in Theoretical +Advanced Study Institute in Elementary Particle Physics: The Obscure Universe: Neutrinos +and Other Dark Matters, 2021, arXiv:2108.09364 [nucl-ex]. +[9] Wikepedia, +Neutrinoless double beta decay, https://www.physics.uoguelph.ca/Nucweb/ +superallowed.html. +[10] A. Nicholson et al., +Phys. Rev. Lett. 121, 172501 (2018), doi:10.1103/PhysRevLett.121. +172501, arXiv:1805.02634 [nucl-th]. +[11] W. Detmold, W. I. Jay, D. J. Murphy, P. R. Oare, and P. E. Shanahan, +(2022), arXiv: +2208.05322 [hep-lat]. +[12] NPLQCD, W. Detmold and D. J. Murphy, (2020), arXiv:2004.07404 [hep-lat]. +[13] X.-Y. Tuo, X. Feng, and L.-C. Jin, Phys. Rev. D 100, 094511 (2019), doi:10.1103/PhysRevD. +100.094511, arXiv:1909.13525 [hep-lat]. +[14] X.-Y. Tuo, X. Feng, and L.-C. Jin, Phys. Rev. D 106, 074510 (2022), doi:10.1103/PhysRevD. +106.074510, arXiv:2206.00879 [hep-lat]. +[15] V. Cirigliano, W. Dekens, J. de Vries, M. L. Graesser, and E. Mereghetti, JHEP 12, 097 +(2018), doi:10.1007/JHEP12(2018)097, arXiv:1806.02780 [hep-ph]. +[16] V. Cirigliano et al., +Phys. Rev. Lett. 120, 202001 (2018), doi:10.1103/PhysRevLett.120. +202001, arXiv:1802.10097 [hep-ph]. +[17] Z. Davoudi and S. V. Kadam, Phys. Rev. Lett. 126, 152003 (2021), doi:10.1103/PhysRevLett. +126.152003, arXiv:2012.02083 [hep-lat]. +[18] Z. Davoudi and S. V. Kadam, Phys. Rev. D 105, 094502 (2022), doi:10.1103/PhysRevD.105. +094502, arXiv:2111.11599 [hep-lat]. +[19] P. Coppi, eConf C040802, L017 (2004). +[20] A. Sakharov, +Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967), doi:10.1070/PU1991v034n05 +ABEH002497. +[21] J. D. Barrow and M. S. Turner, Nature 291, 469 (1981), doi:10.1038/291469a0. +[22] M. Fukugita and T. Yanagida, +Physics Letters B 174, 45 (1986), doi:10.1016/0370- +2693(86)91126-3. +[23] M. Pospelov and A. Ritz, Annals Phys. 318, 119 (2005), doi:10.1016/j.aop.2005.04.002, arXiv: +hep-ph/0504231 [hep-ph]. +[24] The SNS EDM Collaboration, https://nedm.caltech.edu/, A cryogenic EDM Experiment at +Oakridge. +[25] The LANL EDM Collaboration, +https://fsnutown.phy.ornl.gov/fsnufiles/positionpapers/ +nEDM at LANL.pdf, A room temperature EDM Experiment at LANL. + +7 +[26] D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14, 125003 (2012), doi:10.1088/ +1367-2630/14/12/125003, arXiv:1206.2942 [hep-ph]. +[27] C. J. C. Burges and H. J. Schnitzer, +Nucl. Phys. B 228, 464 (1983), doi:10.1016/0550- +3213(83)90555-2. +[28] J. Engel, M. J. Ramsey-Musolf, and U. van Kolck, Prog. Part. Nucl. Phys. 71, 21 (2013), +doi:10.1016/j.ppnp.2013.03.003, arXiv:1303.2371 [nucl-th]. +[29] T. Chupp, P. Fierlinger, M. Ramsey-Musolf, and J. Singh, Rev. Mod. Phys. 91, 015001 (2019), +doi:10.1103/RevModPhys.91.015001, arXiv:1710.02504 [physics.atom-ph]. +[30] The DUNE Collaboration, https://www.dunescience.org/, The Deep Underground Neutrino +Experiment. +[31] P. Huber et al., Snowmass Neutrino Frontier Report, in 2022 Snowmass Summer Study, 2022, +arXiv:2211.08641 [hep-ex]. +[32] A. B. Balantekin et al., (2022), arXiv:2209.06872 [hep-ex]. +[33] D. Akimov et al., The COHERENT Experimental Program, in 2022 Snowmass Summer +Study, 2022, arXiv:2204.04575 [hep-ex]. +[34] J. Asaadi et al., Physics Opportunities in the ORNL Spallation Neutron Source Second Target +Station Era, in 2022 Snowmass Summer Study, 2022, arXiv:2209.02883 [hep-ex]. +[35] R. G. Van de Water et al., LANSCE-PSR Short-Pulse Upgrade for Improved Dark Sector +Particle Searches with the Coherent Captain Mills Experiment, in 2022 Snowmass Summer +Study, 2022, arXiv:2204.01860 [physics.ins-det]. +[36] R. Abdul Khalek et al., Nucl. Phys. A 1026, 122447 (2022), doi:10.1016/j.nuclphysa.2022. +122447, arXiv:2103.05419 [physics.ins-det]. +[37] H.-W. Lin et al., Prog. Part. Nucl. Phys. 100, 107 (2018), doi:10.1016/j.ppnp.2018.01.007, +arXiv:1711.07916 [hep-ph]. +[38] Particle Data Group, R. L. Workman and Others, PTEP 2022, 083C01 (2022), doi:10.1093/ +ptep/ptac097. +[39] C.-Y. Seng, M. Gorchtein, H. H. Patel, and M. J. Ramsey-Musolf, Phys. Rev. Lett. 121, +241804 (2018), doi:10.1103/PhysRevLett.121.241804, arXiv:1807.10197 [hep-ph]. +[40] C. Y. Seng, M. Gorchtein, and M. J. Ramsey-Musolf, +Phys. Rev. D 100, 013001 (2019), +doi:10.1103/PhysRevD.100.013001, arXiv:1812.03352 [nucl-th]. +[41] A. Czarnecki, W. J. Marciano, and A. Sirlin, Phys. Rev. D 100, 073008 (2019), doi:10.1103/ +PhysRevD.100.073008, arXiv:1907.06737 [hep-ph]. +[42] K. Shiells, P. G. Blunden, and W. Melnitchouk, Phys. Rev. D 104, 033003 (2021), doi:10. +1103/PhysRevD.104.033003, arXiv:2012.01580 [hep-ph]. +[43] J. C. Hardy and I. S. Towner, Phys. Rev. C 102, 045501 (2020), doi:10.1103/PhysRevC.102. +045501. +[44] Flavour Lattice Averaging Group (FLAG), Y. Aoki et al., Eur. Phys. J. C 82, 869 (2022), +doi:10.1140/epjc/s10052-022-10536-1, arXiv:2111.09849 [hep-lat]. +[45] X. Feng, M. Gorchtein, L.-C. Jin, P.-X. Ma, and C.-Y. Seng, Phys. Rev. Lett. 124, 192002 +(2020), doi:10.1103/PhysRevLett.124.192002, arXiv:2003.09798 [hep-lat]. +[46] C.-Y. Seng, X. Feng, M. Gorchtein, and L.-C. Jin, Phys. Rev. D 101, 111301 (2020), doi:10. +1103/PhysRevD.101.111301, arXiv:2003.11264 [hep-ph]. +[47] C.-Y. Seng, X. Feng, M. Gorchtein, L.-C. Jin, and U.-G. Meißner, JHEP 10, 179 (2020), +doi:10.1007/JHEP10(2020)179, arXiv:2009.00459 [hep-lat]. +[48] P.-X. Ma, X. Feng, M. Gorchtein, L.-C. Jin, and C.-Y. Seng, +Phys. Rev. D 103, 114503 +(2021), doi:10.1103/PhysRevD.103.114503, arXiv:2102.12048 [hep-lat]. + +8 +[49] J.-S. Yoo, T. Bhattacharya, R. Gupta, S. Mondal, and B. Yoon, (2022), arXiv:2212.12830 +[hep-lat]. +[50] R. Gupta et al., Phys. Rev. D98, 034503 (2018), doi:10.1103/PhysRevD.98.034503, arXiv: +1806.09006 [hep-lat]. +[51] C. Alexandrou et al., Phys. Rev. D 102, 054517 (2020), doi:10.1103/PhysRevD.102.054517, +arXiv:1909.00485 [hep-lat]. +[52] χQCD, L. Liu et al., Phys. Rev. D 104, 094503 (2021), doi:10.1103/PhysRevD.104.094503, +arXiv:2103.12933 [hep-lat]. +[53] Nucleon Matrix Elements (NME), S. Park et al., (2021), arXiv:2103.05599 [hep-lat]. +[54] C. C. Chang et al., Nature 558, 91 (2018), doi:10.1038/s41586-018-0161-8, arXiv:1805.12130 +[hep-lat]. +[55] Flavour Lattice Averaging Group, S. Aoki et al., Eur. Phys. J. C 80, 113 (2020), doi:10.1140/ +epjc/s10052-019-7354-7, arXiv:1902.08191 [hep-lat]. +[56] V. Cirigliano, J. de Vries, L. Hayen, E. Mereghetti, and A. Walker-Loud, Phys. Rev. Lett. +129, 121801 (2022), doi:10.1103/PhysRevLett.129.121801, arXiv:2202.10439 [nucl-th]. +[57] M. Gonz´alez-Alonso, O. Naviliat-Cuncic, and N. Severijns, Prog. Part. Nucl. Phys. 104, 165 +(2019), doi:10.1016/j.ppnp.2018.08.002, arXiv:1803.08732 [hep-ph]. +[58] T. Bhattacharya et al., +Phys.Rev. D85, 054512 (2012), doi:10.1103/PhysRevD.85.054512, +arXiv:1110.6448 [hep-ph]. +[59] H. Saul et al., Phys. Rev. Lett. 125, 112501 (2020), doi:10.1103/PhysRevLett.125.112501, +arXiv:1911.01766 [nucl-ex]. +[60] UCNA, X. Sun et al., Phys. Rev. C 101, 035503 (2020), doi:10.1103/PhysRevC.101.035503, +arXiv:1911.05829 [nucl-ex]. +[61] Fry, J. et al., EPJ Web Conf. 219, 04002 (2019), doi:10.1051/epjconf/201921904002. +[62] V. Cirigliano et al., (2019), arXiv:1907.02164 [nucl-ex]. +[63] A. Garcia et al., http://faculty.washington.edu/agarcia3/Chirality-flipping/. +[64] V. Cirigliano, M. Gonzalez-Alonso, and M. L. Graesser, JHEP 02, 046 (2013), doi:10.1007/ +JHEP02(2013)046, arXiv:1210.4553 [hep-ph]. +[65] ATLAS, G. Aad et al., Phys. Rev. D 100, 052013 (2019), doi:10.1103/PhysRevD.100.052013, +arXiv:1906.05609 [hep-ex]. +[66] CMS, A. Tumasyan et al., +JHEP 07, 067 (2022), doi:10.1007/JHEP07(2022)067, arXiv: +2202.06075 [hep-ex]. +[67] L. Allwicher, D. A. Faroughy, F. Jaffredo, O. Sumensari, and F. Wilsch, +(2022), arXiv: +2207.10756 [hep-ph]. +[68] K. S. Babu, R. N. Mohapatra, and S. Nasri, Phys. Rev. Lett. 97, 131301 (2006), doi:10.1103/ +PhysRevLett.97.131301, arXiv:hep-ph/0606144. +[69] C. Grojean, B. Shakya, J. D. Wells, and Z. Zhang, Phys. Rev. Lett. 121, 171801 (2018), +doi:10.1103/PhysRevLett.121.171801, arXiv:1806.00011 [hep-ph]. +[70] The USQCD Collaboration in Lattice QCD, https://www.usqcd.org/, . +[71] W. Buchmuller and D. Wyler, +Nucl. Phys. B 268, +621 (1986), +doi:10.1016/0550- +3213(86)90262-2. + diff --git a/89E1T4oBgHgl3EQfUAPq/content/tmp_files/load_file.txt b/89E1T4oBgHgl3EQfUAPq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..252f2842ad90b3bac9f63a67f8ce7d2461a32d8b --- /dev/null +++ b/89E1T4oBgHgl3EQfUAPq/content/tmp_files/load_file.txt @@ -0,0 +1,686 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf,len=685 +page_content='SYNERGY BETWEEN NP AND HEP RESEARCH GOALS AND EFFORTS IN FUNDAMENTAL SYMMETRIES AND INTERACTIONS Tanmoy Bhattacharya and Rajan Gupta Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' T-2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' NM 87545,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' USA Kate Scholberg Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Duke University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Durham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 27708,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' USA (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 2023) The aim of this white paper is to highlight several areas for which the Department of Energy’s Office of Nuclear Physics has primary stewardship or significant invest- ment and expertise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' and for which there is also significant interest and expertise within the HEP community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' These areas of overlap offer exciting opportunities for collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The 2021 Snowmass process brought to the fore a remarkable collaboration between nu- clear and high energy physicists to elucidate the potential for significant progress through joint experimental and theoretical efforts in four areas of great interest to the “Fundamental Symmetries” subprogram of the DOE Office of Science, Nuclear Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' This collaboration is evident from the joint authorship of four contributions [1–4], including the associated top- ical group reports [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' These four areas are: (i) neutrinoless double beta decay (0νββ), (ii) the neutron electric dipole moment (nEDM), (iii) tests of CKM unitarity through precision calculations for the extraction of the Vud matrix element, and (iv) lepton-nucleus scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' In addition, there are ongoing searches for novel scalar and tensor interactions at the TeV scale, and NN oscillations for baryon number violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Conclusive results in any of these areas could merit the Nobel prize, and will open new directions in beyond-the-standard- model (BSM) physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' In this short document we summarize the physics goals, the open challenges and why collaborative efforts by multiple communities would greatly accelerate progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1 Neutrinoless double beta decay [8]: A signal in experiments searching for 0νββ will be a clear evidence of lepton-number-violation (LNV) and will demonstrate the Majorana nature of neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' An observation in the next-generation experiments will either identify the neutrino mass ordering or, if oscillation experiments and advances in cosmology will show that neutrinos are organized in the “normal ordering”, may provide decisive evidence of BSM physics, shedding light on the mechanism of neutrino mass generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' There are several experiments worldwide [9], with the US program stewarded by DOE NP pursuing a multi-experiment international strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 1 We note that the areas highlighted here do not represent all possible opportunities for joint NP/HEP collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' For example, instrumentation development challenges are shared between the communities as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='03086v1 [hep-ph] 8 Jan 2023 2 0νββ experiments are sensitive to a variety of LNV mechanisms, from the “standard mechanism” of light-Majorana-neutrino exchange, to contributions mediated by new parti- cles at the TeV scale, or by weakly coupled light particles such as sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Identi- fying the microscopic mechanism behind a signal demands a rich theoretical program over a wide range of energy scales [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' At high energy, particle physics models with LNV need to be further developed, and the complementarity between 0νββ experiments, cosmology, and searches at present and future high-energy colliders needs to be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The 0νββ rates induced by light-Majorana exchange or less minimal LNV models can be computed by using a tower of effective field theories (EFTs), systematically linking the electroweak to the nuclear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Because of the lack of experimental data, the couplings in the nuclear EFTs need to be determined directly from QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lattice QCD is currently the only way to systematically and reliably compute the necessary matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Signif- icant progress has already been achieved in the calculation of LNV pion couplings [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The determination of 0νββ transition operators requires, in addition, nucleon-nucleon LNV couplings [15], even for light-Majorana-neutrino exchange [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Progress on this front will re- quire further theoretical developments to relate lattice QCD results to physical two-nucleon matrix elements [17, 18], coupled with computational advances to obtain precise two-nucleon spectra and matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The results from Lattice QCD will then serve as input for many-body calculations of nu- clear matrix elements (NME) in experimentally relevant isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Here ab initio methods are starting to appear alongside more traditional phenomenological approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' If accompanied by more Lattice QCD and EFT work towards the construction of nuclear interactions and transition operators at the same order and in the same regularization scheme, these meth- ods will provide NMEs, and thus 0νββ rates, with a controlled estimate of the theoretical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Neutron Electric Dipole Moment (nEDM) [3]: One of the profound mysteries of nature is the lack of matter-antimatter symmetry in the universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', the almost total absence of antibaryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The symmetry between baryons and antibaryons is expected to have been broken during the evolution of the universe post inflation [19], and requires CP violation (�� CP) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' If it is in the quark sector, then it has to be larger than that present in the CKM quark mixing matrix [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' In that case, weak-scale Baryogenesis is the favored mechanism for creating the asymmetry [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' If it is in the neutrino mixing matrix, then it would be through Leptogenesis [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Any �� CP interaction in the quark sector necessarily contributes to the nEDM, and most popular BSM models have additional �� CP that would give a dn > 10−28 e-cm [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The DOE NP Flagship SNS EDM experiment being built in the US at Oak Ridge is designed to reach dn ∼ 3×10−28 e-cm [24], and there is a less ambitious effort at LANL [25] using already proven technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' A successful measurement will give credence to electroweak baryogenesis [26] as the mechanism for the baryon asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The value (or the lowering of the bound in case of a null result) for dn will provide stringent constraints on possible BSM theories, provided results for the matrix elements of low energy novel �� CP operators of dimension six or less can be calculated between the neutron ground state with O(20%) accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lattice QCD [3], with effective field theory methods [27] providing the connection between �� CP couplings in BSM theories and the low-energy effective �� CP operators [23, 28, 29], is attempting to reach this precision over the next decade—there are currently multiple col- laborations between nuclear and HEP physicists doing the lattice and the EFT calculations to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' This combined effort is designed to elucidate fundamental symmetries and 3 interactions at far beyond the TeV scale, often complementary to the searches at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lepton-Nucleus scattering [2]: The flagship of the HEP program in the US is the DUNE experiment at Fermilab [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' It is designed to quantify �� CP in the neutrino sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Since there is �� CP in the quark sector, it is important to quantify it in the neutrino sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Reaching the design precision requires accurate measurements of the ν-nucleus cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Essential, but the least constrained, ingredients for this are the nucleon axial vector form fac- tors and transition matrix elements over the range of a few hundred MeV to a couple of GeV incident neutrino energy, and corrections to these from nuclear effects [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' This energy range covers the difficult-to-model quasi-elastic and resonant regions, making the cross-section cal- culations and Monte Carlo event generators challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The most promising approach to reach the required precision is to use lattice QCD to calculate the axial form factors of the nucleons and input them into nuclear many-body calculations of the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' At lower energies (few to few-hundred MeV), neutrino-nucleus interactions are relevant for astrophysical neutrinos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', solar, atmospheric and supernova neutrinos), and their understanding is important both for the interpretation of detected signals and for processes occurring in the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Thus, astrophysical signals provide information on both the sources and the properties of neutrinos themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Neutrino cross-section measurements in this regime are also relevant for the understanding of weak couplings and nuclear transitions, as well as for searches for BSM physics [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Experimental data in this energy regime are sparse and theoretical understanding is also modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Joint HEP-NP efforts for both theory and experiment are underway, for example in the context of experiments at stopped-pion sources [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The planned electron-ion collider (EIC) is designed to provide a detailed 3D tomographic map of the structure of nucleons in terms of quarks and gluons [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Experiments at the EIC will significantly improve the measurements of electric and magnetic form factors that also enter the analysis of ν-nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Similarly, improvements in the extraction of parton distribution functions are of interest to both the NP and HEP communities [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' In all three areas, the ongoing collaborative efforts between HEP and NP physicists again demonstrate that the relevant communities are already working together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Test of CKM unitarity [38–41]: Understanding of nuclear β decays was instrumental in the discovery of the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Even in the era of the LHC, β decay experiments can probe BSM physics at scales of ≳ 10 TeV, highly competitive with direct searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Tests of unitarity of the first row of the Cabibbo-Kobayashi-Maskawa mixing matrix are particularly sensitive to these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Recently, a revaluation of the “inner radiative correction” [39, 41–43] has led to a reduction of the uncertainty in the extraction of Vud from superallowed 0+ → 0+ decays, while progress in lattice QCD resulted in permille accuracy on the form factor f+(0) and on the ratio fK+/fπ+, needed to extract Vus and Vus/Vud from kaon decays [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' These advances revealed a ∼ 3σ tension with the SM [38–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Understand- ing the tension is limited by theoretical errors, with an uncertainty currently dominated by nuclear corrections in 0+ → 0+ decays [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' In the near future, measurements of the neutron lifetime τn with uncertainty ∆τn ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1 s, and of ratio λ = gA/gV of the neutron axial and vector coupling with uncertainty ∆λ/|λ| ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='03%, will allow for the extraction of Vud from neutron decay with accuracy comparable to superallowed β decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Such an extraction will have the advantage of not being affected by nuclear corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lattice QCD can play an important role in validating and reducing the error on the radiative corrections to meson and nucleon decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The first calculations for pion and kaon decays have already appeared [45–49], and work on nucleon decay is ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' In addition to CKM unitarity, decay spectra 4 and correlations also provide tests of new charged-current interactions at scales of about 10 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lattice QCD has provided precise calculations of the scalar and tensor charges [44, 50–53], which are needed to convert bounds on the Fierz interference terms onto bounds on quark-level operators (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Comparing experimental extractions and lattice QCD calculations of the nucleon axial charge gA can provide strong bounds on right-handed charged currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' With lattice QCD approaching the percent level precision [44, 50, 54, 55], these comparisons are now limited by electromagnetic corrections [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Novel Scalar and Tensor Interactions at the TeV scale [57]: The two commu- nities are also working to search for novel scalar and tensor interactions at the TeV scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The low-energy approach requires precision measurements of the neutron or nuclear decay distributions, the calculation of neutron matrix elements using lattice QCD [50, 58], and, in the case of nuclear decays, the ab initio calculation of nuclear matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' At the moment, the best bounds on the neutron Fierz interference term, a probe of both scalar and tensor currents, come from the UCNA and Perkeo III experiments [59, 60], while the Nab experiment will provide bounds of a few per-mil [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The Fierz interference term in- duced by scalar interactions is sensitively probed in 0+ → 0+ superallowed β decays [43], while new experiments such as He6-CRES [62, 63] can investigate TeV-scale tensor cur- rents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' At high energy, scalar and tensor interactions affect the high transverse mass tail of the charged-current Drell-Yan process at the LHC [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The latest high-transverse-mass Drell-Yan dataset from the ATLAS and CMS collaborations [65, 66], which use the full lu- minosity of the LHC Run II, provide constraints on scalar and tensor interactions that are very competitive with present and future β decay experiments [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' ∆B = 2 baryon number violation in NN oscillations: The current limit on the free neutron oscillation time τNN ≳ 108 sec can be converted into new physics scales of 102 −103 TeV, and upcoming experiments at the European Spallation Source will probe parameter space relevant to low-scale baryogenesis scenarios in which the baryon asymmetry is induced by the B violating decays of new particles that mediate NN oscillations [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Synergy in theoretical methods used: There is close synergy and often collaborations between NP and HEP physicists exploiting two theoretical tools needed to achieve physics goals: lattice QCD and effective field theory methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lattice QCD [1, 44, 55]: Large-scale simulations of lattice QCD is the most promising tool for many of the theoretical calculations of matrix elements needed in all physics drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Explicit examples are connecting the 0νββ, nEDM, neutron decay distributions, and N ¯N oscillation experiments to BSM physics, and obtaining crucial input in the the extraction of Vud, Vus and axial vector form factors for lepton-nucleus scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The US lattice QCD communities in both nuclear and high energy physics collaborate and work jointly, for exam- ple, to procure resources that are then allocated by the umbrella USQCD collaboration [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Many of the teams that receive these awards have members from both communities work- ing collaboratively on the above six areas and have a history of producing state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' These efforts would benefit from an increase in computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Effective Field Theory Methods [71] EFT is a systematic method to express interac- tions and their couplings arising in BSM theories in terms of low-energy effective operators composed of quark and gluon fields and organized by symmetries and dimension (roughly translating into importance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' The renomalization group and QCD perturbation theory are used to run the associated couplings from the high scale to the hadronic scale of a few GeV, and in the process integrating out the heavy degrees of freedom systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lattice QCD 5 can then be used to calculate, incorporating full non-perturbative QCD dynamics, the ma- trix elements of these effective operators between hadron states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' These matrix elements then provide the connection between low energy experiments and possible fundamental theories, for example, between bound/value of neutron EDM and allowed values for �� CP couplings in BSM theories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', constraining the space of possible theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Acknowledgments T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Bhattacharya and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gupta, were partly supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' DOE, Office of Sci- ence, HEP under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' DE-AC52-06NA25396 and the LANL LDRD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Scholberg is funded by the Department of Energy Office of Science, HEP and the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 6 [1] USQCD, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Kronfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='07641 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ruso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2022), arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='09030 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Alarcon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Electric dipole moments and the search for new physics, in 2022 Snowmass Summer Study, 2022, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='08103 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Cirigliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2022), arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='12169 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [5] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Davoudi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Report of the Snowmass 2021 Topical Group on Lattice Gauge Theory, in 2022 Snowmass Summer Study, 2022, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='10758 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' de Gouvˆea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='07983 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='08041 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [8] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jones, The Physics of Neutrinoless Double Beta Decay: A Primer, in Theoretical Advanced Study Institute in Elementary Particle Physics: The Obscure Universe: Neutrinos and Other Dark Matters, 2021, arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='09364 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [9] Wikepedia, Neutrinoless double beta decay, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='uoguelph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='ca/Nucweb/ superallowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Nicholson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 121, 172501 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 172501, arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='02634 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Detmold, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jay, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Murphy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Oare, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Shanahan, (2022), arXiv: 2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='05322 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [12] NPLQCD, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Detmold and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Murphy, (2020), arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='07404 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [13] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Tuo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Feng, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 100, 094511 (2019), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='094511, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='13525 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [14] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Tuo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Feng, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 106, 074510 (2022), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='074510, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='00879 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [15] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Cirigliano, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Dekens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' de Vries, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Graesser, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Mereghetti, JHEP 12, 097 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1007/JHEP12(2018)097, arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='02780 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [16] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Cirigliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 120, 202001 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 202001, arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='10097 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [17] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Davoudi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Kadam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 126, 152003 (2021), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='152003, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='02083 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [18] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Davoudi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Kadam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 105, 094502 (2022), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 094502, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='11599 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Coppi, eConf C040802, L017 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Sakharov, Pisma Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 5, 32 (1967), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1070/PU1991v034n05 ABEH002497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Barrow and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Turner, Nature 291, 469 (1981), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1038/291469a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Fukugita and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Yanagida, Physics Letters B 174, 45 (1986), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/0370- 2693(86)91126-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Pospelov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ritz, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 318, 119 (2005), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='002, arXiv: hep-ph/0504231 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [24] The SNS EDM Collaboration, https://nedm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='edu/, A cryogenic EDM Experiment at Oakridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [25] The LANL EDM Collaboration, https://fsnutown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='ornl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='gov/fsnufiles/positionpapers/ nEDM at LANL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='pdf, A room temperature EDM Experiment at LANL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 7 [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Morrissey and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ramsey-Musolf, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 14, 125003 (2012), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1088/ 1367-2630/14/12/125003, arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='2942 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Burges and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Schnitzer, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' B 228, 464 (1983), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/0550- 3213(83)90555-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Engel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ramsey-Musolf, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' van Kolck, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 71, 21 (2013), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='ppnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='003, arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='2371 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [29] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Chupp, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Fierlinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ramsey-Musolf, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Singh, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 91, 015001 (2019), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='015001, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='02504 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='atom-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [30] The DUNE Collaboration, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='dunescience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='org/, The Deep Underground Neutrino Experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Huber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Snowmass Neutrino Frontier Report, in 2022 Snowmass Summer Study, 2022, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='08641 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Balantekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='06872 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [33] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Akimov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', The COHERENT Experimental Program, in 2022 Snowmass Summer Study, 2022, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='04575 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [34] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Asaadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Physics Opportunities in the ORNL Spallation Neutron Source Second Target Station Era, in 2022 Snowmass Summer Study, 2022, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='02883 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Van de Water et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', LANSCE-PSR Short-Pulse Upgrade for Improved Dark Sector Particle Searches with the Coherent Captain Mills Experiment, in 2022 Snowmass Summer Study, 2022, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='01860 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Abdul Khalek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' A 1026, 122447 (2022), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='nuclphysa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 122447, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='05419 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 100, 107 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='ppnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='007, arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='07916 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [38] Particle Data Group, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Workman and Others, PTEP 2022, 083C01 (2022), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1093/ ptep/ptac097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Seng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gorchtein, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Patel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ramsey-Musolf, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 121, 241804 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='241804, arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='10197 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Seng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gorchtein, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ramsey-Musolf, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 100, 013001 (2019), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='013001, arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='03352 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Czarnecki, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Marciano, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Sirlin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 100, 073008 (2019), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/ PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='073008, arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='06737 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [42] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Shiells, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Blunden, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Melnitchouk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 104, 033003 (2021), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='033003, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='01580 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Hardy and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Towner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' C 102, 045501 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 045501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [44] Flavour Lattice Averaging Group (FLAG), Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' C 82, 869 (2022), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1140/epjc/s10052-022-10536-1, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='09849 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [45] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Feng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gorchtein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ma, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Seng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 124, 192002 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='192002, arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='09798 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [46] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Seng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Feng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gorchtein, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 101, 111301 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='111301, arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='11264 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [47] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Seng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Feng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gorchtein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jin, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Meißner, JHEP 10, 179 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1007/JHEP10(2020)179, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='00459 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [48] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Feng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gorchtein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Seng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 103, 114503 (2021), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='114503, arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='12048 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 8 [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Yoo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Bhattacharya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gupta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Mondal, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Yoon, (2022), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='12830 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [50] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D98, 034503 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='034503, arXiv: 1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='09006 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [51] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Alexandrou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 102, 054517 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='054517, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='00485 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [52] χQCD, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 104, 094503 (2021), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='094503, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='12933 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [53] Nucleon Matrix Elements (NME), S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2021), arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='05599 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [54] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Nature 558, 91 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1038/s41586-018-0161-8, arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='12130 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [55] Flavour Lattice Averaging Group, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' C 80, 113 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1140/ epjc/s10052-019-7354-7, arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='08191 [hep-lat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [56] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Cirigliano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' de Vries, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Hayen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Mereghetti, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Walker-Loud, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 129, 121801 (2022), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='121801, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='10439 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [57] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gonz´alez-Alonso, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Naviliat-Cuncic, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Severijns, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 104, 165 (2019), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='ppnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='002, arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='08732 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [58] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D85, 054512 (2012), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='054512, arXiv:1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='6448 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [59] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Saul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 125, 112501 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='112501, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='01766 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [60] UCNA, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' C 101, 035503 (2020), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='035503, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='05829 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [61] Fry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', EPJ Web Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 219, 04002 (2019), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1051/epjconf/201921904002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [62] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Cirigliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', (2019), arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='02164 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [63] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', http://faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='edu/agarcia3/Chirality-flipping/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [64] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Cirigliano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Gonzalez-Alonso, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Graesser, JHEP 02, 046 (2013), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1007/ JHEP02(2013)046, arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='4553 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [65] ATLAS, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D 100, 052013 (2019), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='052013, arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='05609 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [66] CMS, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Tumasyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=', JHEP 07, 067 (2022), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1007/JHEP07(2022)067, arXiv: 2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='06075 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [67] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Allwicher, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Faroughy, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Jaffredo, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Sumensari, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Wilsch, (2022), arXiv: 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='10756 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [68] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Babu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Mohapatra, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Nasri, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 97, 131301 (2006), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/ PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='131301, arXiv:hep-ph/0606144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [69] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Grojean, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Shakya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Wells, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Zhang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' 121, 171801 (2018), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='171801, arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='00011 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [70] The USQCD Collaboration in Lattice QCD, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='usqcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='org/, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' [71] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Buchmuller and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Wyler, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content=' B 268, 621 (1986), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} +page_content='1016/0550- 3213(86)90262-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E1T4oBgHgl3EQfUAPq/content/2301.03086v1.pdf'} diff --git a/8NE4T4oBgHgl3EQf2w06/vector_store/index.pkl b/8NE4T4oBgHgl3EQf2w06/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..af58bc761372d69a2d61b2ba122b2da3476fbf28 --- /dev/null +++ b/8NE4T4oBgHgl3EQf2w06/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d4f50f811fee4c7d713f043bc513e014ed3f317cb4b82a2fd6793a9463e3358b +size 83644 diff --git a/99A0T4oBgHgl3EQfO__U/content/tmp_files/2301.02170v1.pdf.txt b/99A0T4oBgHgl3EQfO__U/content/tmp_files/2301.02170v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e860ed5acf7478f1d7faac8133ebff4cdce448c --- /dev/null +++ b/99A0T4oBgHgl3EQfO__U/content/tmp_files/2301.02170v1.pdf.txt @@ -0,0 +1,3809 @@ +arXiv:2301.02170v1 [math.AP] 4 Jan 2023 +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +IN FINITE-STRAIN ELASTOPLASTICITY +ELISA DAVOLI, CHIARA GAVIOLI, AND VALERIO PAGLIARI +Abstract. This work is devoted to the analysis of the interplay between internal variables and +high-contrast microstructure in inelastic solids. As a concrete case-study, by means of variational +techniques, we derive a macroscopic description for an elastoplastic medium. Specifically, we +consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft +inclusions. We study the Γ-convergence of the related energy functionals as the periodicity tends +to zero. The main challenge is posed by the lack of coercivity brought about by the degeneracy +of the material properties in the soft part. We prove that the Γ-limit, which we compute with +respect to a suitable notion of convergence, is the sum of the contributions resulting from each of +the two components separately. Eventually, convergence of the energy minimizing configurations +is obtained. +2020 Mathematics Subject Classification: 49J45; 74B20; 74C15; 74E30; 74Q05. +Keywords and phrases: finite-strain elastoplasticity, Γ-convergence, homogenization, high-contrast, +two-scale convergence. +Contents +1. +Introduction +2 +Outline +4 +2. +Mathematical setting and results +4 +3. +Preliminaries +10 +3.1. +A decomposition lemma +10 +3.2. +A couple of tools to deal with periodic heterogeneous media +11 +3.3. +Two-scale convergence and the unfolding method +12 +3.4. +Homogenization of connected media in finite plasticity +13 +3.5. +Finsler structure on SL(3) +15 +4. +Compactness and splitting +16 +5. +Γ-limit of the soft component +21 +5.1. +The limiting functional +21 +5.2. +Lower bound for the elastic energy +24 +5.3. +Upper bound for the elastic energy +27 +5.4. +Proof of Proposition 2.10 +28 +6. +Conclusions and a variant +29 +6.1. +Proof of Theorem 2.7 and convergence of minimum problems +29 +6.2. +A non degenerate upper bound for the soft component +31 +6.3. +A variant with plastic dissipation +36 +Acknowledgements +37 +References +38 +Date: January 6, 2023. +1 + +2 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +1. Introduction +The present paper is concerned with the variational analysis of some integral functionals that +model the stored energy of materials governed by finite-strain elastoplasticity with hardening. +Our goal is to derive, by means of Γ-convergence, the effective macroscopic energy of a special +class of heterogeneous materials, those with a so called high-contrast microstructure. The inter- +est in such media stems from the experimental observation of an infinite number of band gaps +in their mechanical behavior. In other words, high-contrast materials exhibit infinitely many +interval of frequencies in which wave propagation is not allowed. This, in turn, makes them +extremely interesting for possible cloaking applications. Some recent ones in civil engineering, +e.g. in seismic waves cloaking, and in the modeling of advanced sensor and actuator devices call +for advancements in the mathematical modeling of those classes of high-contrast materials that +have not been fully studied yet, like the ones we consider here. +The mathematical literature on high-contrast materials is vast. +To keep our presentation +concise, we only point out that, besides results for stratified elastoplastic composites [14, 15, 22, +25], the only additional available contributions in the inelastic setting concern the study of brittle +fracture problems [5, 6, 42]. For the modeling of nonlinear elastic high-contrast composites we +single out the works [10, 13]. +When undertaking the analysis of high-contrast media beyond the elastic purview, hurdles +are posed by the mathematical treatment of possible internal variables and dissipative effects, as +well as by their interplay with the high-contrast microstructure. In this paper we initiate such +task by focusing on the case-study of finite elastoplasticity (see, e.g., [37]). At this first stage we +neglect both the difficulties due to possible lack of coercivity for the dissipative effects and those +associated with time evolution. Thus, we focus here on a static model for a single time-step with +a global regularization on the gradient of the plastic strain, and leave the analysis of different +regimes and the passage to the limit in the quasistatic evolutions for future investigations. +The present study grounds on a previous result that we obtained in [24], where we addressed +the static homogenization of elastoplastic microstructures in the large strain regime. +As in +that work, our starting point is the description of the medium at the microscopic level. We let +Ω ⊂ R3 be an open, bounded, connected set with Lipschitz boundary, and we suppose it to be +the reference configuration of an elastoplastic body that exhibits the following microstructure: +denoting by ε > 0 the microscale, we suppose that a stiff perforated matrix Ω1 +ε sits in Ω and +that its pores are filled by soft inclusions, which form the set Ω0 +ε (see Figure 2). Let us denote +by SL(3) the group of 3 × 3 real matrices with determinant equal to 1. When the matrix and +the inclusions exhibit the same plastic-hardening H, the functionals encoding the stored energy +associated with the deformation y ∈ W 1,2(Ω; R3) and the plastic strain P ∈ W 1,q(Ω; SL(3)) read +Jε(y, P) := +ˆ +Ω0ε +W 0 +ε +� +ε∇y(x)P −1(x) +� +dx + +ˆ +Ω1ε +W 1 � +∇y(x)P −1(x) +� +dx ++ +ˆ +Ω +H +�P(x) +� dx + +ˆ +Ω +|∇P(x)|q dx, +(1.1) +where {W 0 +ε }ε>0 and W 1 are, respectively, the elastic energy densities of the inclusions and of +the matrix. +Let us briefly comment on some modeling choices underlying position (1.1). The factor ε +multiplying the argument of W 0 +ε encodes the high-contrast between the two components, and +it results in a loss of coercivity in the problem. From a modeling perspective, this heuristically +means that very large deformations of the inclusions are allowed or, in other words, that the + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +3 +inclusions are very soft – whence the expression high-contrast to describe the difference between +the phases. +As for the hardening term, note that also additional hardening variables have been taken into +account in the literature, see [38, 39] for a modeling overview. Here, to the purpose of putting +the high-contrast behavior to the foreground, we give up full generality and restrict ourselves to +the case in which only a hardening dependence on the plastic strain is given. A discussion on +alternative modeling choices is also presented in Remark 2.3. +Our main result describes the asymptotics of the functionals Jε, and it is presented in Theo- +rem 2.7. The precise mathematical framework of our analysis is described in Section 2, where +further details on the definitions and on the roles of the terms in Jε may be found. +We work under the classical assumption that the elastic behavior of our sample Ω is indepen- +dent of preexistent plastic distortions. Then, the deformation gradient ∇y associated with any +deformation y: Ω → R3 of the body decomposes into an elastic strain and a plastic one. In the +framework of linearized elastoplasticity the decomposition would take an additive form. In the +case at stake, that of finite plasticity [34, 36, 39, 38], the existence of an intermediate configura- +tion determined by purely plastic distortions is instead assumed. It is then supposed that elastic +deformations are applied on such intermediate configuration. Mathematically, these hypotheses +amount to a multiplicative decomposition of the gradient of any deformation y ∈ W 1,2(Ω; R3): +∇y(x) = Fel(x)P(x) +for a. e. x ∈ Ω, +for a suitable elastic strain Fel ∈ L2(Ω; R3×3) and a plastic strain P ∈ L2(Ω; SL(3)). On one +hand, such multiplicative structure has recently found an atomistic validation in the framework +of crystal plasticity by means of a discrete-to-continuum analysis [18, 19]. On the other hand, +alternative models for finite plasticity have been proposed. However, since a discussion of fine +modeling issues goes beyond the scopes of our work, we do not dwell here on a comparison of the +various modeling theories. We refer the reader interested on this point to, e.g., [23, 31, 32, 40]. +We finally comment on the regularizing term in ∇P in the energy (1.1). As mentioned before, +at this stage we assume it to provide coercivity of the energy with respect to the plastic-strain +variables on the whole set Ω. From a modeling point of view, we note that this regularization +is common in engineering models, for it prevents the formation of microstructures, see [7, 11]. +Alternative higher order regularizations are discussed in [27]. +Let us conclude our introduction with a few words on the proofs. A delicate point is choosing +a convergence that ensures effective compactness properties. Indeed, the fact that the energy +contributions in the soft inclusions are evaluated in terms of ε∇y leads to a loss of coercivity for +which compactness in classical weak Sobolev topologies is prevented. On the other hand, arguing +with strong two-scale convergence of the gradients, as in [13] does not guarantee convergence of +minimizers of Jε to minimizers of the limiting functional. To cope with this difficulty, we adapt +the approach in [26] and introduce an ad hoc notion of convergence for deformations, to which +we refer as convergence in the sense of extensions. Roughly speaking, a sequence of deformations +converges in the sense of extensions if it is bounded in L2 and can be extended in W 1,2 in such +a way that the extensions are weakly compact in the Sobolev sense, cf. +Definition 2.4 and +Remarks 2.5 and 2.6 for the precise definition and some basic properties. For the plastic strains, +we argue instead with the weak convergence in W 1,q. This choice is motivated by the fact that +sequences of deformations and plastic strains with uniformly bounded energies are precompact +with respect to the above topology. Thus our Γ-convergence analysis directly entails convergence +of minimizers. We observe that this result easily extends to functionals which take into account +also plastic dissipation. On this point we refer to Section 6. + +4 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +The strategy relies on extension results on perforated domains, on two-scale convergence and +periodic unfolding techniques, as well as on equiintegrability arguments to control the behavior +of the microstructure close to the boundary of the set Ω. A key-step is a splitting procedure +that allows to treat the soft and the stiff parts separately. +Outline. The setup of our analysis and the main result, Theorem 2.7, are presented in Section 2. +Section 3 contains some preliminary useful facts. In Section 4 we discuss the equicoercivity of the +energy functionals under consideration and the splitting procedure. The asymptotic behavior +of the soft inclusions is characterized in Section 5. The ground is then laid for the proof of the +Theorem 2.7, which is contained in Section 6 together with a variant including plastic dissipation +and a comparison with an aforementioned result from [13]. +2. Mathematical setting and results +Hereafter, Ω is an open, bounded, and connected set with Lipschitz boundary in R3. Working +in the 3-dimensional space is not essential, and our analysis can be easily adapted to the setting +of Rd with d = 2 or d > 3. Real-valued 3 × 3 and 3 × 3 × 3 tensors are denoted by R3×3 and +by R3×3×3, respectively. We adopt the symbol I for the identity matrix. With | · | we denote +indiscriminately the Euclidean norms in R3, R3×3 and R3×3×3. To deal with plastic strains, we +recall the classical notation +SL(3) := {F ∈ R3×3 : det F = 1}. +If A ⊂ R3 is a measurable set, we will denote by L3(A) its three-dimensional Lebesgue +measure. +A fundamental role in our study is played by the following notion of variational convergence, +see the monograph [21] for a thorough treatment: +Definition 2.1. Let X be a set endowed with a notion of convergence. We say that the family +{Gε}, with Gε : X → [−∞, +∞], Γ-converges as ε → 0 to G : X → [−∞, +∞] if for all x ∈ X +and all infinitesimal sequences {εk}k∈N the following holds: +(1) for every sequence {xk}k∈N ⊂ X such that xk → x, we have +G(x) ≤ lim inf +k→+∞ Gεk(xk); +(2) there exists a sequence {xk}k∈N ⊂ X such that xk → x and +lim sup +k→+∞ +Gεk(xk) ≤ G(x). +When X is equipped with a topology τ, we write e.g. Γ(τ)-convergence to stress what the +underlying convergence for sequences in X is. In what follows, for notational convenience, we +indicate the dependence on εk by means of the subscript k alone, e.g. Jk := Jεk. +Our aim is to study elastoplastic media with high-contrast periodic microstructure in the case +of soft inclusions inserted in a perforated stiff matrix. To describe the geometry in precise terms, +let Q := [0, 1)3 be the periodicity cell, and let Q0 ⊂ Q be an open set such that Q1 := Q \ Q0 is +connected and has a Lipschitz boundary (see Figure 1). The set Ω, which represents the region +of space occupied by the composite, is then subdivided by means of the sets +Ω0 +ε := +� +t∈Tε +ε(t + Q0), +with Tε := {t ∈ Z3 : ε(t + Q0) ⊂ Ω}, +(2.1) +Ω1 +ε := Ω \ Ω0ε, +(2.2) + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +5 +Figure 1. The periodicity cell Q and its partition into the soft inclusion Q0 +(white) and the stiff matrix Q1 (gray). +Q0 +Q0 +Q1 +Figure 2. The microstructure of the composite in Ω. The soft inclusions that +form Ω0 +ε correspond to the white holes, while the grey region represents the matrix +Ω1 +ε. +ε +which stand respectively for the collection of the inclusions and for the matrix (see Figure 2). +We also define the Q-periodic set +E1 := +� +t∈Z3 +(t + Q1), +(2.3) +where we say that a set E ⊂ R3 is Q-periodic if E + t = E for all t ∈ Z3. Note that the set +Ω1 +ε is connected and Lipschitz, because (2.1) ensures that the inclusions are well separated from +∂Ω. Our assumptions allow for some flexibility on the geometry of the inclusions, which could +for instance form interconnected fibers (see Figure 3). +Our Γ-convergence result deals with the asymptotic behavior, as ε tends to 0, of the family +{Jε} defined by (1.1). Before stating the result, we collect the hypotheses we use in the following +lines. +The elastic energy density of the stiff matrix W 1 : R3×3 → [0, +∞] satisfies the following: +E1: It is 2-coercive and has at most quadratic growth, i.e., there exist 0 < c1 ≤ c2 such that +for all F ∈ R3×3 +c1|F|2 ≤ W 1(F) ≤ c2 +� +|F|2 + 1 +� +. + +6 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Figure 3. In the 3-dimensional space, interconnected soft fibers do not discon- +nect the matrix. A simple case is depicted here: the cylindrical perforation Q0 +runs through the periodicity cell and its complement Q1 is connected. +Q0 +Q1 +E2: It is 2-Lipschitz: there exists c3 > 0 such that for all F1, F2 ∈ R3×3 +|W 1(F1) − W 1(F2)| ≤ c3 (1 + |F1| + |F2|) |F1 − F2|. +The assumptions on the soft densities W 0 +ε : R3×3 → [0, +∞] are analogous: +E3: There exist 0 < c1 ≤ c2 such that for all F ∈ R3×3, and all ε > 0, +c1|F|2 ≤ W 0 +ε (F) ≤ c2 +� +|F|2 + 1 +� +. +E4: There exists c3 > 0 such that for all F1, F2 ∈ R3×3, and all ε > 0, +���W 0 +ε (F1) − W 0 +ε (F2) +��� ≤ c3 (1 + |F1| + |F2|) |F1 − F2|. +E5: There exists W 0 : R3×3 → [0, +∞] such that for all F ∈ R3×3 +lim +ε→0 W 0 +ε (F) = W 0(F). +Remark 2.2. The function W 0 possesses the same growth and regularity properties of W 0 +ε . +Our assumptions rule out non-impenetrability constraints at the level of the energy. A blow +up of the energy on matrices with non-positive determinant is desirable from a modeling point of +view, but it is at the same time very hard to be handled with in the context of homogenization. +Frame indifference is instead compatible with our hypotheses and we point out that, up to a +normalization, we can require all energy densities to vanish on the identity. +We list next the assumptions on the hardening H : R3×3 → [0, +∞]. +H1: Assume that a Finsler structure on SL(3) is assigned. H(F) is finite if and only if F ∈ K, +where K ⊂ SL(3) is a geodesically convex, compact neighborhood of I. +H2: The restriction of H to K is Lipschitz continuous. +The requirement that K is geodesically convex with respect to the Finsler structure assigned +on SL(3) is the crucial ingredient to invoke [24, Theorem 2.2], which in our context is employed +to capture the asymptotic behaviour of the stiff matrix, see Theorem 3.8. We refer to [24] for a +discussion on the role of the Finsler geometry for the homogenization of elastoplastic media, and +to Subsection 3.5 for a summary of the tools from that theory that we need here. In particular, +the existence of a set K complying with H1 is settled in Corollary 3.11 below. +Requirement H1 prescribes that the effective domain of H coincides with a compact set K +containing I. It follows then there exists cK > 0 such that +|F| + |F −1| ≤ cK +for every F ∈ K, +(2.4) + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +7 +because SL(3) is by definition well separated from 0. As a consequence, plastic strains with +finite hardening are uniformly bounded in L∞, and, in particular, we infer that for any F ∈ K +and G ∈ R3×3 +|G| = +���GF −1F +��� ≤ cK +���GF −1��� . +(2.5) +Remark 2.3. Note that in principle it would be reasonable to suppose that the soft and the stiff +components feature different hardening behaviors. For instance, it could be imposed that the +soft hardening is evaluated on an ε-rescaling of the plastic stress, thus replicating the structure +of the elastic contribution. As the only available tool to deal with periodic homogenization at +finite strains is [24, Theorem 2.2], we leave such scenarios for possible future investigation and +we restrain ourselves to a simpler setting, namely we choose to model both hardening terms +by a single function satisfying H1 and H2. We point out that under these assumptions making +a distinction between Hi = Hi(P), i = 0, 1 would not require any substantial change in our +approach, therefore we dispense with it. Qualitatively, keeping the soft hardening contribution +of order 1 amounts to the situation in which, for small ε, elastic deformations of a much larger +magnitude than the plastic ones are allowed. +We can now state the homogenization result for high-contrast elastoplastic media. Since we +want our analysis to yield convergence of minima and minimizers of Jε to the ones of the limiting +energy, we need to introduce a convergence that is compliant with the degeneracy of the soft +inclusions. For shortness, we refer to it as convergence in the sense of extensions, even though +the name is not at all standard. +Definition 2.4. Let {εk} be an infinitesimal sequence. We say that {yk} ⊂ W 1,2(Ω; R3) con- +verges to y ∈ W 1,2(Ω; R3) in the sense of extensions with respect to the scales εk if the following +hold: +(1) {yk} is bounded in L2(Ω; R3); +(2) there exists a sequence {˜yk} ⊂ W 1,2(Ω; R3) such that yk = ˜yk in Ω1 +k := Ω1 +εk and ˜yk ⇀ y +weakly in W 1,2(Ω; R3). +Remark 2.5. Let ˜yk = ˜y′ +k a.e. in Ω1 +k. Let as well ˜yk ⇀ y and ˜y′ +k ⇀ y′ weakly in W 1,2(Ω; R3). +Then, recalling (2.2)–(2.3) and observing that Ω ∩ εkE1 ⊂ Ω1 +k, +0 = +lim +k→+∞ +ˆ +Ω1 +k +|˜yk − ˜y′ +k| dx ≥ +lim +k→+∞ +ˆ +Ω +χεkE1(x)|˜yk − ˜y′ +k| dx = c +ˆ +Ω +|y − y′| dx, +for a constant c > 0. From this, we conclude that y = y′ a.e. in Ω. In particular, if the limit in +the sense of extensions exists, then it is unique. +Remark 2.6. By the definition of Ω1 +k, there exists a tubular neighborhood O of ∂Ω such that +Ω1 +k ∩ O ≡ Ω ∩ O. Therefore, if y and ˜y coincide in Ω1 +k, their traces on ∂Ω are also equal. +The asymptotic behavior of the family {Jε} with respect to the notion of convergence that +we have just introduced is described in the next theorem: +Theorem 2.7. Let {W 1} and {W 0 +ε } satisfy E1–E5, and let H satisfy H1–H2. For all y ∈ +L2(Ω; R3) and P ∈ Lq(Ω; SL(3)) there exists +J (y, P) := Γ- lim +ε→0 Jε(y, P), +where the underlying convergences are the one in the sense of extensions and the uniform one, +respectively for the first and for the second argument. The Γ-limit is characterized as follows: +J (y, P) = J 0(0, P) + J 1(y, P), + +8 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +where +J 0(y, P) := + + + + + + + + + + + +L3(Q0) +ˆ +Ω +� +Q′W 0�∇y(x), P −1(x) +�+H +�P(x) +�� +dx +if y = 0 and P ∈ W 1,q(Ω; K), ++∞ +otherwise in L2(Ω; R3) × Lq(Ω; SL(3)), +(2.6) +and +J 1(y, P) := + + + + + + + + + + + +ˆ +Ω +� +� +W 1 +hom +�∇y(x), P(x) +� + L3(Q1)H +�P(x) +� + |∇P(x)|q� +dx +if (y, P) ∈ W 1,2(Ω; R3) × W 1,q(Ω; K), ++∞ +otherwise in L2(Ω; R3) × Lq(Ω; SL(3)). +(2.7) +Here, for F, G ∈ R3×3, +Q′W 0(F, G) := inf +�ˆ +Q +W 0��F + ∇v(z) +�G +� +dz : v ∈ W 1,2 +0 (Q; R3) +� +, +(2.8) +while +� +W 1 +hom(F, G) := +lim +λ→+∞ +1 +λ3 inf +� ˆ +(0,λ)3∩E1 W 1��F + ∇y(x) +�G−1� +dx : y ∈ W 1,2 +0 ((0, λ)3; R3) +� +. +The formula defining Q′W 0 provides a variant of the classical quasiconvex envelope of W 0. +We refer to Section 5 for further discussion on this point. +Remark 2.8. In principle, it cannot be excluded that some nontrivial energy densities W 0 +ε +do not contribute to the elastic homogenized energy, in the sense that, when finite, for the +corresponding J 0 we have +J 0(0, P) = L3(Q0) +ˆ +Ω +H +�P(x) +� dx. +As an instance of this phenomenon, we consider the following example. For any F ∈ R3×3, we +let W 0 +ε (F) = W 0(F) := |F|2. Conditions E3–E5 are satisfied by definition. Since for any fixed +G ∈ R3×3 the function F �→ W 0 +G(F) := W 0(FG) is convex, it is, in particular, also quasiconvex. +Hence, Q′W 0(0, G) = W 0(0, G) = W 0(0) = 0. +As a byproduct of our asymptotic analysis, we are in a position to infer convergence of the +minimum problems associated with the energy functionals and of the related (quasi) minimizers. +Corollary 2.9. Let the same assumptions and notation of Theorem 2.7 hold, and let {(yk, Pk)} ⊂ +W 1,2 +0 (Ω; R3) × W 1,q(Ω; SL(3)) be a sequence of almost minimizers, that is, +lim +k→+∞ +� +Jk(yk, Pk) − inf Jk(y, P) +� += 0, +where the infimum is taken over W 1,2 +0 (Ω; R3) × W 1,q(Ω; SL(3)). Then, there exists a minimizer +(y, P) ∈ W 1,2 +0 (Ω; R3) × W 1,q(Ω; SL(3)) of J such that, up to subsequences, yk → y in the sense +of extensions and Pk → P uniformly. Moreover, +inf Jk → min J . + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +9 +The proof of Theorem 2.7 consists of three steps. First, we study the compactness properties +of sequences {(yε, Pε)} satisfying supε Jε(yε, Pε) ≤ C, and characterize their limits. Second, we +show that the two components of the material can be studied independently. Finally, we perform +the analysis of each single component. In view of this approach, it is useful to introduce the +functionals that account for the two different contributions, namely +E0 +ε (y, P) := +ˆ +Ω +χ0 +ε(x) +� +W 0 +ε +� +ε∇y(x)P −1(x) +� ++ H +�P(x) +�� +dx, +(2.9) +E1 +ε (y, P) := +ˆ +Ω +χ1 +ε(x) +� +W 1 � +x, ∇y(x)P −1(x) +� ++ H +�P(x) +�� +dx, +(2.10) +where, for i = 0, 1, χi +ε(x) denotes the characteristic function of Ωi +ε, i.e. χi +ε(x) = 1 if x ∈ Ωi +ε and +χi +ε(x) = 0 otherwise. We also decompose the functional Jε accordingly: +Jε = J 0 +ε + J 1 +ε , +with +J 0 +ε (y, P) := +� +E0 +ε (y, P) +if (y, P) ∈ W 1,2 +0 (Ω0 +ε; R3) × W 1,q(Ω; K), ++∞ +otherwise in L2(Ω; R3) × Lq(Ω; SL(3)), +(2.11) +J 1 +ε (y, P) := +� +E1 +ε (y, P) + ∥∇P∥q +Lq(Ω;R3×3×3) +if (y, P) ∈ W 1,2(Ω; R3) × W 1,q(Ω; K), ++∞ +otherwise in L2(Ω; R3) × Lq(Ω; SL(3)). +(2.12) +In contrast to J 1 +ε (y, P), whose asymptotic behavior is derived from [24, Theorem 2.2], the soft +part requires a dedicated treatment. This happens already in the setting of nonlinear elasticity +(see [13]). Recall the topology τ in (3.5). We obtain the following: +Proposition 2.10. Let (v, P) ∈ W 1,2 +0 (Ω; R3) × W 1,q(Ω; SL(3)). For an infinitesimal sequence +{εk}, consider J 0 +k and J 0 as in (2.11) and (2.6), respectively. +(1) For every sequence {(vk, Pk)} ⊂ W 1,2 +0 (Ω0 +k; R3) × W 1,q(Ω; SL(3)) such that (εkvk, Pk) +τ→ +(v, P) we have +J 0(v, P) ≤ lim inf +k→+∞ J 0 +k (vk, Pk), +provided that {vk} is bounded in L2(Ω; R3) and that {εk∇vk} is 2-equiintegrable. +(2) There exists a sequence {vk} ⊂ W 1,2 +0 (Ω0 +k; R3) such that εkvk → v in L2(Ω; R3) and that +lim sup +k→+∞ +J 0 +k (vk, Pk) ≤ J 0(v, P), +provided Pk → P uniformly. +In the statement above, the space W 1,2 +0 (Ω0 +ε; R3) is regarded for each ε as a subset of W 1,2(Ω; R3) +by extending its elements to 0 on Ω1 +ε. +Remark 2.11. Let Ω ⊂ R3 be bounded Lipschitz domain and, for p > 1, let us consider the +local integral functionals on W 1,p(Ω; R3) +v �→ +ˆ +Ω +Wk(∇v) dx. +If the energy densities {Wk} satisfy standard p-growth conditions, as a consequence of Rellich- +Kondrachov theorem, the Γ-limits with respect to the strong Lp-convergence and with respect +to the weak W 1,p-convergence coincide (if they exist). + +10 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +For the sequence of functionals +v �→ +ˆ +Ω +Wk(εk∇v) dx, +(2.13) +again under standard growth conditions for {Wk}, the analysis is more delicate. The natural +bound that follows from the p-coercivity is ∥εk∇vk∥Lp ≤ C, and it suggests the use of weak two- +scale convergence (see Subsection 3.3). However, this estimate alone is not enough to deduce +convergence of the sequence {vk}: a further control on the ε-difference quotients is required to +guarantee that a two-scale variant of Rellich-Kondrachov theorem holds (see [44, Theorem 4.4]). +In other words, in our degenerate setting, compactness of sequences of gradients, say {εk∇vk}, +does not bring compactness of {vk}. This explains why in Proposition 2.10 we need to require +a bound also for ∥vk∥L2 in order to establish the lower limit inequality. +We note incidentally that, by means of Lemma 3.6(4) below, it can be shown that the Γ-limit +of the functionals (2.13) with respect to the strong two-scale convergence in Lp of {vk} is the same +as the one computed by combining the latter convergence and the weak two-scale convergence of +{εk∇vk}. Those are not suitable choices for our goals, though, because, as we commented above, +they do not match the natural compactness of the problem. This explains why in [13], where +strong two-scale convergence is considered, the asymptotic behavior of minimum problems is not +immediately determined by the Γ-convergence (see [13, Sec. 10]). We also refer to the Appendix +for a comparison between our findings and the ones in [13]. +3. Preliminaries +We gather in this section the technical tools to be employed in the sequel. +3.1. A decomposition lemma. In our analysis of heterogeneous media it will be often desir- +able to disregard the energy contributions arising from the region close to ∂Ω, for the composite +fails to be periodic there (recall positions (2.1)–(2.2)). To this aim, it is natural to resort to +p-equiintegrability arguments, because such boundary strip has small measure. We recall that a +family C ⊂ Lp(Ω; R3) is said to be p-equiintegrable if for all δ > 0 there exists m > 0 such that +sup +u∈C +ˆ +E +|u|p dx < δ +whenever E ⊂ Ω satisfies L3(E) < m. +The ensuing lemma grants that for any bounded sequence in Lp we can always find another +one which is p-equiintegrable and “does not differ too much” from the given one. +Lemma 3.1 (Theorem 2.20 in [3]; see also Lemma 1.2 in [29]). Let Ω be as in Section 2. For any +sequence {vk} ⊂ W 1,2(Ω; R3) such that vk ⇀ v weakly in W 1,2(Ω; R3) there exist a subsequence +{kj} and a sequence {uj} ⊂ W 1,2(Ω; R3) satisfying the following: +(1) uj ⇀ v weakly in W 1,2(Ω; R3); +(2) uj = v in a neighborhood of ∂Ω; +(3) {∇uj} is 2-equiintegrable; +(4) limj→+∞ L3({x ∈ Ω : vkj(x) ̸= uj(x)}) = 0. +Point (4) yields limj→+∞ L3({∇vkj ̸= ∇uj}) = 0, because by standard properties of Sobolev +functions (see e.g. [30, Lemma 7.7]) the inclusion {vkj ̸= uj} ⊇ {∇vkj ̸= ∇uj} holds true. + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +11 +3.2. A couple of tools to deal with periodic heterogeneous media. The periodic geome- +try of the composite calls for an extension result for Sobolev maps on perforated domains. Since +the perforations of the matrix are well detached from the boundary, by applying [9, Lemma B.7] +the following can be proved: +Lemma 3.2 (Lemma 8 in [13]). Let Ω be open and bounded, and let Ω1 +ε be as in Section 1. +There exists a linear and continuous extension operator +Tε: W 1,2(Ω1 +ε; R3) → W 1,2(Ω; R3) +such that for all y ∈ W 1,2(Ω1 +ε; R3) +Tεy = y +a. e. in Ω1 +ε, +∥Tεy∥L2(Ω;R3) ≤ c ∥y∥L2(Ω1ε;R3), +∥∇(Tεy)∥L2(Ω;R3×3) ≤ c ∥∇y∥L2(Ω1ε;R3×3), +where c is independent of ε and Ω. +Remark 3.3. Even though the lemma above is a classical result, it is worth clarifying the way +we employ it. +In the sequel, we always work with sequences which are already defined on the whole Ω. When +we apply Lemma 3.2 to such a sequence, say {yε} ⊂ W 1,2(Ω; R3), it is tacitly understood that the +functions that are extended are the restrictions yε⌞Ω1 +ε. So, in a sense, the process modifies yε on +the region occupied by the soft inclusions rather than extending it. Note that the modification +is a true one, because Tε cannot be the identity. The two crucial points for our analysis are that +(1) if {yε⌞Ω1 +ε} and {∇yε⌞Ω1 +ε} are bounded in L2, then {Tεyε} is bounded in W 1,2(Ω; R3); +(2) if {yε} is bounded in L2(Ω; R3) and {∇yε} is a 2-equiintegrable sequence, then {∇(Tεyε)} +is 2-equiintegrable as well. +The second point follows from the construction of Tε, which is modeled on the proof of [9, +Lemma B.8] by patching together the extensions from W 1,2(Q1; R3) to W 1,2(Q; R3) given by [9, +Lemma B.7] via partitions of unity (this is also the reason why the constant c above depends +only on Q1 and Q). The extensions in [9, Lemma B.7] preserve equiintegrability, because they +rely on the classical reflection procedure. +The first application of the extension lemma is the following Poincaré inequality on periodic +heterogeneous media (cf. formula (4.5) in [2] where, however, the proof is not provided). +Proposition 3.4. Let Ω, Ω0 +ε and Ω1 +ε be as in Section 1. There exists a constant c independent +of ε, and such that for every y ∈ W 1,2 +0 (Ω; R3) +∥y∥L2(Ω;R3) ≤ c +� +ε∥∇y∥L2(Ω0ε;R3×3) + ∥∇y∥L2(Ω1ε;R3×3) +� +. +Proof. For ε fixed, we use the extension operator Tε from Lemma 3.2 to obtain +∥y∥L2 ≤ ∥y − Tεy∥L2 + ∥Tεy∥L2 += ∥y − Tεy∥L2(Ω0ε) + ∥Tεy∥L2. +(3.1) +Observe that Tεy ∈ W 1,2 +0 (Ω; R3), as Tεy = y a. e. in Ω1 +ε and there exists a tubular neighborhood +O of ∂Ω such that Ω1 +ε ∩ O ≡ Ω ∩ O. Then, by the standard Poincaré’s inequality, +∥Tεy∥L2 ≤ c∥∇(Tεy)∥L2 ≤ c∥∇y∥L2(Ω1ε). +(3.2) + +12 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Observe that y − Tεy ∈ W 1,2 +0 (Ω0 +ε; R3) as well. In view of the periodic structure of Ω0 +ε and of +Poincaré inequality on each cube, we infer +∥y − Tεy∥2 +L2(Ω0ε) = +� +t∈Tε +∥y − Tεy∥2 +L2(ε(t+D0)) += +� +t∈Tε +ε3 +ˆ +D0 +|y(ε(t + z)) − Tεy(ε(t + z))|2 dz +≤ c +� +t∈Tε +ε5 +ˆ +D0 +|∇(y − Tεy)(ε(t + z))|2 dz += cε2∥∇(y − Tεy)∥2 +L2(Ω0ε), +where c depends only on D0. By applying again Lemma 3.2 we find +∥y − Tεy∥L2(Ω0ε) ≤ c +� +ε∥∇y∥L2(Ω0ε) + ∥∇y∥L2(Ω1ε) +� +. +This, together with (3.1) and (3.2), yields the result. +□ +3.3. Two-scale convergence and the unfolding method. From a mathematical perspective, +the high-contrast structure of the functional Jε results in the absence of uniform bounds in L2 +for sequences with equibounded energy; indeed, only bounds on {ε∇yεP −1 +ε +} are available. Such +degenerate bounds are conveniently dealt with by means of two-scale convergence [2, 41], whose +definition we recall next. Hereafter, the subscript per denotes spaces of Q-periodic functions, +e.g. +W 1,2 +per(R3) := {u ∈ W 1,2 +loc (R3) : u(x + t) = u(x) a.e. for all t ∈ Z3}. +Definition 3.5. Let {εk} ⊂ (0, +∞) be infinitesimal. A sequence {yk} ⊂ L2(Ω; R3) weakly two- +scale converges in L2 to a function y ∈ L2(Ω; L2 +per(R3; R3)) if for every v ∈ L2(Ω; Cper(R3; R3)) +lim +k→+∞ +ˆ +Ω +yk(x) · v +� +x, x +εk +� +dx = +ˆ +Ω +ˆ +Q +y(x, z) · v(x, z) dz dx. +A sequence {yk} ⊂ L2(Ω; R3) strongly two-scale converges in L2 to y ∈ L2(Ω; L2 +per(R3; R3)) if +yk +2⇀ y in L2 and ∥yk∥L2(Ω;R3) → ∥y∥L2(Ω×Q;R3). We use the notations yk +2⇀ y and yk +2→ y for +the weak and strong two-scale convergence, respectively. +Recalling that for i = 0, 1 χi +k(x) = 1 if x ∈ Ωi +k and χi +k(x) = 0 otherwise, an example of strong +two-scale convergence is provided by the sequences {χ0 +k} and {χ1 +k}. Indeed, +χi +k +2→ χi +strongly two-scale in L2, +(3.3) +where χi(x, z) := χQi(z) for all (x, z) ∈ Ω × Q. +We collect in the next lemma some basic properties of two-scale convergence which we will +resort to in the following. Proofs and more details can be found in [2, 43, 44]. +Lemma 3.6. Let {εk} ⊂ (0, +∞) be infinitesimal and consider {yk} ⊂ L2(Ω; R3). +(1) If {yk} is weakly two-scale convergent, then it is bounded in L2(Ω; R3); conversely, if +{yk} is bounded in L2(Ω; R3), then it admits a weakly two-scale convergent subsequence. +(2) If yk +2⇀ y weakly two-scale in L2, then yk ⇀ +´ +Q y( · , z) dz weakly in L2(Ω; R3). +(3) If yk +2⇀ y weakly two-scale in L2 and if {uk} ⊂ L2(Ω; R3) converges to u strongly two- +scale in L2, then ykuk +2⇀ yu weakly two-scale in L2. + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +13 +(4) Suppose that {yk} ⊂ W 1,2(Ω; R3) and that {yk} and {εk∇yk} are bounded in L2. Then, +there exists y ∈ L2(Ω; W 1,2 +per(R3; R3)) such that, up to subsequences, yk +2⇀ y and εk∇yk +2⇀ +∇zy weakly two-scale in L2. +Two-scale convergence in L2 can be related to L2 convergence by means of unfolding operator, +which, for ε > 0, is the map Sε : L2(Ω) → L2(R3; L2 +per(R3; R3)) defined as +Sεy(x, z) := ˆy +� +ε +�x +ε +� ++ εz +� +, +(3.4) +where ˆy denotes the extension of y by 0 outside Ω. +Lemma 3.7. If {yε} ⊂ L2(Ω; R3) is bounded, the following hold: +(1) yε +2⇀ y weakly two-scale in L2 if and only if Sεyε ⇀ y weakly in L2(R3 × Q; R3); +(2) yε +2→ y strongly two-scale in L2 if and only if Sεyε → y strongly in L2(R3 × Q; R3). +In addition, if {yε} is 2-equiintegrable, the family of unfoldings {Sεyε} is as well 2-equiintegrable +on R3 × Q. Lastly, if y ∈ W 1,2(Ω; R3), then +Sε(ε∇y)(x, z) = ∇z(Sεy)(x, z). +For a proof of Lemma 3.7 and for further reading on the unfolding operator we refer to +[43, 44, 16, 17]. +3.4. Homogenization of connected media in finite plasticity. We present a variant of [24, +Theorem 2.2] that is instrumental in dealing with the analysis of the stiff matrix. Its proof is an +adaptation of the one in [24], the most substantial difference being the use of [9, Theorem 19.1] +instead of [9, Theorem 14.5]. +We work in the space W 1,2(Ω; R3)×W 1,q(Ω; SL(3)) endowed with the topology τ characterized +by +(yk, Pk) τ→ (y, P) +if and only if + + + +yk → y +strongly in L2(Ω; R3), +Pk → P +uniformly. +(3.5) +Theorem 3.8. Let E be an open and connected set that is Q-periodic and that has Lipschitz +boundary. For every (y, P) ∈ W 1,2(Ω; R3) × W 1,q(Ω; K), let +� +W(x, F) := χE(x)W 1(F), +�H(x, P) := χE(x)H(P), +and define +Fε(y, P) := +ˆ +Ω +� +W +�x +ε , ∇y(x)P −1(x) +� +dx + +ˆ +Ω +�H +�x +ε , P(x) +� +dx + +ˆ +Ω +|∇P(x)|q dx, +(3.6) +which we extend by setting +Fε(y, P) = +∞ +on +�L2(Ω; R3) × Lq(Ω; SL(3)) +� \ +�W 1,2(Ω; R3) × W 1,q(Ω; K) +�. +If W 1 and H satisfy E1–E2 and H1–H2, respectively, then for all (y, P) ∈ L2(Ω; R3)×Lq(Ω; SL(3)) +the Γ-limit +F(y, P) := Γ(τ)- lim +ε→0 Fε(y, P) +exists and we have that +F(y, P) = + + + + + + + + + + + +ˆ +Ω +� +� +Whom(∇y(x), P(x)) + �Hhom(P(x)) + |∇P(x)|q� +dx +if (y, P) ∈ W 1,2(Ω; R3) × W 1,q(Ω; K), ++∞ +otherwise in L2(Ω; R3) × Lq(Ω; SL(3)), + +14 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +where � +Whom : R3×3 × K → [0, +∞) and �Hhom : K → [0, +∞) are defined as +� +Whom(F, G) := +lim +λ→+∞ +1 +λ3 inf +�ˆ +(0,λ)3 +� +W +�x, (F + ∇y(x))G−1� dx : y ∈ W 1,2 +0 ((0, λ)3; R3) +� +, +�Hhom(F) := +ˆ +Q +�H(z, F) dz. +We observe that the theorem above is similar in spirit to homogenization results for perforated +domains. The case at stake is however different, in that later we will deal with functions defined +on the nonperforated domain Ω. This makes the analysis simpler because it spares us the need +of extending SL(3)-valued Sobolev maps. +Thanks to Lemma 3.1, we are able to refine the choice of recovery sequences for F. This will +come in handy in the proof of Corollary 2.9. +Corollary 3.9. Under the same assumptions of Theorem 3.8, for any (y, P) ∈ W 1,2(Ω; R3) × +W 1,q(Ω; K) there exists a recovery sequence (yk, Pk) for F(y, P) satisfying the following: +(1) yk ⇀ y weakly in W 1,2(Ω; R3); +(2) yk = y in a neighborhood of ∂Ω; +(3) {∇yk} is 2-equiintegrable. +Proof. Let {(wk, Pk)} be a recovery sequence for F(y, P) as provided by Theorem 3.8. We apply +Lemma 3.1 to {wk}. We deduce the existence of sequences {kj} and {uj} ⊂ W 1,2(Ω; R3) such +that the sequence defined by +yk := +� +uj +if k = kj for some j ∈ N, +y +otherwise +satisfies properties (1)–(3) and (yk, Pk) τ→ (y, P). Moreover +lim +j→+∞ L3(Nj) = 0, +where Nj := {x ∈ Ω : wkj(x) ̸= uj(x)}. +We are left to prove that {(yk, Pk)} satisfies the upper limit inequality. Loosely speaking, +this is a consequence of the fact that passing to a 2-equiintegrable sequence “does not increase +the energy”. +Upon passing to a subsequence, which we do not relabel, we can assume that +{Fk(yk, Pk)} is convergent. We provisionally focus just on the elastic and hardening parts of +the energy Fkj. It holds +ˆ +Ω +� +W +� +x +εkj +, ∇wkjP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx += +ˆ +Nj +� +W +� +x +εkj +, ∇wkjP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx ++ +ˆ +Ω\Nj +� +W +� +x +εkj +, ∇ujP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx +≥ +ˆ +Ω\Nj +� +W +� +x +εkj +, ∇ujP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx, + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +15 +so that +lim sup +j→+∞ +ˆ +Ω +� +W +� +x +εkj +, ∇wkjP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx +≥ lim sup +j→+∞ +ˆ +Ω\Nj +� +W +� +x +εkj +, ∇ujP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx += lim sup +j→+∞ +ˆ +Ω +� +W +� +x +εkj +, ∇ujP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx, +where the equality follows from the growth condition E1 and from the 2-equiintegrability of +{∇uj} (recall that supk∈N ∥P −1 +k ∥∞ ≤ C), together with the boundedness of H. +Therefore, +coming back to the full functional Fkj, +lim +j→+∞ Fkj(wkj, Pkj) +≥ lim sup +j→+∞ +ˆ +Ω +� +W +� +x +εkj +, ∇wkjP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx + lim inf +j→+∞ +ˆ +Ω +|∇Pkj|q dx +≥ lim sup +j→+∞ +ˆ +Ω +� +W +� +x +εkj +, ∇ujP −1 +kj +� ++ H +� +x +εkj +, Pkj +�� +dx + lim inf +j→+∞ +ˆ +Ω +|∇Pkj|q dx +≥ +lim +j→+∞ Fkj(uj, Pkj). +(3.7) +Recalling that {(wk, Pk)} is a recovery sequence, we find +lim +k→+∞ Fk(yk, Pk) = +lim +j→+∞ Fkj(uj, Pkj) ≤ +lim +j→+∞ Fkj(wkj, Pkj) = F(y, P), +which in turn yields that {(yk, Pk)} is also a recovery sequence. +□ +3.5. Finsler structure on SL(3). In order to apply the results on homogenization of elasto- +plastic media in [24] we endow SL(3) with a Finsler structure. In doing so, we follow [38], whose +approach is based on the notion of plastic dissipation. Such line of thought links the geometry +of SL(3) to the physics of the system under consideration, and allows to conveniently include +dissipation effects in the model, see Subsection 6.3. +We start from the observation that SL(3) is a smooth manifold with respect to the topology +induced by the inclusion in R3×3. For every F ∈ SL(3) the tangent space at F is characterized +as +TFSL(3) = Fsl(3) := {FM ∈ R3×3 : trM = 0}, +and, in particular, TISL(3) coincides with sl(3) := {M ∈ R3×3 : trM = 0}. To the purpose of +endowing SL(3) with a Finsler structure, we first consider a C2 function ∆I : sl(3) → [0, +∞), +on which we make the following assumptions: +D1: It is positively 1-homogeneous: ∆I(cM) = c∆I(M) for all c ≥ 0 and M ∈ sl(3); +D2: It is 1-coercive and has at most linear growth: there exist 0 < c4 ≤ c5 such that for all +M ∈ sl(3) +c4|M| ≤ ∆I(M) ≤ c5|M|. +D3: It is strictly convex. +Note that we consider more restrictive regularity assumptions than the ones in [38], because +we appeal to results of differential geometry, where smoothness is customarily required. The +drawback of this choice is that in our analysis we cannot encompass some models, such as single +crystal plasticity. However, on the positive side, our assumptions cover Von Mises plasticity, see +[33, 38]. + +16 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Let TSL(3) denote the tangent bundle to SL(3). We can “translate” ∆I to the tangent spaces +other than sl(3) by setting +∆: +TSL(3) +→ +[0, +∞) +(F, M) +�→ +∆I(F −1M). +(3.8) +Then, it can be proved that (SL(3), ∆) is a C2 Finsler manifold. For an introduction to Finsler +geometry we refer to the monograph [4]. +Next, we introduce the family C(F0, F1) of piecewise C2 curves Φ: [0, 1] → SL(3) such that +Φ(0) = F0 and Φ(1) = F1. We set +D(F0, F1) := inf +�ˆ 1 +0 +∆ +�Φ(t), ˙Φ(t) +�dt : Φ ∈ C(F0, F1) +� +, +(3.9) +where ˙Φ is the velocity of the curve. The function D provides a non-symmetric distance on +SL(3): it is positive, attains 0 if and only if it is evaluated on the diagonal of SL(3) × SL(3), and +satisfies the triangular inequality; in general, however, D(F0, F1) ̸= D(F1, F0). +By the direct method of the calculus of variations (cf. [38, Theorem 5.1]) it can be proved +that for every F0, F1 ∈ SL(3) there exists a curve Φ ∈ C1,1([0, 1]; SL(3)) such that Φ(0) = F0, +Φ(1) = F1 and +D(F0, F1) = +ˆ 1 +0 +∆ +�Φ(t), ˙Φ(t) +� dt. +(3.10) +We call such Φ a shortest path between F0 and F1. We need the following local uniqueness +result for shortest paths, which wraps up the content of [4, Exercises 6.3.3]. +Proposition 3.10. For any point F in the Finsler manifold SL(3) there exists a relatively +compact neighborhood U of F such that for any F0, F1 ∈ U there exists a unique shortest path Φ +joining F0 and F1, and such path depends smoothly on its endpoints F0 and F1. +From Proposition 3.10 we deduce the existence of a set K as in H1, but we first need to recall +some terminology from differential geometry. A geodesic between F0 and F1 is a path that is a +critical point of the length functional under variations that do not alter the endpoints. When +for any couple of points in a given subset S of a Finsler manifold there is a unique shortest path +contained in S joining those points, we say that S is geodesically convex. +Corollary 3.11. Assume that a C2 Finsler structure on SL(3) is assigned. Then, there exists +a geodesically convex, compact neighborhood of I. +Proof. Owing to Proposition 3.10, there exists a relatively compact neighborhood U of I ∈ SL(3) +such that for any F0, F1 ∈ U there is a unique shortest path Φ joining F0 and F1. Thanks to +a Finsler variant of a theorem by Whitehead [4, Exercise 6.4.3], there is an open neighborhood +V of I that is compactly contained in U and geodesically convex. Let us set K := ¯V . Since +K ⊂ U, there is a unique shortest path Φ from F0 to F1 for any F0, F1 ∈ K. The fact that K is +geodesically convex as well may be proved by the same argument that proves that the closure +of a convex set is still convex. +□ +4. Compactness and splitting +From now on we turn to the analysis of the high-contrast energy in (1.1). We investigate in +this section the compactness properties of sequences with equibounded energy. We will see that, +as a consequence of the behavior of the hardening functional H, we can reduce the problem to +the case of pure elasticity addressed by K. Cherdantsev & M. Cherednichenko [13], and +we adapt their approach. + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +17 +Lemma 4.1 (Compactness). Let {εk} be an infinitesimal sequence. We suppose that {(yk, Pk)}k∈N ⊂ +W 1,2(Ω; R3) × W 1,q(Ω; SL(3)) satisfies +∥yk∥L2(Ω;R3) ≤ C, +Jk(yk, Pk) ≤ C +for some C ≥ 0, uniformly in k. +Let us denote by ˜yk the extension of yk in the sense of +Remark 3.3 above. Then, there exist subsequences of {εk}, {yk}, and {Pk}, which we do not +relabel, as well as y ∈ L2(Ω; W 1,2 +per(R3; R3)), y1 ∈ W 1,2(Ω; R3), v ∈ L2(Ω; W 1,2 +0 (Q0; R3)), and +P ∈ W 1,q(Ω; SL(3)) such that the following hold: +y(x, z) = y1(x) + v(x, z) +for a. e. (x, z) ∈ Ω × Q, +(4.1) +yk +2⇀ y +weakly two-scale in L2, +(4.2) +εk∇yk +2⇀ ∇zv +weakly two-scale in L2, +(4.3) +˜yk ⇀ y1 +weakly in W 1,2(Ω; R3), +(4.4) +Pk → P, +P −1 +k +→ P −1 +weakly in W 1,q(Ω; SL(3)) and uniformly in C(¯Ω; SL(3)), +∇˜ykP −1 +k +⇀ ∇y1P −1 +weakly in L2(Ω; R3×3). +(4.5) +Proof. From the definition of Jk, for all k ∈ N +∥∇Pk∥Lq ≤ C. +(4.6) +Besides, for all k, hypothesis E3, the definition of H and the bound (2.4) imply +���εkχ0 +k∇ykP −1 +k +��� +L2 + +���χ1 +k∇ykP −1 +k +��� +L2 ≤ C, +(4.7) +∥Pk∥L∞ + +���P −1 +k +��� +L∞ ≤ C. +(4.8) +Thanks to (2.5), from the first estimate we deduce +���εkχ0 +k∇yk +��� +L2 + +���χ1 +k∇yk +��� +L2 ≤ C, +(4.9) +which is precisely formula (21) in [13]. Thus, for what concerns the sequence of deformations, +the same bounds as the purely elastic case are retrieved. While referring to [13] for details, here +we limit ourselves to sketch how (4.9) entails two-scale compactness. +The boundedness of {yk} in L2 and Lemma 3.6(4) yield the existence of a function y ∈ +L2(Ω; W 1,2 +per(R3; R3)) such that, up to subsequences, (4.2) holds and +εk∇yk +2⇀ ∇zy +weakly two-scale in L2. +(4.10) +Thanks to (3.3) and Lemma 3.6(3), we also infer that +χ1 +kyk +2⇀ χ1y, +εkχ1 +k∇yk +2⇀ χ1∇zy +weakly two-scale in L2. +Moreover, there exist y1 ∈ W 1,2(Ω; R3) and v ∈ L2(Ω; W 1,2 +0 +(Q0; R3)) such that the decomposi- +tion (4.1) and the convergence (4.4) hold. By combining (4.1) and (4.10), (4.3) follows. +We now turn to the sequence of plastic strains. +By (4.6) and (4.8), we see that {Pk} is +bounded in W 1,q(Ω; SL(3)). Since q > 3, Morrey’s embedding yields the uniform convergence +of (a subsequence of) {Pk} to some P ∈ W 1,q(Ω; SL(3)). Therefore, by definition of the inverse +matrix +P −1 +k += (cofPk)T +det Pk += (cofPk)T , +we also deduce that P −1 +k +→ P −1 uniformly. + +18 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Finally, we observe that, thanks to (4.4) and the uniform convergence of {P −1 +k }, (4.5) is also +inferred. +□ +It is well-known that Γ-limits are not additive. In our case, however, we are able to show +that the asymptotic behavior of the functionals Jε is given exactly by the sum of the Γ-limits +of the soft and of the stiff contributions. Such splitting will enable us to treat the Γ-limits of +J 0 +ε and of J 1 +ε separately. We premise a simple lemma, which deals with the hardening part of +the energy. We recall that, for i = 0, 1, χi +k is the characteristic function of Ωi +k. +Lemma 4.2. Under assumptions H1–H2, for any sequence {Pk} ⊂ W 1,q(Ω; K) converging +uniformly to P ∈ W 1,q(Ω; K) it holds +lim +k→+∞ +ˆ +Ω +χi +k(x)H +�Pk(x) +� dx = L3(Qi) +ˆ +Ω +H +�P(x) +� dx +for i = 0, 1. +Proof. Let us focus on the case i = 0 first. We set +E0 := +� +t∈Z3 +(t + Q0) = R3 \ E1, +ˆΩ0 +k := +� +t∈ ˆTk +εk(t + Q0), +where +ˆTk := {t ∈ Z3 : εk(t + Q) ⊂ Ω} ⊂ Tk. +(4.11) +By definition of Ω0 +k (see (2.1)), we have +εkE0 \ Ω0 +k ⊂ εkE0 \ ˆΩ0 +k. +Note that Ω∩(εkE0 \ ˆΩ0 +k) is contained in the strip {x ∈ Ω : dist(x, ∂Ω) < +√ +3εk}. Since {H(Pk)} +is uniformly bounded by H1 and H2, we see that +lim +k→+∞ +ˆ +Ω +χ0 +k(x)H +�Pk(x) +� dx += +lim +k→+∞ +ˆ +Ω +χεkE0(x)H +�Pk(x) +� dx − +lim +k→+∞ +ˆ +Ω +�χεkE0(x) − χ0 +k(x) +�H +�Pk(x) +� dx += +lim +k→+∞ +ˆ +Ω +χεkE0(x)H +�Pk(x) +� dx. +Then, by the Lipschitz continuity of H on its domain, +lim +k→+∞ +ˆ +Ω +χεkE0(x)H +�Pk(x) +� dx = +lim +k→+∞ +ˆ +Ω +χεkE0(x)H +�P(x) +� dx += L3(Q0) +ˆ +Ω +H +�P(x) +� dx. +The case i = 1 follows from the previous one by the identities χ1 +k = χΩ − χ0 +k and L3(Q1) = +1 − L3(Q0). +□ +The splitting process is explained by the ensuing proposition. +Proposition 4.3 (Splitting). Let {εk} be an infinitesimal sequence, and let {(yk, Pk)}k∈N ⊂ +W 1,2(Ω; R3) × W 1,q(Ω; SL(3)) be a sequence satisfying +∥yk∥L2(Ω;R3) ≤ C, +Jk(yk, Pk) ≤ C + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +19 +for some C ≥ 0, uniformly in k. Let ˜yk be the extension of yk in the sense of Remark 3.3, and +let v ∈ L2(Ω; W 1,2 +0 +(Q0; R3)) be as in Lemma 4.1. Then, defining vk := yk − ˜yk, the following +hold: +{vk} ⊂ W 1,2 +0 (Ω0 +k; R3), +∥vk∥L2(Ω;R3) ≤ C, +εk∇vk +2⇀ ∇zv +weakly two-scale in L2, +(4.12) +lim inf +k→+∞ J 0 +k (vk, Pk) + lim inf +k→+∞ J 1 +k (˜yk, Pk) ≤ lim inf +k→+∞ Jk(yk, Pk), +(4.13) +lim sup +k→+∞ +Jk(yk, Pk) ≤ lim sup +k→+∞ +J 0 +k (vk, Pk) + lim sup +k→+∞ +J 1 +k (˜yk, Pk). +(4.14) +Moreover, in (4.13), {vk} may be replaced with another sequence {wk} ⊂ W 1,2 +0 (Ω0 +k; R3) such that +{εk∇wk} is 2-equiintegrable and εk∇wk ⇀ 0 weakly in L2(Ω; R3×3). +Proof. We first prove that (4.12) – (4.14) hold for the sequence {vk}. Afterwards, we will show +how to recover the equiintegrability for the sequence of gradients. +We split the functional Jk evaluated on (yk, Pk) as follows: +Jk(yk, Pk) = J 0 +k (yk, Pk) + J 1 +k (yk, Pk) += J 0 +k (vk, Pk) + J 1 +k (˜yk, Pk) + Rk, +(4.15) +where J 0 +k and J 1 +k are as in (2.11) and (2.12), and +Rk := J 0 +k (yk, Pk) − J 0 +k (vk, Pk) += +ˆ +Ω +χ0 +k +� +W 0 +ε +� +εk∇ykP −1 +k +� +− W 0 +ε +� +εk∇vkP −1 +k +�� +dx. +We next show that Rk is asymptotically negligible. +Hypothesis E4 yields +|Rk| ≤ c3 +ˆ +Ω +χ0 +k +� +1 + +���εk∇ykP −1 +k +��� + +���εk∇vkP −1 +k +��� +� ���εk∇˜ykP −1 +k +��� dx. +(4.16) +Since {(yk, Pk)} is equibounded in energy, the sequences {εkχ0 +k∇ykP −1 +k }, {χ1 +k∇ykP −1 +k }, and +{P −1 +k } are bounded in suitable Lebesgue spaces (see (4.7) and (4.8)). By the properties of the +extension operator Tε in Lemma 3.2, we deduce that +ˆ +Ω +���∇˜ykP −1 +k +��� +2 dx ≤ c +ˆ +Ω +|∇˜yk|2 dx ≤ c +ˆ +Ω +���χ1 +k∇yk +��� +2 dx ≤ c +ˆ +Ω +���χ1 +k∇ykP −1 +k +��� +2 dx ≤ C +(recall estimate (2.5)). So, thanks to (4.3), we deduce that +εk∇vk = εk∇yk − εk∇˜yk +2⇀ ∇zv +weakly two-scale in L2, +In particular, by Lemma 3.6(1), {εkχ0 +k∇vkP −1 +k } is bounded in L2(Ω; R3×3). By applying Hölder’s +inequality to the right-hand side of (4.16), we then find Rk = O(εk). Owing to (4.15) we conclude +that (4.13) and (4.14) hold. +To complete the proof, we are only left to establish the existence of the sequence {wk}. Upon +extraction of a subsequence, which we do not relabel, we may assume that in (4.13) the lower +limit involving J 0 +k is a limit. From the equiboundedness of the energy, by arguing as in the lines +before (4.9), we get +∥εk∇yk∥L2 ≤ C, +∥χ1 +k∇yk∥L2 ≤ C. +(4.17) + +20 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Then, (4.3) holds and, by Lemma 3.6(2), we obtain +εk∇yk ⇀ 0 +weakly in L2(Ω; R3×3). +Lemma 3.1 applied to the sequence {εk∇yk} yields two sequences, {kj} and {uj} ⊂ W 1,2(Ω; R3), +such that {εkj∇uj} is 2-equiintegrable, +εkj∇uj ⇀ 0 +weakly in L2(Ω; R3×3), +(4.18) +lim +j→+∞L3(Nj) = 0, +with Nj := {x ∈ Ω : ykj(x) ̸= uj(x)}. +Besides, we have +εkjχ1 +kj∇uj → 0 +strongly in L2(Ω; R3×3). +(4.19) +Indeed, it holds +∥εkjχ1 +kj∇uj∥L2 = ∥εkjχ1 +kj∇uj∥L2(Nj) + ∥εkjχ1 +kj∇ykj∥L2(Ω\Nj) +≤ ∥εkj∇uj∥L2(Nj) + εkj∥χ1 +kj∇ykj∥L2, +and the conclusion follows by the 2-equiintegrability of {εkj∇uj} and from (4.17). +We now define ˜uj := Tkjuj, with Tkj as in Lemma 3.2. From Remark 3.3 it follows that +{εkj∇˜uj} is 2-equiintegrable as well. Thus, the sequence defined by +wk := +� +uj − ˜uj +if k = kj for some j ∈ N, +0 +otherwise +has the properties that wk ∈ W 1,2 +0 (Ω0 +k; R3) and {εk∇wk} is 2-equiintegrable. Moreover, +εk∇wk ⇀ 0 +weakly in L2(Ω; R3×3). +To see this, we write +εkj∇wkj = εkj∇uj − εkj∇˜uj. +The first term converges to 0 weakly in L2(Ω; R3×3), as stated in (4.18). Additionally, Lemma 3.2 +entails +∥εkj∇˜uj∥L2 ≤ c∥εkjχ1 +kj∇uj∥L2, +and the weak convergence of {εk∇wk} follows from (4.19). +We are now ready to prove the validity of (4.13) when {εk∇vk} is replaced by the 2-equiintegrable +sequence {εk∇wk}. By the definition of the sequence at stake, we have +εkj(∇vkj − ∇wkj) = εkj(∇ykj − ∇uj) − εkj(∇˜ykj − ∇˜uj) +a. e. in Ω. +(4.20) +Lemma 3.2 yields +εkj∥∇˜ykj − ∇˜uj∥L2 = εkj∥∇ +�Tkj(ykj − uj) +�∥L2 +≤ cεkj∥χ1 +kj∇(ykj − uj)∥L2 += cεkj∥χ1 +kj(∇ykj − ∇uj)∥L2(Nj) +≤ c +� +εkj∥χ1 +kj∇ykj∥L2 + ∥εkj∇uj∥L2(Nj) +� +. +Thus, (4.17) and the 2-equiintegrability of {εkj∇uj} entail +εkj +� +∇˜ykj − ∇˜uj +� +→ 0 +strongly in L2(Ω; R3×3). +(4.21) + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +21 +Therefore, using (4.20) and the fact that the densities W 0 +kj are bounded from below, we have +ˆ +Ω +χ0 +kj(x)W 0 +kj +�εkj∇vkj(x)P −1 +kj (x) +� dx += +ˆ +Nj +χ0 +kj(x)W 0 +kj +�εkj∇vkj(x)P −1 +kj (x) +� dx ++ +ˆ +Ω\Nj +χ0 +kj(x)W 0 +kj +��εkj∇wkj(x) − εkj(∇˜ykj(x) − ∇˜uj(x)) +�P −1 +kj (x) +� +dx +− +ˆ +Ω\Nj +χ0 +kj(x)W 0 +kj +�εkj∇wkj(x)P −1 +kj (x) +� dx + +ˆ +Ω\Nj +χ0 +kj(x)W 0 +kj +�εkj∇wkj(x)P −1 +kj (x) +� dx +≥ −c +�ˆ +Ω\Nj +|εkj(∇˜ykj(x) − ∇˜uj(x))|2 dx +�1/2 ++ +ˆ +Ω\Nj +χ0 +kj(x)W 0 +kj +�εkj∇wkj(x)P −1 +kj (x) +� dx, +where the Lipschitz regularity E4 and Hölder’s inequality were employed to derive the last bound +(recall that supk∈N ∥P −1 +k ∥∞ ≤ C). We now take the limit in the inequality above. According to +Lemma 4.2, the hardening term has a limit. Therefore, also the elastic contribution is convergent, +and it satisfies +lim +k→+∞ J 0 +k (vk, Pk) = +lim +j→+∞ +ˆ +Ω +χ0 +kj(x)W 0 +kj +�εkj∇vkj(x)P −1 +kj (x) +� dx + L3(Q0) +ˆ +Ω +H +�P(x) +� dx. +The strong converge (4.21) implies +lim +j→+∞ +ˆ +Ω +χ0 +kj(x)W 0 +kj +�εkj∇vkj(x)P −1 +kj (x) +� dx +≥ lim inf +j→+∞ +ˆ +Ω\Nj +χ0 +kj(x)W 0 +kj +�εkj∇wkj(x)P −1 +kj (x) +� dx += lim inf +j→+∞ +ˆ +Ω +χ0 +kj(x)W 0 +kj +�εkj∇wkj(x)P −1 +kj (x) +� dx, +where the equality follows from the growth condition E3 and from the equiintegrability of +{εkj∇wkj}. We thereby infer +lim inf +k→+∞ J 0 +k (wk, Pk) ≤ lim inf +j→+∞ J 0 +kj(wkj, Pkj) ≤ +lim +k→+∞ J 0 +k (vk, Pk), +and this concludes the proof. +□ +5. Γ-limit of the soft component +We devote this section to the study of the asymptotics of the functional J 0 +ε in (2.11), which +encodes the energy of the inclusions. After some observations on the limiting functional J 0 in +(2.6), in the second and third subsections we deal respectively with the lower and with the upper +limit inequality for the elastic part of the energy. The other contributions will be taken into +account in Subsection 5.4, where we prove Proposition 2.10. +5.1. The limiting functional. The definition of Q′W 0 in (2.8), which encodes the limiting +elastic contribution of the soft inclusions, may be regarded as a variant of the well known +Dacorogna’s formula for the quasiconvex envelope [20, Theorem 6.9]. As such, the infimum in +(2.8) does not depend on Q, and we may rewrite Q′W 0 as follows: +Q′W 0(F, G) = inf +� +Q0 W 0��F + ∇v(z) +�G +� +dz : v ∈ W 1,2 +0 (Q0; R3) +� +. +(5.1) + +22 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Note that here quasiconvexification occurs just with respect to the first argument, since a +very strong convergence is considered for the second one. The fact that different variables in a +problem may call for different relaxation procedures has been already observed. As an example, +we mention the concept of cross-quasiconvexity introduced by Le Dret & Raoult [35] to deal +with dimension reduction problems in elasticity. +For the sake of completeness, we explicitly mention some basic properties of Q′W 0. +Lemma 5.1. Let W 0 : R3×3 → R, and assume there exist 0 < c1 ≤ c2 such that for all F ∈ R3×3 +c1|F|2 ≤ W 0(F) ≤ c2 +� +|F|2 + 1 +� +. +Let Q′W 0 be as in (2.8). +(1) For all F, G ∈ R3×3 +c1|FG|2 ≤ Q′W 0(F, G) ≤ c2 +� +|FG|2 + 1 +� +, +and for all G ∈ R3×3 there exists c := c(G) > 0 such that for all F1, F2 ∈ R3×3 +���Q′W 0(F1, G) − Q′W 0(F2, G) +��� ≤ c (1 + |F1| + |F2|) |F1 − F2|. +Suppose further that there exists c3 > 0 such that for all F1, F2 ∈ R3×3 +���W 0(F1) − W 0(F2) +��� ≤ c3 (1 + |F1| + |F2|) |F1 − F2|. +(5.2) +(2) Then, Q′W 0(F, · ) is continuous for all F ∈ R3×3. +(3) If {Pk} ⊂ W 1,q(Ω; SL(3)) converges weakly to P ∈ W 1,q(Ω; SL(3)), then for any V ∈ +L2(Ω; R3×3) +lim +k→+∞ +ˆ +Ω +Q′W 0�V (x), P −1 +k (x) +� dx = +ˆ +Ω +Q′W 0�V (x), P −1(x) +� dx. +Proof. The growth conditions on Q′W 0 are an immediate consequence of the ones on W 0 and +of the definition of Q′W 0. +For what concerns the 2-Lipschitz property, let us set W 0 +G(F) := W 0(FG) for any fixed +G ∈ R3×3. Then, Q′W 0( · , G) coincides with the quasiconvex envelope of W 0 +G. By [20, Remark +5.3(iii)] it follows that Q′W 0( · , G) is separately convex, and hence, in view of the growth +assumptions on W 0, the proof of item (1) is concluded by [20, Proposition 2.32]. +As for point (2), let Gk → G in R3×3. In view of (5.2), for every δ > 0 there exists cδ > 0 +such that +Q′W 0(F, Gk) − Q′W 0(F, G) ≤ cδ|Gk − G| + δ. +Similarly, for any k ∈ N there exists vk ∈ W 1,p +0 (Q; R3×3) such that +Q′W 0(F, Gk) − Q′W 0(F, G) +≥ −c3|Gk − G| +ˆ +Q +�1 + |(F + ∇vk)Gk| + |(F + ∇vk)G| +�|F + ∇vk| dx − 1 +k. +Thanks to the coercivity of the integrand, it follows that {∇vk} is bounded in L2, whence +Q′W 0(F, Gk) − Q′W 0(F, G) ≥ −c |Gk − G| − 1 +k +for a constant c independent of k. The continuity of Q′W 0(F, · ) is then proved by letting first +k → +∞ and then δ → 0. + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +23 +Eventually, taking into account points (1) and (2), as well as the compact embedding of +W 1,q(Ω) into C(¯Ω), we can employ the dominated convergence theorem to obtain the continuity +property in (3). +□ +We now exhibit an alternative expression for the soft limiting elastic energy, which is to be +exploited in the proof of Proposition 5.7. +Lemma 5.2. For every couple (V, P) ∈ L2(Ω; R3×3) × W 1,q(Ω; SL(3)) we have +ˆ +Ω +Q′W 0�V (x), P −1(x) +� dx += inf +�ˆ +Ω + +Q0 W 0��V (x) + ∇zw(x, z) +�P −1(x) +� +dz : w ∈ L2(Ω; W 1,2 +0 (Q0; R3)) +� +. +(5.3) +The identity above rests on a measurable selection criterion that we recall next. +Lemma 5.3 (Lemma 3.10 in [28]). Let S be a multifunction defined on the measurable space +X and taking values in the collection of subsets of the separable metric space Y . If S(x) is +nonempty and open in Y for every x ∈ X, and if the set { x ∈ X : y ∈ S(x) } is measurable for +every y ∈ Y , then S admits a measurable selection, that is, there exists a measurable function +s: X → Y such that s(x) ∈ S(x) for all x ∈ X. +The previous lemma is a variant of [12, Theorem III.6], and we refer to that monograph for +a comprehensive treatment of measurable selection principles. +Proof of Lemma 5.2. The argument follows the one proposed in [28, Corollary 3.2]. +Let us fix w ∈ L2(Ω; W 1,2 +0 (Q0; R3)), so that, for almost every x ∈ Ω, w(x, · ) ∈ W 1,2 +0 (Q0; R3). +Hence, according to (5.1), we have +Q′W 0�V (x), P −1(x) +� ≤ + +Q0 W 0��V (x) + ∇zw(x, z) +�P −1(x) +� +dz +for a. e. x ∈ Ω, +whence, after integration over Ω, we deduce that in (5.3) the left-hand side is smaller that the +righ-hand one. +In order to establish the opposite inequality, we first observe that, by (5.1), for every x ∈ Ω +and every δ > 0 there exists vx,δ ∈ W 1,2 +0 (Q0; R3) such that + +Q0 W 0��V (x) + ∇vx,δ(z) +�P −1(x) +� +dz − Q′W 0�V (x), P −1(x) +� < δ. +(5.4) +We introduce the multifunction S defined for x ∈ Ω by +S(x) := +� +v ∈ W 1,2 +0 (Q0; R3) : (5.4) holds for vx,δ = v +� +. +We show that it admits a measurable selection. To this purpose, observe that, as a consequence of +the growth assumptions on W 0 and of the dominated convergence theorem, S(x) is a nonempty, +open subset of W 1,2 +0 (Q0; R3) for every x ∈ Ω. +Second, for every v ∈ W 1,2 +0 (Q0; R3) the set +{x ∈ Ω : v ∈ S(x)} is measurable, because it is the sublevel set of a measurable function. +Thanks to Lemma 5.3, for every δ > 0 we retrieve a measurable function wδ : Ω → W 1,2 +0 (Q0; R3) +that satisfies +ˆ +Ω + +Q0 W 0��V (x) + ∇zwδ(x, z) +�P −1(x) +� +dz dx ≤ +ˆ +Ω +Q′W 0�V (x), P −1(x) +� + O(δ). + +24 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +In particular, by the growth conditions on W 0, wδ must belong to L2(Ω; W 1,2 +0 +(Q0; R3)). There- +fore, since δ is arbitrary, we conclude that the left-hand side in (5.3) bounds from above the +right-hand one. +□ +5.2. Lower bound for the elastic energy. The goal of this subsection is to prove the ensuing: +Proposition 5.4. Let {W 0 +k }k satisfy assumptions E3–E5, and let P ∈ W 1,q(Ω; SL(3)). For +every sequence {(vk, Pk)} ⊂ W 1,2 +0 (Ω0 +k; R3)×W 1,q(Ω; SL(3)) such that {εk∇vk} is 2-equiintegrable +and Pk → P uniformly, it holds +L3(Q0) +ˆ +Ω +Q′W 0�0, P −1(x) +� dx ≤ lim inf +k→+∞ +ˆ +Ω +χ0 +k(x)W 0 +k +�εk∇vk(x)P −1 +k (x) +� dx. +(5.5) +At a first glance, it may look bizarre that no convergence for the sequence {εk∇vk} is pre- +scribed. The statement becomes clearer once we recall that if Qf is the quasiconvex envelope +of f : R3×3 → R, then +Qf(0) ≤ + +Ω +f +�∇v(x) +� dx +for any v ∈ W 1,∞ +0 +(Ω; R3). +In order to establish (5.5), it is convenient to unfold the elastic energy. +Lemma 5.5. Let {W 0 +k }k satisfy assumptions E3–E5. For any (v, P) ∈ W 1,2(Ω; R3)×W 1,q(Ω; SL(3)) +it holds +ˆ +Ω +χ0 +k(x)W 0 +k +� +εk∇v(x)P −1(x) +� +dx = +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 W 0 +k +� +∇zˆv(x, z) ˆP −1(x, z) +� +dz dx +(5.6) +where ˆv := Skv, ˆP := SkP and Sk := Sεk is the unfolding operator introduced in Lemma 3.7. +Proof. According to the definition of Ω0 +k in (2.1), the left-hand side of (5.6) equals +ε3 +k +� +t∈Tk +ˆ +Q0 W 0 +k +� +εk∇v +�εk(t + z) +�P −1�εk(t + z) +�� +dz. +We use the unfolding operator to rewrite this quantity as a double integral. Recalling Lemma 3.7, +we firstly observe that for every t ∈ Tk and z ∈ Q0 we have the identities +Sk(εk∇v)(εkt, z) = εk∇v +�εk(t + z) +�, +SkP −1(εkt, z) = P −1�εk(t + z) +�. +Then, we also have +Sk(εk∇v) = ∇z +�Skv +� = ∇zˆv, +SkP −1 = (SkP)−1 = ˆP −1. +We obtain +ˆ +Ω +χ0 +k(x)W 0 +k +�εk∇v(x)P −1(x) +� dx += ε3 +k +� +t∈Tk +ˆ +Q0 W 0 +k +�Sk(εk∇v)(εkt, z)Sk(P −1)(εkt, z) +� dz += +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 W 0 +k +� +∇zˆv +� +εk +� x +εk +� +, z +� +ˆP −1 +� +εk +� x +εk +� +, z +�� +dz dx, +because ⌊x/εk⌋ = t for all x ∈ εk(t + Q). Since, in general, it holds +Sku +� +εk +� x +εk +� +, z +� += u +� +εk +� x +εk +� ++ εkz +� += Sku(x, z), +(5.6) follows. +□ + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +25 +A crucial ingredient in the proof of Proposition 5.4 is a sort of lower semicontinuity result for +the elastic contribution to the energy. +Lemma 5.6. Let {W 0 +k }k satisfy assumptions E3–E5. Let also {wk} ⊂ L2(Ω; W 1,2 +0 (Q0; R3)) be +such that {∇zwk} is 2-equiintegrable. Then, for all P ∈ W 1,q(Ω; SL(3)), +L3(Q0) +ˆ +Ω +Q′W 0�0, P −1(x) +� dx ≤ lim inf +k→+∞ +ˆ +Ω +ˆ +Q0 W 0 +k +�∇zwk(x, z)P −1 +k (x) +� dz dx, +whenever Pk → P uniformly. +Proof. From (5.1) it follows that for all k ∈ N +L3(Q0) +ˆ +Ω +Q′W 0�0, P −1 +k (x) +� dx ≤ +ˆ +Ω +ˆ +Q0 W 0�∇zwk(x, z)P −1 +k (x) +� dz dx. +(5.7) +Next, relying on the pointwise convergence of {W 0 +k } to W 0, we adapt the argument in the proof +of [21, Theorem 5.14] to pass from W 0 to W 0 +k on the right-hand side (see also [26, Lemma 5.2] for +a similar result in the context of A -quasiconvexity). Fix δ > 0. If {∇zwk} is 2-equiintegrable, +then so is {∇zwkP −1 +k }. Therefore, since the 2-growth assumptions on {W 0 +k } transfer to the +pointwise limit W 0, there exists r > 0 such that +sup +k∈N +ˆ +{(x,z)∈Ω×Q0:|∇zwk(x,z)P −1 +k +(x)|>r} +W 0�∇zwk(x, z)P −1 +k (x) +� dz dx ≤ δ. +(5.8) +Owing to assumption E4 and Remark 2.2, we can find ρ > 0 such that for every F, G ∈ R3×3 +contained in the open ball B(0, ρ) +sup +k∈N +|W 0 +k (F) − W 0 +k (G)| + |W 0(F) − W 0(G)| ≤ δ. +(5.9) +Let now F1, . . . , Fn ∈ B(0, r) be such that +B(0, r) ⊂ +n +� +i=1 +B (Fi, ρ) . +(5.10) +Due to the pointwise convergence of W 0 +k to W 0, for any such Fi there exist ¯ki ∈ N such that +|W 0 +k (Fi)−W 0(Fi)| ≤ δ if k > ¯ki. Letting ¯k := max{¯k1, . . . , ¯kn}, it follows that for any i = 1, . . . , n +|W 0 +k (Fi) − W 0(Fi)| ≤ δ +if k > ¯k. +(5.11) +By (5.10), for every G ∈ B(0, r) there exists i ∈ {1, . . . , n} such that G ∈ B(Fi, ρ). For this +particular i, the combination of the triangle inequality, (5.9) and (5.11) yields +|W 0 +k (G) − W 0(G)| ≤ |W 0 +k (G) − W 0 +k (Fi)| + |W 0 +k (Fi) − W 0(Fi)| + |W 0(G) − W 0(Fi)| ≤ 3δ, (5.12) +for every G ∈ B(0, r) and every k > ¯k. + +26 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Thanks to Lemma 5.1(3) and (5.7) we deduce +L3(Q0) +ˆ +Ω +Q′W 0�0, P −1(x) +� dx += L3(Q0) lim +k→+∞ +ˆ +Ω +Q′W 0�0, P −1 +k (x) +� dx +≤ lim inf +k→+∞ +ˆ +Ω +ˆ +Q0 W 0�∇zwk(x, z)P −1(x) +� dz dx +≤ lim inf +k→+∞ +ˆ +{(x,z)∈Ω×Q0:|∇zwk(x,z)P −1 +k +(x)|≤r} +W 0�∇zwk(x, z)P −1 +k (x) +� dz dx + δ +≤ lim inf +k→+∞ +ˆ +Ω +ˆ +Q0 W 0 +k +�∇zwk(x, z)P −1 +k (x) +� dz dx + 3δL6(Ω × Q0) + δ, +where the second inequality is due to (5.8), and the last one to (5.12). The arbitrariness of δ > 0 +yields the conclusion. +□ +We are now ready to prove the lower bound for the elastic contribution of the soft part. +Proof of Proposition 5.4. Let ˆvk := Skvk. +In view of the 2-equiintegrability of the sequence +{εk∇vk} and of Lemma 3.7, {∇zˆvk} is 2-equiintegrable as well. Hence it is a fortiori bounded +in L2. From Lemma 5.5, restricting the summation in (5.6) to the set of translations in (4.11), +we deduce +lim inf +k→+∞ +ˆ +Ω +χ0 +k(x)W 0 +k +�εk∇vk(x)P −1 +k (x) +� dx +≥ +lim inf +k→+∞ +ˆ +ΩQ +k +ˆ +Q0 W 0 +k +�∇zˆvk(x, z)P −1 +k (x) +� dz dx, +where +ΩQ +k := +� +t∈ ˆTk +εk(t + Q). +(5.13) +We rewrite the right-hand side of the previous inequality as the difference between the integrals +I′ +k := +ˆ +Ω +ˆ +Q0 W 0 +k +�∇zˆvk(x, z)P −1 +k (x) +� dz dx, +I′′ +k := +ˆ +Ω\ΩQ +k +ˆ +Q0 W 0 +k +�∇zˆvk(x, z)P −1 +k (x) +� dz dx. +Being {∇zˆvk} 2-equiintegrable, the sequence {∇zˆvkP −1 +k } is still 2-equiintegrable. Thus, by the +growth condition E3, we obtain +lim +k→+∞ I′′ +k = 0. +Taking into account Lemma 5.6 we conclude +lim inf +k→+∞ +ˆ +Ω +χ0 +k(x)W 0 +k +�εk∇vk(x)P −1 +k (x) +� dx ≥ lim inf +k→+∞ I′ +k ≥ L3(Q0) +ˆ +Ω +Q′W 0�0, P −1(x) +� dx. +□ + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +27 +5.3. Upper bound for the elastic energy. In this subsection we address the proof of Γ-upper +limit inequality for the elastic contribution of the soft component. Differently from the previous +subsection, in order to establish the desired inequality we perform an analysis that is genuinely +two-scale, in the sense that we interpret 0 as the average with respect to the periodic variable +of the two-scale limit of the sequence {εk∇vk}. +Proposition 5.7. Let {W 0 +k }k satisfy assumptions E3–E5, and let P ∈ W 1,q(Ω; SL(3)). For all +δ > 0 there exists a sequence {vk} ⊂ W 1,2 +0 (Ω0 +k; R3) such that εk∇vk ⇀ 0 weakly in L2(Ω; R3×3) +and that +lim sup +k→+∞ +ˆ +Ω +χ0 +k(x)W 0 +k +� +εk∇vk(x)P −1 +k (x) +� +dx < L3(Q0) +ˆ +Ω +Q′W 0�0, P −1(x) +� dx + δ, +(5.14) +whenever Pk → P uniformly. +We begin with a lemma that provides a strong two-scale approximation of any sufficiently +regular function. The result has already appeared in [13], where, however, the proof is just +sketched. In order to keep the exposition self-contained, we include it in the Appendix, where +we also compare our result with the one in [13]. +Lemma 5.8. Let w ∈ L2(Ω; W 1,2 +0 (Q0; R3)) ∩ C2(Ω × Q0; R3). Then, there exists a sequence +{vk} ⊂ L2(Ω; R3) such that, letting ˆvk := Skvk, it holds +∇zˆvk → ∇zw +strongly in L2(Ω × Q; R3×3). +(5.15) +We are now ready to prove the Γ-limsup inequality for the soft inclusions functional. +Proof of Proposition 5.7. According +to +Lemma +5.2, +for +every +δ +> +0 +there +exists +wδ ∈ L2(Ω; W 1,2 +0 (Q0; R3)) satisfying +ˆ +Ω +ˆ +Q0 W 0�∇zwδ(x, z)P −1(x) +� dz dx < L(Q0) +ˆ +Ω +Q′W 0�0, P −1(x) +� dx + δ +(5.16) +We would like to apply Lemma 5.8 which, however, requires wδ ∈ L2(Ω; W 1,2 +0 (Q0; R3))∩C2(Ω× +Q0; R3). We therefore establish the inequality first for a sufficiently regular wδ, and we then +extend the result by a density argument. +Case 1: wδ regular +Let wδ ∈ L2(Ω; W 1,2 +0 (Q0; R3)) ∩ C2(Ω × Q0; R3). We consider the recovery sequence {vk} +coming from Lemma 5.8. +Lemmas 3.7 and 3.6(2) yield εk∇vk ⇀ 0 weakly in L2(Ω; R3×3). +Assumption E4 and Hölder’s inequality entail +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 +���W 0 +k +� +∇zˆvk(x, z)P −1 +k (x) +� +− W 0 +k +� +∇zwδ(x, z)P −1 +k (x) +���� dz dx +≤ c +� +t∈Tk +�ˆ +εk(t+Q) +ˆ +Q0 |∇zˆvk(x, z) − ∇zwδ(x, z)|2 dz dx +�1/2 +, + +28 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +where the constant c bounds ∥P −1 +k ∥L∞. Thanks to the strong convergence of {∇zˆvk}, we obtain +that the term above is infinitesimal when k → +∞. From Lemma 5.5 we then deduce +lim sup +k→+∞ +ˆ +Ω +χ0 +k(x)W 0 +k +� +εk∇vk(x)P −1 +k (x) +� +dx += lim sup +k→+∞ +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 W 0 +k +� +∇zwδ(x, z)P −1 +k (x) +� +dz dx += lim sup +k→+∞ +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 W 0 +k +� +∇zwδ(x, z)P −1(x) +� +dz dx += lim sup +k→+∞ +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 W 0 � +∇zwδ(x, z)P −1(x) +� +dz dx, +where the second identity follows from E4 and the last one from E5. Note also that, by absolute +continuity of the Lebesgue integral, +lim sup +k→+∞ +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 W 0 � +∇zwδ(x, z)P −1(x) +� +dz dx += +ˆ +Ω +ˆ +Q0 W 0 � +∇zwδ(x, z)P −1(x) +� +dz dx. +Therefore, by combining the equalities that we have just found with (5.16), we achieve the +conclusion in the case under consideration. +Case 2: wδ generic +Let +now +wδ +∈ +L2(Ω; W 1,2 +0 (Q0; R3)). +By +mollification, +we +retrieve +a +function +˜wδ ∈ L2(Ω; W 1,2 +0 (Q0; R3)) ∩ C2(Ω × Q0; R3) such that +ˆ +Ω +ˆ +Q0 W 0�∇z ˜wδ(x, z)P −1(x) +� dz ≤ +ˆ +Ω +ˆ +Q0 W 0�∇zwδ(x, z)P −1(x) +� dz + δ. +To achieve the conclusion, it only suffices to repeat the argument in Case 1 for ˜wδ. +□ +5.4. Proof of Proposition 2.10. We are eventually in a position to reap the fruits of the +previous subsections. +Proof of Proposition 2.10. Let us start with the lower limit inequality. +If the lower limit of +J 0 +k (vk, Pk) is not finite, there is nothing to prove. Otherwise, recalling Lemma 3.6(4), we deduce +that εk∇vk +2⇀ ∇z˜v weakly two-scale in L2 for some ˜v ∈ L2(Ω; W 1,2 +per(R3; R3)). In particular, by +Lemma 3.6(2), it must be +∇v(x) = +ˆ +Q +∇z˜v(x, z) dz = 0 +for a. e. x ∈ Ω, +whence, being Ω connected, must v be identically zero. Statement (1) in Proposition 2.10 then +follows by combining Proposition 5.4 and Lemma 4.2. +We now turn to the upper bound. The only nontrivial case corresponds to v = 0. Propo- +sition 5.7 provides for all δ > 0 a sequence {vk} ⊂ W 1,2 +0 (Ω0 +k; R3) such that εk∇vk ⇀ 0 = ∇v +weakly in L2(Ω; R3×3) and (5.14) holds. By the Rellich-Kondrachov theorem in W 1,2 +0 (Ω; R3), it +follows that εkvk → 0 strongly in L2(Ω; R3) (up to subsequences). We employ again Lemma 4.2 +to deduce that +lim sup +k→+∞ +J 0 +k (vk, Pk) < J 0(v, P) + δ. + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +29 +This inequality is actually equivalent to the desired one (cf. [8, Section 1.2]), and the proof is +therefore concluded. +□ +6. Conclusions and a variant +We devote this final section to the proof of the homogenization result for high-contrast com- +posites and to the discussion of a variant of the problem featuring plastic dissipation. +6.1. Proof of Theorem 2.7 and convergence of minimum problems. As we outlined be- +fore, the proof of Theorem 2.7 is achieved by combining the splitting procedure in Proposition 4.3 +with Theorem 3.8 and Proposition 2.10, which account for the asymptotics of the stiff and the +soft components, respectively. Once the homogenization theorem is on hand, the convergence +of the minimum problems and of their minimizers will follow thanks to the compactness result +in Lemma 4.1. +Proof of Theorem 2.7. Let {εk} be an infinitesimal sequence and let us fix y ∈ L2(Ω; R3) and +P ∈ Lq(Ω; SL(3)). We separate the proof of the lower and of the upper limit inequalities. +Lower bound +We consider a sequence {(yk, Pk)} ⊂ L2(Ω; R3) × Lq(Ω; SL(3)) such that yk → y in the sense of +extensions and that Pk → P uniformly. The only case to discuss is the one in which the lower +limit of Jk(yk, Pk) is finite, and we may thus assume that {Jk(yk, Pk)} is bounded. Keeping in +force the notation of Definition 2.4, we let {˜yk} ⊂ W 1,2(Ω; R3) be a sequence such that yk = ˜yk +in Ω1 +k and ˜yk ⇀ y weakly in W 1,2(Ω; R3). In the light of (4.4) and Remark 2.5, we may without +loss of generality assume that ˜yk := Tkyk, with Tk as in Lemma 3.2. +We now apply Proposition 4.3, which yields {vk} ⊂ W 1,2 +0 (Ω0 +k; R3) satisfying (4.13) and such +that {vk} is bounded in L2, {εk∇vk} is 2-equiintegrable and εkvk → 0 strongly in L2. +In +particular, (εkvk, Pk) τ→ (0, P) and Proposition 2.10 yields +J 0(0, P) ≤ lim inf +k→+∞ J 0 +k (vk, Pk). +At this stage, recalling (4.13), the proof of the lower bound is concluded as soon as we show +that +J 1(y, P) ≤ lim inf +k→+∞ J 1 +k (˜yk, Pk) = lim inf +k→+∞ J 1 +k (yk, Pk) +(6.1) +with J 1(y, P) given by (2.7). This is what we prove next. +Let us set +� +W 1(x, F) := χE1(x)W 1(F), +�H(x, P) := χE1(x)H(P), +� +J 1 +k (y, P) := +ˆ +Ω +� +� +W 1 +� x +εk +, ∇˜yP −1 +� ++ �H +� x +εk +, P +� ++ |∇P|q +� +dx. +(6.2) +It holds +lim inf +k→+∞ +� +J 1 +k (˜yk, Pk) ≤ lim inf +k→+∞ J 1 +k (˜yk, Pk). +Since (˜yk, Pk) τ→ (y, P), by applying Theorem 3.8 to the left-hand side of the previous inequality, +(6.1) is deduced. +Upper bound +If P /∈ W 1,q(Ω; K) there is nothing to prove; let us then assume that P ∈ W 1,q(Ω; K). + +30 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +As we have already observed, { � +J 1 +k } satisfies the requirements of Theorem 3.8. In view of +Corollary 3.9, for any (y, P) ∈ W 1,2(Ω; R3) × W 1,q(Ω; K) there exists a sequence {(uk, Pk)} ⊂ +W 1,2(Ω; R3) × W 1,q(Ω; K) such that {∇uk} is 2-equiintegrable, (uk, Pk) τ→ (y, P), and +lim sup +k→+∞ +� +J 1 +k (uk, Pk) ≤ J 1(y, P). +Note that +0 ≤ J 1 +k (uk, Pk) − � +J 1 +k (uk, Pk) += +ˆ +Ω +�χ1 +k(x) − χεkE1(x) +��W 1(∇ukP −1 +k ) + H(Pk) +� dx +≤ c +ˆ +Ω +�χ1 +k(x) − χεkE1(x) +��|∇uk|2 + 1 +� dx +for all k ∈ N. Thanks to the 2-equiintegrability of {∇uk}, we deduce +lim sup +k→+∞ +J 1 +k (uk, Pk) = lim sup +k→+∞ +� +J 1 +k (uk, Pk) ≤ J 1(y, P). +(6.3) +We focus now on the soft part. Proposition 2.10 grants the existence of a sequence {vk} ⊂ +W 1,2 +0 (Ω0 +k; R3) such that εkvk → 0 strongly in L2(Ω; R3×3) and that +lim sup +k→+∞ +J 0 +k (vk, Pk) ≤ J 0(0, P), +(6.4) +where {Pk} is as in (6.3). Notice that if yk := uk + vk, then {Jk(yk, Pk)} is bounded and {yk} +converges to y in the sense of extensions. Letting ˜yk := Tkyk, thanks to (4.14) we conclude the +proof of the upper limit inequality: +lim sup +k→+∞ +Jk(yk, Pk) ≤ lim sup +k→+∞ +J 0 +k (yk − ˜yk, Pk) + lim sup +k→+∞ +J 1 +k (˜yk, Pk) += lim sup +k→+∞ +J 0 +k (vk, Pk) + lim sup +k→+∞ +J 1 +k (uk, Pk) +≤ J (y, P). +In the previous lines, the equality is a consequence of the facts that {∇uk} and {∇˜yk} are +bounded and that uk = yk on Ω1 +k, whereas the last bound accounts for (6.3) and (6.4). +□ +Finally, we are only left to establish the convergence of the minimum problems associated with +the energy functionals Jε. What we need is an adaptation of the Γ-convergence statement that +we have just proved so as to make it comply with Dirichlet boundary conditions. To this aim, +as it is customary (see e.g. [9, Proposition 11.7]), we could employ the fundamental estimate +derived in [24] on the functionals { � +J 1 +k } in (6.2); indeed, boundary data concern only the stiff +part, cf. Remark 2.6. In the light of Corollary 3.9 we can adopt an alternative strategy. +Proof of Corollary 2.9. Since {(yk, Pk)} is a sequence of almost-minimizers, there exists C such +that Jk(yk, Pk) ≤ C. The 2-growth condition from below, together with Lemma 3.4, provides a +bound on ∥yk∥L2. By Proposition 4.1, there exists (y, P) ∈ W 1,2 +0 (Ω; R3) × W 1,q(Ω; K) such that, +up to subsequences, yk → y in the sense of extensions and Pk → P uniformly. Theorem 2.7 +ensures that +J (y, P) ≤ lim inf +k→+∞ Jk(yk, Pk). +We now prove the existence of a recovery sequence meeting the boundary conditions. As sug- +gested by Remark 2.6, we focus on the stiff part. Let us consider again the functional � +J 1 +k in (6.2). +Since the sequence { � +J 1 +k } falls within the scopes of Theorem 3.8, for any (�y, �P) ∈ W 1,2 +0 (Ω; R3) × + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +31 +W 1,q(Ω; K) Corollary 3.9 provides a sequence {(uk, �Pk)} ⊂ W 1,2 +0 (Ω; R3) × W 1,q(Ω; K) such that +{∇uk} is 2-equiintegrable, (uk, �Pk) τ→ (�y, �P) and +lim sup +k→+∞ +� +J 1 +k (uk, �Pk) ≤ J 1(y, P). +By reasoning as in the proof of the upper bound in Theorem 2.7 we retrieve a sequence {�yk, �Pk} ∈ +W 1,2 +0 (Ω; R3) × W 1,q(Ω; K) such that �yk → �y in the sense of extensions, �Pk → �P uniformly and +lim sup +k→+∞ +Jk(�yk, �Pk) ≤ J (�y, �P), +whence +lim sup +k→+∞ +(inf Jk) ≤ inf J . +Recalling that {(yk, Pk)} is a sequence of almost minimizers, we conclude +inf J ≤ J (y, P) ≤ lim inf +k→+∞ Jk(yk, Pk) = lim inf +k→+∞ inf Jk ≤ inf J , +as desired. +□ +6.2. A non degenerate upper bound for the soft component. We proved in Section 5 +that the limiting behavior of the soft inclusions is described by a degenerate functional. However, +under our assumptions, a non-degenerate upper bound may still be established, as we prove in +the remainder. The argument follows [13], where Cherdantsev & Cherednichenko derived +the effective energy of high-contrast nonlinear elastic materials. Differently from us, the Γ-limit +that they retrieve keeps track of both the macro- and the microscopic variable, and this roots in +the choice of a stronger notion of convergence. The drawback of such an approach is the lack of +compactness for sequences with equibounded energy. It was shown in [26, Example 2.12] that, +when weaker topologies are considered, the quasiconvex envelope does not provide the correct +limiting energy density for the Γ-lower limit. +We start by proving a more detailed version of Lemma 5.8. +Lemma 6.1 (cf. Lemma 22 in [13]). Let w ∈ L2(Ω; W 1,2 +0 +(Q0; R3)) ∩ C2(Ω × Q0; R3). Then, +there exists a sequence {wk} ⊂ L2(Ω; W 1,2 +per(R3; R3)) such that ∇zwk → ∇zw strongly in L2(Ω × +Q; R3×3). Besides, setting for x ∈ Ω +vk(x) := wk +� +x, x +εk +� +, +(6.5) +{vk} converges strongly two-scale to w in L2 and (5.15) holds. +Proof. We extend w by setting it equal to 0 on Q\Q0, so as to obtain a function in L2(Ω; W 1,2 +per(R3; R3)) +which, by a slight abuse of notation, we denote again by w. +Keeping in mind the definition of ΩQ +k (see (5.13)), for (¯x, ¯z) ∈ Ω × R3 we define wk(¯x, ¯z) in +terms of the averages of w( · , ¯z) on the cubes that form ΩQ +k : +wk(¯x, ¯z) := + + + + + + +εk(t+Q) +w(x, ¯z) dx +if ¯x ∈ εk(t + Q) for some t ∈ ˆTk, +0 +for any other ¯x ∈ Ω. +(6.6) +By definition, wk( · , z) is piecewise constant for all z ∈ ¯Q. Moreover, for almost every x ∈ Ω, +wk(x, · ) is Q-periodic as well as weakly differentiable, and ∇zwk → ∇zw strongly in L2(Ω × + +32 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +Q; R3×3). Indeed, from (6.6) and Jensen’s inequality, we have that +ˆ +Ω +ˆ +Q +|∇zwk(x, z) − ∇zw(x, z)|2 dz dx += +ˆ +ΩQ +k +ˆ +Q +|∇zwk(x, z) − ∇zw(x, z)|2 dz dx + +ˆ +Ω\ΩQ +k +ˆ +Q +|∇zw(x, z)|2 dz dx += +� +t∈ ˆTk +ˆ +εk(t+Q) +ˆ +Q +|∇zwk(x, z) − ∇zw(x, z)|2 dz dx + o(1) +≤ +� +t∈ ˆTk +ˆ +εk(t+Q) +ˆ +Q + +εk(t+Q) +��∇zw(ξ, z) − ∇zw +�x, z) +��2 dξ dz dx + o(1), +and the last term is infinitesimal for k → +∞ (recall that w ∈ C2 and the mean value theorem +applies). +We now turn to the functions vk given by (6.5). First of all, we point out that, thanks to +the regularity of w, vk is measurable and vanishes on Ω1 +k. Besides, it belongs to W 1,2 +0 (Ω0 +k; R3). +Secondly, we show that {vk} converges weakly two-scale to w in L2. To this aim, let us fix +φ ∈ C(¯Ω; Cper(R3; R3)). We find +ˆ +Ω +vk(x) · φ +� +x, x +εk +� +dx = +ˆ +Ω0 +k +wk +� +x, x +εk +� +· φ +� +x, x +εk +� +dx += +� +t∈Tk +ˆ +εk(t+Q0) +wk +� +x, x +εk +� +· φ +� +x, x +εk +� +dx += ε3 +k +� +t∈Tk +ˆ +Q0 wk +�εk(t + z), z +� · φ +�εk(t + z), z +� dz += +� +t∈ ˆTk +ˆ +Q0 +ˆ +εk(t+Q) +w(x, z) · φ +�εk(t + z), z +� dx dz += +ˆ +ΩQ +k +ˆ +Q0 w(x, z) · φk(x, z) dz dx, +where φk(x, z) := φ(εk(t + z), z) if x ∈ εk(t + Q) with t ∈ ˆTk. By the dominated convergence +theorem, we infer +lim +k→+∞ +ˆ +Ω +vk(x) · φ +� +x, x +εk +� +dx = +ˆ +Ω +ˆ +Q0 w(x, z) · φ(x, z) dz dx, +that is, vk +2⇀ w weakly two-scale in L2 (recall that w(x, z) = 0 if z ∈ Q1). +In order to prove that strong two-scale convergence actually holds, we study the limiting +behavior of the L2 norm of {vk}. On one hand, the weak two-scale convergence yields +∥w∥L2(Ω×Q) ≤ lim inf +k→+∞∥vk∥L2(Ω). +(6.7) +On the other hand, from the properties of {wk} and a change of variables we have the identities +ˆ +Ω +|vk(x)|2 dx = +ˆ +Ω0 +k +����wk +� +x, x +εk +����� +2 +dx = +� +t∈Tk +ˆ +εk(t+Q0) +����wk +� +x, x +εk +����� +2 +dx += +� +t∈Tk +ε3 +k +ˆ +Q0 +��wk +�εk(t + z), z +���2 dz = +� +t∈ ˆTk +ε3 +k +ˆ +Q0 +����� + +εk(t+Q) +w(x, z) dx +����� +2 +dz. + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +33 +Thanks to Jensen’s inequality we deduce +ˆ +Ω +|vk(x)|2 dx ≤ +� +t∈ ˆTk +ε3 +k +ˆ +Q0 + +εk(t+Q) +|w(x, z)|2 dx dz = +ˆ +Q0 +ˆ +ΩQ +k +|w(x, z)|2 dx dz. +This, combined with (6.7), ensures that +lim +k→+∞∥vk∥L2(Ω) = ∥w∥L2(Ω×Q). +In view of Definition 3.5 the conclusion is achieved. +Finally, the strong convergence (5.15) follows by observing that, if x ∈ εk(t + Q), it holds +∇zˆvk(x, z) = ∇zwk +�εk(t + z), z +�. +□ +We are now in a position to prove a non-degenerate Γ-upper limit inequality that is the +counterpart of the one in Proposition 5.7 under the current stronger convergence assumptions. +Proposition 6.2. Let {W 0 +k }k satisfy assumptions E3–E5. For any (w, P) ∈ L2(Ω; W 1,2 +0 (Q0; R3))× +W 1,q(Ω; SL(3)). there exists a sequence {vk} ⊂ W 1,2 +0 (Ω0 +k; R3) such that: +(1) vk +2→ w strongly two-scale in L2; +(2) εk∇vk +2⇀ ∇zw weakly two-scale in L2; +(3) whenever Pk → P uniformly, it holds +lim sup +k→+∞ +ˆ +Ω +χ0 +k(x)W 0 +k +�εk∇vk(x)P −1 +k (x) +� dx +≤ +ˆ +Ω +ˆ +Q0 Q′W 0�∇zw(x, z), P −1(x) +� dz dx, +where Q′W 0 is given by (2.8). +The conclusion is not a straightforward consequence of Lemma 6.1, because along the sequence +{vk} in (6.5) we would not end up with the correct limiting energy density. Therefore, the actual +recovery sequence is obtained by adding a “correction” to vk. +Proof of Proposition 6.2. The proof consists of several steps. At first, to circumvent measura- +bility issues, it is convenient to consider a sufficiently regular w. Under such assumption, we +are able to construct a recovery sequence of the form vk = ˜vk + ˜wk, where {˜vk} is provided by +Lemma 6.1 and { ˜wk} allows to pass from the densities W 0 +k to Q′W 0 +k . The definition of ˜wk is +given in Step 1, while Step 2 deals with the upper limit inequality in the regular case. The +general statement is eventually retrieved by approximation. +Step 1: construction of ˜wk for a regular w +Let us assume that w ∈ L2(Ω; W 1,2 +0 (Q0; R3)) ∩ C2(Ω × Q0; R3). +We consider a cover of Q0 +made of cubes whose edge length is εk. We set ˆΣk := { s ∈ Z3 : εk(s + Q) ⊂ ¯Q0 } and, for all +(t, s) ∈ ˆTk × ˆΣk, we introduce the averages +Ak(t, s) := + +εk(t+Q) + +εk(s+Q) +∇zw(x, z) dz dx +(6.8) +and the piecewise constant functions +Ak(x, z) := + + + +Ak(t, s) +if (x, z) ∈ εk(t + Q) × εk(s + Q), (t, s) ∈ ˆTk × ˆΣk, +0 +otherwise. + +34 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +We record here for later use that, by means of Lebesgue differentiation and dominated conver- +gence theorems, it follows +lim +k→+∞∥Ak − ∇zw∥2 +L2(Ω×Q) += +lim +k→+∞ +� +t∈ ˆTk +� +s∈ˆΣk +ˆ +εk(t+Q) +ˆ +εk(s+Q) +|Ak(t, s) − ∇zw(x, z)|2 dz dx += 0. +(6.9) +By the definition of Q′W 0 +k , for all k ∈ N there exists ψk ∈ W 1,2 +0 (Q; R3) such that +ˆ +Q +χ0(z)W 0 +k +��Ak(t, s) + ∇ψk(z) +�P −1 +k (x) +� +dz ≤ Q′W 0 +k +�Ak(t, s), P −1(x) +� + 1 +k. +(6.10) +Note that, due to the smoothness of w, the averages Ak are bounded uniformly in k, t and +s. +In the light of Lemma 5.1, the values Q′W 0 +k +�Ak(t, s), P −1(x) +� are uniformly bounded as +well. Therefore, by combining (6.10) with assumption E3, we deduce that {ψk} is bounded in +W 1,2 +0 (Q; R3). +A change of variables in (6.10) yields +ˆ +εk(s+Q) +χ0 +� z +εk +− s +� +W 0 +k +�� +Ak(t, s) + ∇ψk +� z +εk +− s +�� +P −1(x) +� +dz +≤ ε3 +k +� +Q′W 0 +k +�Ak(t, s), P −1(x) +� + 1 +k +� +, +(6.11) +and that suggests us to introduce the functions +˜ψk(x, z) := + + + +εkψk +� z +εk +− s +� +if (x, z) ∈ εk(t + Q) × εk(s + Q), (t, s) ∈ ˆTk × ˆΣk, +0 +otherwise. +Note that, for each k and x ∈ Ω, ˜ψk(x, · ) admits a weak derivative with respect to z; thus, by +summing over (t, s) ∈ ˆTk × ˆΣk, from (6.11) we may write +� +(t,s)∈ ˆTk׈Σk +ˆ +εk(t+Q) +ˆ +εk(s+Q) +χ0 +� z +εk +− s +� +W 0 +k +��Ak(x, z) + ∇z ˜ψk(x, z) +�P −1(x) +� +dz dx +≤ +� +(t,s)∈ ˆTk׈Σk +ˆ +εk(t+Q) +ε3 +k +� +Q′W 0 +k +�Ak(t, s), P −1 +k (x) +� + 1 +k +� +dx. +(6.12) +We also observe that, since {ψk} is bounded, ˜ψk → 0 strongly in L2(Ω × Q; R3). Then, given +that {∇z ˜ψ} is bounded L2(Ω × Q; R3×3), it must converge weakly in L2 to 0. It follows that, if +wk is as in Lemma 6.1 and if (x, z) ∈ εk(t + Q) × εk(s + Q) with (t, s) ∈ ˆTk × ˆΣk, +∇z(wk + ˜ψk) ⇀ ∇zw +weakly in L2(Ω × Q; R3×3). +(6.13) +We further notice that +˜wk(x) := ˜ψk +� +x, x +εk +� += +� +(t,s)∈ ˆTk׈Σk +εkψk +� +x +ε2 +k +− s +� +χεk(t+Q)(x)χεk(s+Q) +� x +εk +� +is a measurable function. A quick application of the definition of weak derivative proves also +that ˜wk belongs to W 1,2 +0 (Ω0 +k; R3). + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +35 +Step 2: w regular +We now turn to the proof of the limsup inequality along the sequence {vk} defined as +vk := ˜vk + ˜wk, +(6.14) +where +˜vk(x) := wk +� +x, x +εk +� +with wk as in Lemma 6.1, and where { ˜wk} was introduced in Step 1. We have +ˆvk(x, z) := Skvk(x, z) = wk +� +εk +� x +εk +� ++ εkz, z +� ++ ˜ψk +� +εk +� x +εk +� ++ εkz, z +� +, +so that if (x, z) ∈ εk(t + Q) × εk(s + Q) +∇zˆvk(x, z) = ∇zwk +�εk(t + z), z +� + ∇ψk +� z +εk +− s +� +. +(6.15) +Taking into account (6.13), (6.15) and Lemma 3.7(1), it follows that +εk∇vk +2⇀ ∇zw +weakly two-scale in L2. +Recalling Lemma 5.5, we have that +lim sup +k→+∞ +ˆ +Ω +χ0 +k(x)W 0 +k +�εk∇vk(x)P −1 +k (x) +� dx += lim sup +k→+∞ +� +t∈Tk +ˆ +εk(t+Q) +ˆ +Q0 W 0 +k +� +∇zˆvk(x, z)P −1 +k (x) +� +dz dx += lim sup +k→+∞ +Ik, +where +Ik := +� +(t,s)∈ ˆTk׈Σk +ˆ +εk(t+Q) +ˆ +εk(s+Q) +W 0 +k +�∇zˆvk(x, z)P −1 +k (x) +� dz dx. +Indeed, ˆvk vanishes if x ∈ Ω\ΩQ +k or if z ∈ Q0\∪{εk(s+Q) : s ∈ ˆΣk}, and the sequence {W 0 +k (0)} is +bounded by virtue of E3. Therefore, since the measure of Ω\ΩQ +k and of Q0\∪{εk(s+Q) : s ∈ ˆΣk} +vanish for k → +∞, the second equality holds. +Being the value of ∇zˆvk (x, z) expressed by formula (6.15), we introduce +I′ +k := +� +t,s +ˆ +εk(t+Q) +ˆ +εk(s+Q) +W 0 +k +�� +Ak(t, s) + ∇ψk +� z +εk +− s +�� +P −1 +k (x) +� +dz dx, +where the summation runs over ˆTk × ˆΣk. By exploiting assumption E4 and Hölder’s inequality, +we obtain the estimate +��Ik − I′ +k +�� ≤ c +� +t,s +ˆ +εk(t+Q) +ˆ +εk(s+Q) +��� +� +∇zwk +�εk(t + z), z +� − Ak(t, s) +� +P −1 +k (x) +��� +2 +dz dx. +In view of Lemma 6.1 and (6.9) we deduce +lim +k→+∞ +��Ik − I′ +k +�� = 0. +(6.16) +Next, let us set +I′′ +k := +ˆ +ΩQ +k +ˆ +Q0 Q′W 0 +k +�Ak(x, z), P −1 +k (x) +� dz dx. + +36 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +According to (6.12), the difference between the integrands of I′ +k and I′′ +k is of order k−1: +lim +k→+∞ +��I′ +k − I′′ +k +�� = 0. +(6.17) +Finally, we compare I′′ +k and the limiting functional. We have +�����I′′ +k − +ˆ +Ω +ˆ +Q0 Q′W 0�∇zw(x, z), P −1(x) +� dz dx +����� +≤ +ˆ +ΩQ +k +ˆ +Q0 +���Q′W 0 +k +�Ak(x, z), P −1 +k (x) +� − Q′W 0 +k +�∇zw(x, z), P −1 +k +(x) +���� dz dx ++ +ˆ +ΩQ +k +ˆ +Q0 +���Q′W 0 +k +�∇zw(x, z), P −1 +k (x) +� − Q′W 0 +k +�∇zw(x, z), P −1(x) +���� dz dx ++ +ˆ +ΩQ +k +ˆ +Q0 +���Q′W 0 +k +�∇zw(x, z), P −1(x) +� − Q′W 0�∇zw(x, z), P −1(x) +���� dz dx ++ +ˆ +Ω\ΩQ +k +ˆ +Q0 Q′W 0�∇zw(x, z), P −1(x) +� dz dx. +All the terms on the right-hand side vanish as k → +∞. Indeed, by using the Lipschitz continuity +of Q′W 0 +k (see Lemma 5.1(1)) and the uniform bound on {Pk}, the first summand is controlled +by the norm of Ak − ∇zv, which, according to (6.9), is infinitesimal. For what concerns the +second term, Lemma 5.1(2) and the uniform convergence of {Pk} imply that the integrand is +infinitesimal for k → +∞. The third quantity vanishes because {Q′W 0 +k } pointwise converges +to Q′W 0 (recall that they are just variants of the quasiconvex envelopes). Lastly, the fourth +summand is negligible since L3(Ω \ ΩQ +k ) tends to 0. +On the whole, taking into account (6.16) and (6.17), we conclude +lim +k→+∞ Ik = +ˆ +Ω +ˆ +Q0 Q′W 0�∇zw(x, z), P −1(x) +� dz dx. +Step 3: w generic +The argument follows the one of Case 2 in the proof of Proposition 5.7. +□ +6.3. A variant with plastic dissipation. With a view to applying Theorem 2.7 to time- +dependent problems, it is useful to modify the functionals Jε by adding a term that encodes the +plastic dissipation mechanism of the system. Precisely, we take into account the non-symmetric +distance D: R3×3 × R3×3 → [0, +∞] in (3.9) and we define the dissipation between P0, P1 : Ω → +SL(3) as +D(P0; P1) := +ˆ +Ω +D(P0, P1) dx. +From a physical viewpoint, if P0, P1 : Ω → SL(3) are admissible plastic strains, D(P0, P1) is +interpreted as the minimum amount of energy that is dissipated when the system moves from +a plastic configuration to another. Then, assuming that ¯P ∈ W 1,q(Ω; SL(3)) represents a pre- +existent plastic strain of the body, we set +J diss +ε +(y, P) := Eε(y, P) + D( ¯P; P) + ∥∇P∥q +Lq(Ω;R3×3×3). +(6.18) + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +37 +In the same spirit of (2.9) and (2.10), we distinguish between the dissipation of the inclusions +and the one of the matrix, respectively +D0 +ε( ¯P; P) := +ˆ +Ω +χ0 +ε(x)D( ¯P, P) dx, +D1 +ε( ¯P; P) := +ˆ +Ω +χ1 +ε(x)D( ¯P, P) dx. +For what concerns the compactness of sequences with equibounded energy, we notice that the +presence of the dissipation D does not affect Lemma 4.1: the same conclusions hold if the bound +on Jk(yk, Pk) is replaced by a bound on J diss +k +(yk, Pk). +Also our Γ-convergence results easily extend to the family {J diss +ε +}. The dissipation is indeed +a continuous perturbation: +Lemma 6.3. Let P, ¯P ∈ C(Ω; K) be given. If {Pk} ⊂ C(Ω; K) converges uniformly to P, then +lim +k→+∞ Di +k( ¯P; Pk) = L3(Qi)D( ¯P; P) +for i = 0, 1. +Proof. We firstly observe that if Pk → P pointwise, then +D +�Pk(x), P(x) +� → 0, +D +�P(x), Pk(x) +� → 0. +(6.19) +To see this, let γ be such that for all (t, F, G) ∈ [0, 1]×SL(3)×SL(3), γ(t, F, G) is the evaluation +at t of the unique minimizing geodesic connecting F and G, cf. Corollary 3.11. Then, by (3.9) +and the definition of γ, +D +�Pk(x), P(x) +� = +ˆ 1 +0 +∆ +� +γ +�t, Pk(x), P(x) +�, ˙γ +�t, Pk(x), P(x) +�� +dt +≤ c +ˆ 1 +0 +|˙γ +�t, Pk(x), P(x) +�| dt, +where the inequality follows from the definition of ∆ in (3.8) and (2.4). Since ˙γ is continuous +and bounded, by dominated convergence we deduce that the last term vanishes as k → +∞. In +a similar fashion, we show that D(P, Pk) → 0 as well. +As second step, we notice that +D +� ¯P(x), Pk(x) +� → D +� ¯P(x), P(x) +�. +(6.20) +Indeed, the triangular inequality yields +D +� ¯P(x), P(x) +� − D +�Pk(x), P(x) +� ≤ D +� ¯P(x), Pk(x) +� ≤ D +� ¯P(x), P(x) +� + D +�P(x), Pk(x) +�, +and the assertion follows as a consequence of (6.19). +Finally, we observe that (6.20) grants that +lim +k→+∞Di +k( ¯P; Pk) = +lim +k→+∞ +ˆ +Ω +χi +k(x)D +� ¯P(x), P(x) +�dx, +and the conclusion is achieved by arguing as in Lemma 4.2. +□ +Acknowledgements +We acknowledge support from the Austrian Science Fund (FWF) projects F65, V662, Y1292, +from the FWF-GAČR project I 4052/19-29646L, and from the OeAD-WTZ project CZ04/2019 +(MŠMTČR 8J19AT013). + +38 +E. DAVOLI, C. GAVIOLI, AND V. PAGLIARI +References +[1] E. Acerbi, G. Buttazzo. On the limits of periodic Riemannian metrics. J. Anal. Math. 43 (1983), 183–201. +[2] G. Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6) (1992), 1482–1518. +[3] M. Baía, I. Fonseca. The limit behavior of a family of variational multiscale problems. Indiana Univ. Math. +J. 56(1) (2007), 1–50. +[4] D. Bao, S.-S. Chern, Z. Shen. An Introduction to Riemann-Finsler Geometry. Springer Science & Business +Media, 2000. +[5] M. Barchiesi. Toughening by Crack Deflection in the Homogenization of Brittle Composites with Soft +Inclusions. Arch. Ration. Mech. Anal. 227 (2018), 749–766. +[6] M. Barchiesi, G. Lazzaroni, C.I. Zeppieri. A bridging mechanism in the homogenization of brittle +composites with soft inclusions. SIAM J. Math. Anal. 48 (2016), 1178–1209. +[7] S. Bartels, C. Carstensen, K. Hackl, U. Hoppe. Effective relaxation for microstructure simulations: +algorithms and applications. Comput. Methods Appl. Mech. Eng. 193 (2004), 5143–5175. +[8] A. Braides. Γ-convergence for beginners. Oxford University Press, 2002. +[9] A. Braides, A. Defranceschi. Homogenization of Multiple Integrals. Oxford University Press, 1998. +[10] A. Braides, A. Garroni. Homogenization of periodic nonlinear media with stiff and soft inclusions. Math. +Models Methods Appl. Sci. 5 (1995), 543–564. +[11] C. Carstensen, K. Hackl, A. Mielke. Non-convex potentials and microstructures in finite-strain plasticity. +Proc. R. Soc. Lond. Ser. A. 458 (2002), 299–317. +[12] C. Castaing, M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, +Vol. 580, Springer-Verlag, 1977. +[13] M. Cherdantsev, K.D. Cherednichenko. Two-scale Γ-convergence of integral functionals and its appli- +cation to homogenisation of nonlinear high-contrast periodic composites. Arch. Ration. Mech. Anal. 204(2) +(2012), 445–478. +[14] F. Christowiak, C. Kreisbeck. Homogenization of layered materials with rigid components in single-slip +finite crystal plasticity. Calc. Var. Partial Differential Equations 56 (2018), 75. +[15] F. Christowiak, C. Kreisbeck. Asymptotic rigidity of layered structures and its application in homoge- +nization theory. Arch. Ration. Mech. Anal. 235 (2020), 51–98. +[16] D. Cioranescu, A. Damlamian, G. Griso. Periodic unfolding and homogenization. C. R. Math. Acad. Sci. +Paris 335(1) (2002), 99–104. +[17] D. Cioranescu, A. Damlamian, G. Griso. The periodic unfolding method in homogenization. SIAM J. +Math. Anal. 40(4) (2008), 1585–1620. +[18] S. Conti, C. Reina. Kinematic description of crystal plasticity in the finite kinematic framework: a mi- +cromechanical understanding of F = FeFp. J. Mech. Phys. Solids 67 (2014), 40–61. +[19] S. Conti, C. Reina, A. Schlömerkemper. Derivation of F = FeFp as the continuum limit of crystalline +slip. J. Mech. Phys. Solids 89 (2016), 231–254. +[20] B. Dacorogna. Direct Methods in the Calculus of Variations. Springer Science & Business Media, 2008. +[21] G. Dal Maso. An Introduction to Γ-convergence. Springer Science & Business Media, 2012. +[22] E. Davoli, R. Ferreira, C. Kreisbeck. Homogenization in BV of a model for layered composites in finite +crystal plasticity. Adv. Calc. Var. 14 (2021), 441–473. +[23] E. Davoli, G.A. Francfort. A Critical Revisiting of Finite elastoplasticity. SIAM J. Math. Anal. 47 +(2015), 526–565. +[24] E. Davoli, C. Gavioli, V. Pagliari. A homogenization result in finite plasticity. Submitted. ArXiv preprint +2204.09084. +[25] E. Davoli, C. Kreisbeck. On static and evolutionary homogenization in crystal plasticity for stratified +composites. In: Research in the Mathematics of Materials Science. Springer AWM series, 2022, to appear. +[26] E. Davoli, M. Kružík, V. Pagliari. Homogenization of high-contrast composites under differential con- +straints. Adv. Calc. Var., to appear. DOI: 10.1515/acv-2022-0009. +[27] E. Davoli, T. Roubíček, U. Stefanelli. A note about hardening-free viscoelastic models in Maxwellian- +type rheologies at large strains. Math. Mech. Solids 26 (2021), 1483–1497. +[28] I. Fonseca, S. Krömer. Multiple integrals under differential constraints: two-scale convergence and homog- +enization. Indiana Univ. Math. J. 59(2) (2010), 427–457. +[29] I. Fonseca, S. Müller, P. Pedregal. Analysis of concentration and oscillation effects generated by +gradients. SIAM J. Math. Anal. 29 (1998), 736–756. +[30] D. Gilbarg, N.S. Trudinger. Elliptic Differential Equations of Second Order, 2nd ed., Springer-Verlag, +Berlin, New York, 1983. + +HOMOGENIZATION OF HIGH-CONTRAST MEDIA +39 +[31] D. Grandi, U. Stefanelli. Finite plasticity in P T P. Part I: constitutive model. Contin. Mech. Thermodyn. +29 (2017), 97–116. +[32] D. Grandi, U. Stefanelli. Finite plasticity in P T P. Part II: quasi-static evolution and linearization. SIAM +J. Math. Anal. 49 (2017), 1356–1384. +[33] K. Hackl, A. Mielke, D. Mittenhuber. Dissipation distances in multiplicative elastoplasticity. In: W. +Wendland, M. Efendiev (eds.) Analysis and Simulation of Multifield Problems, Springer, New York, 2003, +87–100. +[34] E. Kröner. Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Ration. Mech. +Anal. 4 (1959), 273–334. +[35] H. Le Dret, A. Raoult. Variational convergence for nonlinear shell models with directors and related +semicontinuity and relaxation results. Arch. Ration. Mech. Anal. 154 (2000), 101–134. +[36] E.H. Lee. Elastic-plastic deformation at finite strains. J. Appl. Mech. 36 (1969), 1–6. +[37] J. Lubliner. Plasticity theory. Macmillan Publishing Company, New York, 1990. +[38] A. Mielke. Finite Elastoplasticity Lie Groups and Geodesics on SL(d). In: P. Newton, A. Weinstein, and +P.J. Holmes (eds.) Geometry, Mechanics, and Dynamics, Springer-Verlag, New York, 2002, 61–90. +[39] A. Mielke. Energetic formulation of multiplicative elastoplasticity using dissipation distances. Contin. Mech. +Thermodyn. 15 (2003), 351–382. +[40] P.M. Naghdi. A critical review of the state of finite plasticity. Z. Angew. Math. Phys. 41 (1990), 315–394. +[41] G. Nguetseng. A general convergence result for a functional relate to the theory of homogenization. SIAM +J. Math. Anal. 20(3) (1989), 608–623. +[42] X. Pellet, L. Scardia, C.I. Zeppieri. Homogenization of high-contrast Mumford-Shah energies. SIAM J. +Math. Anal. 51 (2019), 1696–1729. +[43] A. Visintin. Some properties of two-scale convergence. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. +Rend. Lincei (9) Mat. Appl. 15(2) (2004), 93–107. +[44] A. Visintin. Towards a two-scale calculus. ESAIM Control Optim. Calc. Var. 12(3) (2006), 371–397. +E. Davoli, C. Gavioli, V. Pagliari. +Institute of Analysis and Scientific Computing, TU Wien, +Wiedner Hauptstraße 8-10, 1040 Vienna, Austria. +E-mails: elisa.davoli@tuwien.ac.at, chiara.gavioli@tuwien.ac.at, valerio.pagliari@tuwien.ac.at. + diff --git a/99A0T4oBgHgl3EQfO__U/content/tmp_files/load_file.txt b/99A0T4oBgHgl3EQfO__U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3627f91389989b19fd67e49e245bd0904d1c14b --- /dev/null +++ b/99A0T4oBgHgl3EQfO__U/content/tmp_files/load_file.txt @@ -0,0 +1,1917 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf,len=1916 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='02170v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='AP] 4 Jan 2023 HOMOGENIZATION OF HIGH-CONTRAST MEDIA IN FINITE-STRAIN ELASTOPLASTICITY ELISA DAVOLI, CHIARA GAVIOLI, AND VALERIO PAGLIARI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This work is devoted to the analysis of the interplay between internal variables and high-contrast microstructure in inelastic solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As a concrete case-study, by means of variational techniques, we derive a macroscopic description for an elastoplastic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We study the Γ-convergence of the related energy functionals as the periodicity tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The main challenge is posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We prove that the Γ-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Eventually, convergence of the energy minimizing configurations is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: 49J45;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 74B20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 74C15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 74E30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 74Q05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Keywords and phrases: finite-strain elastoplasticity, Γ-convergence, homogenization, high-contrast, two-scale convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Introduction 2 Outline 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mathematical setting and results 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Preliminaries 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A decomposition lemma 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A couple of tools to deal with periodic heterogeneous media 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Two-scale convergence and the unfolding method 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization of connected media in finite plasticity 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finsler structure on SL(3) 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Compactness and splitting 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Γ-limit of the soft component 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The limiting functional 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lower bound for the elastic energy 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Upper bound for the elastic energy 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10 28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Conclusions and a variant 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 and convergence of minimum problems 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A non degenerate upper bound for the soft component 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A variant with plastic dissipation 36 Acknowledgements 37 References 38 Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 1 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Introduction The present paper is concerned with the variational analysis of some integral functionals that model the stored energy of materials governed by finite-strain elastoplasticity with hardening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Our goal is to derive, by means of Γ-convergence, the effective macroscopic energy of a special class of heterogeneous materials, those with a so called high-contrast microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The inter- est in such media stems from the experimental observation of an infinite number of band gaps in their mechanical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In other words, high-contrast materials exhibit infinitely many interval of frequencies in which wave propagation is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This, in turn, makes them extremely interesting for possible cloaking applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Some recent ones in civil engineering, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' in seismic waves cloaking, and in the modeling of advanced sensor and actuator devices call for advancements in the mathematical modeling of those classes of high-contrast materials that have not been fully studied yet, like the ones we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The mathematical literature on high-contrast materials is vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To keep our presentation concise, we only point out that, besides results for stratified elastoplastic composites [14, 15, 22, 25], the only additional available contributions in the inelastic setting concern the study of brittle fracture problems [5, 6, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For the modeling of nonlinear elastic high-contrast composites we single out the works [10, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' When undertaking the analysis of high-contrast media beyond the elastic purview, hurdles are posed by the mathematical treatment of possible internal variables and dissipative effects, as well as by their interplay with the high-contrast microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In this paper we initiate such task by focusing on the case-study of finite elastoplasticity (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=', [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' At this first stage we neglect both the difficulties due to possible lack of coercivity for the dissipative effects and those associated with time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thus, we focus here on a static model for a single time-step with a global regularization on the gradient of the plastic strain, and leave the analysis of different regimes and the passage to the limit in the quasistatic evolutions for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The present study grounds on a previous result that we obtained in [24], where we addressed the static homogenization of elastoplastic microstructures in the large strain regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As in that work, our starting point is the description of the medium at the microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We let Ω ⊂ R3 be an open, bounded, connected set with Lipschitz boundary, and we suppose it to be the reference configuration of an elastoplastic body that exhibits the following microstructure: denoting by ε > 0 the microscale, we suppose that a stiff perforated matrix Ω1 ε sits in Ω and that its pores are filled by soft inclusions, which form the set Ω0 ε (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us denote by SL(3) the group of 3 × 3 real matrices with determinant equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' When the matrix and the inclusions exhibit the same plastic-hardening H, the functionals encoding the stored energy associated with the deformation y ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and the plastic strain P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) read Jε(y, P) := ˆ Ω0ε W 0 ε � ε∇y(x)P −1(x) � dx + ˆ Ω1ε W 1 � ∇y(x)P −1(x) � dx + ˆ Ω H �P(x) � dx + ˆ Ω |∇P(x)|q dx, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) where {W 0 ε }ε>0 and W 1 are, respectively, the elastic energy densities of the inclusions and of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us briefly comment on some modeling choices underlying position (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The factor ε multiplying the argument of W 0 ε encodes the high-contrast between the two components, and it results in a loss of coercivity in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From a modeling perspective, this heuristically means that very large deformations of the inclusions are allowed or, in other words, that the HOMOGENIZATION OF HIGH-CONTRAST MEDIA 3 inclusions are very soft – whence the expression high-contrast to describe the difference between the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As for the hardening term, note that also additional hardening variables have been taken into account in the literature, see [38, 39] for a modeling overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Here, to the purpose of putting the high-contrast behavior to the foreground, we give up full generality and restrict ourselves to the case in which only a hardening dependence on the plastic strain is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A discussion on alternative modeling choices is also presented in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Our main result describes the asymptotics of the functionals Jε, and it is presented in Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The precise mathematical framework of our analysis is described in Section 2, where further details on the definitions and on the roles of the terms in Jε may be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We work under the classical assumption that the elastic behavior of our sample Ω is indepen- dent of preexistent plastic distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, the deformation gradient ∇y associated with any deformation y: Ω → R3 of the body decomposes into an elastic strain and a plastic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the framework of linearized elastoplasticity the decomposition would take an additive form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the case at stake, that of finite plasticity [34, 36, 39, 38], the existence of an intermediate configura- tion determined by purely plastic distortions is instead assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' It is then supposed that elastic deformations are applied on such intermediate configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mathematically, these hypotheses amount to a multiplicative decomposition of the gradient of any deformation y ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3): ∇y(x) = Fel(x)P(x) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' x ∈ Ω, for a suitable elastic strain Fel ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3) and a plastic strain P ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On one hand, such multiplicative structure has recently found an atomistic validation in the framework of crystal plasticity by means of a discrete-to-continuum analysis [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On the other hand, alternative models for finite plasticity have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' However, since a discussion of fine modeling issues goes beyond the scopes of our work, we do not dwell here on a comparison of the various modeling theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We refer the reader interested on this point to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=', [23, 31, 32, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We finally comment on the regularizing term in ∇P in the energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As mentioned before, at this stage we assume it to provide coercivity of the energy with respect to the plastic-strain variables on the whole set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From a modeling point of view, we note that this regularization is common in engineering models, for it prevents the formation of microstructures, see [7, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Alternative higher order regularizations are discussed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us conclude our introduction with a few words on the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A delicate point is choosing a convergence that ensures effective compactness properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Indeed, the fact that the energy contributions in the soft inclusions are evaluated in terms of ε∇y leads to a loss of coercivity for which compactness in classical weak Sobolev topologies is prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On the other hand, arguing with strong two-scale convergence of the gradients, as in [13] does not guarantee convergence of minimizers of Jε to minimizers of the limiting functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To cope with this difficulty, we adapt the approach in [26] and introduce an ad hoc notion of convergence for deformations, to which we refer as convergence in the sense of extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Roughly speaking, a sequence of deformations converges in the sense of extensions if it is bounded in L2 and can be extended in W 1,2 in such a way that the extensions are weakly compact in the Sobolev sense, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4 and Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6 for the precise definition and some basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For the plastic strains, we argue instead with the weak convergence in W 1,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This choice is motivated by the fact that sequences of deformations and plastic strains with uniformly bounded energies are precompact with respect to the above topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thus our Γ-convergence analysis directly entails convergence of minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We observe that this result easily extends to functionals which take into account also plastic dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On this point we refer to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI The strategy relies on extension results on perforated domains, on two-scale convergence and periodic unfolding techniques, as well as on equiintegrability arguments to control the behavior of the microstructure close to the boundary of the set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A key-step is a splitting procedure that allows to treat the soft and the stiff parts separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The setup of our analysis and the main result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7, are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Section 3 contains some preliminary useful facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In Section 4 we discuss the equicoercivity of the energy functionals under consideration and the splitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The asymptotic behavior of the soft inclusions is characterized in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The ground is then laid for the proof of the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7, which is contained in Section 6 together with a variant including plastic dissipation and a comparison with an aforementioned result from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mathematical setting and results Hereafter, Ω is an open, bounded, and connected set with Lipschitz boundary in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Working in the 3-dimensional space is not essential, and our analysis can be easily adapted to the setting of Rd with d = 2 or d > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Real-valued 3 × 3 and 3 × 3 × 3 tensors are denoted by R3×3 and by R3×3×3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We adopt the symbol I for the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' With | · | we denote indiscriminately the Euclidean norms in R3, R3×3 and R3×3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To deal with plastic strains, we recall the classical notation SL(3) := {F ∈ R3×3 : det F = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If A ⊂ R3 is a measurable set, we will denote by L3(A) its three-dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A fundamental role in our study is played by the following notion of variational convergence, see the monograph [21] for a thorough treatment: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let X be a set endowed with a notion of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We say that the family {Gε}, with Gε : X → [−∞, +∞], Γ-converges as ε → 0 to G : X → [−∞, +∞] if for all x ∈ X and all infinitesimal sequences {εk}k∈N the following holds: (1) for every sequence {xk}k∈N ⊂ X such that xk → x, we have G(x) ≤ lim inf k→+∞ Gεk(xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) there exists a sequence {xk}k∈N ⊂ X such that xk → x and lim sup k→+∞ Gεk(xk) ≤ G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' When X is equipped with a topology τ, we write e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Γ(τ)-convergence to stress what the underlying convergence for sequences in X is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In what follows, for notational convenience, we indicate the dependence on εk by means of the subscript k alone, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Jk := Jεk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Our aim is to study elastoplastic media with high-contrast periodic microstructure in the case of soft inclusions inserted in a perforated stiff matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To describe the geometry in precise terms, let Q := [0, 1)3 be the periodicity cell, and let Q0 ⊂ Q be an open set such that Q1 := Q \\ Q0 is connected and has a Lipschitz boundary (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The set Ω, which represents the region of space occupied by the composite, is then subdivided by means of the sets Ω0 ε := � t∈Tε ε(t + Q0), with Tε := {t ∈ Z3 : ε(t + Q0) ⊂ Ω}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) Ω1 ε := Ω \\ Ω0ε, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2) HOMOGENIZATION OF HIGH-CONTRAST MEDIA 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The periodicity cell Q and its partition into the soft inclusion Q0 (white) and the stiff matrix Q1 (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Q0 Q0 Q1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The microstructure of the composite in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The soft inclusions that form Ω0 ε correspond to the white holes, while the grey region represents the matrix Ω1 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ε which stand respectively for the collection of the inclusions and for the matrix (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We also define the Q-periodic set E1 := � t∈Z3 (t + Q1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) where we say that a set E ⊂ R3 is Q-periodic if E + t = E for all t ∈ Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note that the set Ω1 ε is connected and Lipschitz, because (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) ensures that the inclusions are well separated from ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Our assumptions allow for some flexibility on the geometry of the inclusions, which could for instance form interconnected fibers (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Our Γ-convergence result deals with the asymptotic behavior, as ε tends to 0, of the family {Jε} defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Before stating the result, we collect the hypotheses we use in the following lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The elastic energy density of the stiff matrix W 1 : R3×3 → [0, +∞] satisfies the following: E1: It is 2-coercive and has at most quadratic growth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=', there exist 0 < c1 ≤ c2 such that for all F ∈ R3×3 c1|F|2 ≤ W 1(F) ≤ c2 � |F|2 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the 3-dimensional space, interconnected soft fibers do not discon- nect the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A simple case is depicted here: the cylindrical perforation Q0 runs through the periodicity cell and its complement Q1 is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Q0 Q1 E2: It is 2-Lipschitz: there exists c3 > 0 such that for all F1, F2 ∈ R3×3 |W 1(F1) − W 1(F2)| ≤ c3 (1 + |F1| + |F2|) |F1 − F2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The assumptions on the soft densities W 0 ε : R3×3 → [0, +∞] are analogous: E3: There exist 0 < c1 ≤ c2 such that for all F ∈ R3×3, and all ε > 0, c1|F|2 ≤ W 0 ε (F) ≤ c2 � |F|2 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' E4: There exists c3 > 0 such that for all F1, F2 ∈ R3×3, and all ε > 0, ���W 0 ε (F1) − W 0 ε (F2) ��� ≤ c3 (1 + |F1| + |F2|) |F1 − F2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' E5: There exists W 0 : R3×3 → [0, +∞] such that for all F ∈ R3×3 lim ε→0 W 0 ε (F) = W 0(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The function W 0 possesses the same growth and regularity properties of W 0 ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Our assumptions rule out non-impenetrability constraints at the level of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A blow up of the energy on matrices with non-positive determinant is desirable from a modeling point of view, but it is at the same time very hard to be handled with in the context of homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Frame indifference is instead compatible with our hypotheses and we point out that, up to a normalization, we can require all energy densities to vanish on the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We list next the assumptions on the hardening H : R3×3 → [0, +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' H1: Assume that a Finsler structure on SL(3) is assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' H(F) is finite if and only if F ∈ K, where K ⊂ SL(3) is a geodesically convex, compact neighborhood of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' H2: The restriction of H to K is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The requirement that K is geodesically convex with respect to the Finsler structure assigned on SL(3) is the crucial ingredient to invoke [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2], which in our context is employed to capture the asymptotic behaviour of the stiff matrix, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We refer to [24] for a discussion on the role of the Finsler geometry for the homogenization of elastoplastic media, and to Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5 for a summary of the tools from that theory that we need here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In particular, the existence of a set K complying with H1 is settled in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Requirement H1 prescribes that the effective domain of H coincides with a compact set K containing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' It follows then there exists cK > 0 such that |F| + |F −1| ≤ cK for every F ∈ K, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) HOMOGENIZATION OF HIGH-CONTRAST MEDIA 7 because SL(3) is by definition well separated from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As a consequence, plastic strains with finite hardening are uniformly bounded in L∞, and, in particular, we infer that for any F ∈ K and G ∈ R3×3 |G| = ���GF −1F ��� ≤ cK ���GF −1��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note that in principle it would be reasonable to suppose that the soft and the stiff components feature different hardening behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For instance, it could be imposed that the soft hardening is evaluated on an ε-rescaling of the plastic stress, thus replicating the structure of the elastic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As the only available tool to deal with periodic homogenization at finite strains is [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2], we leave such scenarios for possible future investigation and we restrain ourselves to a simpler setting, namely we choose to model both hardening terms by a single function satisfying H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We point out that under these assumptions making a distinction between Hi = Hi(P), i = 0, 1 would not require any substantial change in our approach, therefore we dispense with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Qualitatively, keeping the soft hardening contribution of order 1 amounts to the situation in which, for small ε, elastic deformations of a much larger magnitude than the plastic ones are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We can now state the homogenization result for high-contrast elastoplastic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since we want our analysis to yield convergence of minima and minimizers of Jε to the ones of the limiting energy, we need to introduce a convergence that is compliant with the degeneracy of the soft inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For shortness, we refer to it as convergence in the sense of extensions, even though the name is not at all standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {εk} be an infinitesimal sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We say that {yk} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) con- verges to y ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) in the sense of extensions with respect to the scales εk if the following hold: (1) {yk} is bounded in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) there exists a sequence {˜yk} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that yk = ˜yk in Ω1 k := Ω1 εk and ˜yk ⇀ y weakly in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let ˜yk = ˜y′ k a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' in Ω1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let as well ˜yk ⇀ y and ˜y′ k ⇀ y′ weakly in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) and observing that Ω ∩ εkE1 ⊂ Ω1 k, 0 = lim k→+∞ ˆ Ω1 k |˜yk − ˜y′ k| dx ≥ lim k→+∞ ˆ Ω χεkE1(x)|˜yk − ˜y′ k| dx = c ˆ Ω |y − y′| dx, for a constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From this, we conclude that y = y′ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In particular, if the limit in the sense of extensions exists, then it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By the definition of Ω1 k, there exists a tubular neighborhood O of ∂Ω such that Ω1 k ∩ O ≡ Ω ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, if y and ˜y coincide in Ω1 k, their traces on ∂Ω are also equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The asymptotic behavior of the family {Jε} with respect to the notion of convergence that we have just introduced is described in the next theorem: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {W 1} and {W 0 ε } satisfy E1–E5, and let H satisfy H1–H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For all y ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and P ∈ Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) there exists J (y, P) := Γ- lim ε→0 Jε(y, P), where the underlying convergences are the one in the sense of extensions and the uniform one, respectively for the first and for the second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The Γ-limit is characterized as follows: J (y, P) = J 0(0, P) + J 1(y, P), 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI where J 0(y, P) := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 L3(Q0) ˆ Ω � Q′W 0�∇y(x), P −1(x) �+H �P(x) �� dx if y = 0 and P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K), +∞ otherwise in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) and J 1(y, P) := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˆ Ω � � W 1 hom �∇y(x), P(x) � + L3(Q1)H �P(x) � + |∇P(x)|q� dx if (y, P) ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K), +∞ otherwise in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7) Here, for F, G ∈ R3×3, Q′W 0(F, G) := inf �ˆ Q W 0��F + ∇v(z) �G � dz : v ∈ W 1,2 0 (Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8) while � W 1 hom(F, G) := lim λ→+∞ 1 λ3 inf � ˆ (0,λ)3∩E1 W 1��F + ∇y(x) �G−1� dx : y ∈ W 1,2 0 ((0, λ)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The formula defining Q′W 0 provides a variant of the classical quasiconvex envelope of W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We refer to Section 5 for further discussion on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In principle, it cannot be excluded that some nontrivial energy densities W 0 ε do not contribute to the elastic homogenized energy, in the sense that, when finite, for the corresponding J 0 we have J 0(0, P) = L3(Q0) ˆ Ω H �P(x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As an instance of this phenomenon, we consider the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For any F ∈ R3×3, we let W 0 ε (F) = W 0(F) := |F|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Conditions E3–E5 are satisfied by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since for any fixed G ∈ R3×3 the function F �→ W 0 G(F) := W 0(FG) is convex, it is, in particular, also quasiconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hence, Q′W 0(0, G) = W 0(0, G) = W 0(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As a byproduct of our asymptotic analysis, we are in a position to infer convergence of the minimum problems associated with the energy functionals and of the related (quasi) minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let the same assumptions and notation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 hold, and let {(yk, Pk)} ⊂ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) be a sequence of almost minimizers, that is, lim k→+∞ � Jk(yk, Pk) − inf Jk(y, P) � = 0, where the infimum is taken over W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, there exists a minimizer (y, P) ∈ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) of J such that, up to subsequences, yk → y in the sense of extensions and Pk → P uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Moreover, inf Jk → min J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 9 The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 consists of three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' First, we study the compactness properties of sequences {(yε, Pε)} satisfying supε Jε(yε, Pε) ≤ C, and characterize their limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Second, we show that the two components of the material can be studied independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finally, we perform the analysis of each single component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In view of this approach, it is useful to introduce the functionals that account for the two different contributions, namely E0 ε (y, P) := ˆ Ω χ0 ε(x) � W 0 ε � ε∇y(x)P −1(x) � + H �P(x) �� dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) E1 ε (y, P) := ˆ Ω χ1 ε(x) � W 1 � x, ∇y(x)P −1(x) � + H �P(x) �� dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10) where, for i = 0, 1, χi ε(x) denotes the characteristic function of Ωi ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' χi ε(x) = 1 if x ∈ Ωi ε and χi ε(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We also decompose the functional Jε accordingly: Jε = J 0 ε + J 1 ε , with J 0 ε (y, P) := � E0 ε (y, P) if (y, P) ∈ W 1,2 0 (Ω0 ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K), +∞ otherwise in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) J 1 ε (y, P) := � E1 ε (y, P) + ∥∇P∥q Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3×3×3) if (y, P) ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K), +∞ otherwise in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12) In contrast to J 1 ε (y, P), whose asymptotic behavior is derived from [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2], the soft part requires a dedicated treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This happens already in the setting of nonlinear elasticity (see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Recall the topology τ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We obtain the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let (v, P) ∈ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For an infinitesimal sequence {εk}, consider J 0 k and J 0 as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (1) For every sequence {(vk, Pk)} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) such that (εkvk, Pk) τ→ (v, P) we have J 0(v, P) ≤ lim inf k→+∞ J 0 k (vk, Pk), provided that {vk} is bounded in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and that {εk∇vk} is 2-equiintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) There exists a sequence {vk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that εkvk → v in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and that lim sup k→+∞ J 0 k (vk, Pk) ≤ J 0(v, P), provided Pk → P uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the statement above, the space W 1,2 0 (Ω0 ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) is regarded for each ε as a subset of W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) by extending its elements to 0 on Ω1 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let Ω ⊂ R3 be bounded Lipschitz domain and, for p > 1, let us consider the local integral functionals on W 1,p(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) v �→ ˆ Ω Wk(∇v) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If the energy densities {Wk} satisfy standard p-growth conditions, as a consequence of Rellich- Kondrachov theorem, the Γ-limits with respect to the strong Lp-convergence and with respect to the weak W 1,p-convergence coincide (if they exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI For the sequence of functionals v �→ ˆ Ω Wk(εk∇v) dx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) again under standard growth conditions for {Wk}, the analysis is more delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The natural bound that follows from the p-coercivity is ∥εk∇vk∥Lp ≤ C, and it suggests the use of weak two- scale convergence (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' However, this estimate alone is not enough to deduce convergence of the sequence {vk}: a further control on the ε-difference quotients is required to guarantee that a two-scale variant of Rellich-Kondrachov theorem holds (see [44, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In other words, in our degenerate setting, compactness of sequences of gradients, say {εk∇vk}, does not bring compactness of {vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This explains why in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10 we need to require a bound also for ∥vk∥L2 in order to establish the lower limit inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We note incidentally that, by means of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(4) below, it can be shown that the Γ-limit of the functionals (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) with respect to the strong two-scale convergence in Lp of {vk} is the same as the one computed by combining the latter convergence and the weak two-scale convergence of {εk∇vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Those are not suitable choices for our goals, though, because, as we commented above, they do not match the natural compactness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This explains why in [13], where strong two-scale convergence is considered, the asymptotic behavior of minimum problems is not immediately determined by the Γ-convergence (see [13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We also refer to the Appendix for a comparison between our findings and the ones in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Preliminaries We gather in this section the technical tools to be employed in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A decomposition lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In our analysis of heterogeneous media it will be often desir- able to disregard the energy contributions arising from the region close to ∂Ω, for the composite fails to be periodic there (recall positions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To this aim, it is natural to resort to p-equiintegrability arguments, because such boundary strip has small measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We recall that a family C ⊂ Lp(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) is said to be p-equiintegrable if for all δ > 0 there exists m > 0 such that sup u∈C ˆ E |u|p dx < δ whenever E ⊂ Ω satisfies L3(E) < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The ensuing lemma grants that for any bounded sequence in Lp we can always find another one which is p-equiintegrable and “does not differ too much” from the given one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='20 in [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' see also Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 in [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let Ω be as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For any sequence {vk} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that vk ⇀ v weakly in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) there exist a subsequence {kj} and a sequence {uj} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) satisfying the following: (1) uj ⇀ v weakly in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) uj = v in a neighborhood of ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3) {∇uj} is 2-equiintegrable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4) limj→+∞ L3({x ∈ Ω : vkj(x) ̸= uj(x)}) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Point (4) yields limj→+∞ L3({∇vkj ̸= ∇uj}) = 0, because by standard properties of Sobolev functions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [30, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7]) the inclusion {vkj ̸= uj} ⊇ {∇vkj ̸= ∇uj} holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A couple of tools to deal with periodic heterogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The periodic geome- try of the composite calls for an extension result for Sobolev maps on perforated domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since the perforations of the matrix are well detached from the boundary, by applying [9, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7] the following can be proved: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 (Lemma 8 in [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let Ω be open and bounded, and let Ω1 ε be as in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' There exists a linear and continuous extension operator Tε: W 1,2(Ω1 ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) → W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that for all y ∈ W 1,2(Ω1 ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) Tεy = y a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' in Ω1 ε, ∥Tεy∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3) ≤ c ∥y∥L2(Ω1ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3), ∥∇(Tεy)∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3×3) ≤ c ∥∇y∥L2(Ω1ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3×3), where c is independent of ε and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Even though the lemma above is a classical result, it is worth clarifying the way we employ it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the sequel, we always work with sequences which are already defined on the whole Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' When we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 to such a sequence, say {yε} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), it is tacitly understood that the functions that are extended are the restrictions yε⌞Ω1 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' So, in a sense, the process modifies yε on the region occupied by the soft inclusions rather than extending it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note that the modification is a true one, because Tε cannot be the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The two crucial points for our analysis are that (1) if {yε⌞Ω1 ε} and {∇yε⌞Ω1 ε} are bounded in L2, then {Tεyε} is bounded in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) if {yε} is bounded in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and {∇yε} is a 2-equiintegrable sequence, then {∇(Tεyε)} is 2-equiintegrable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The second point follows from the construction of Tε, which is modeled on the proof of [9, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8] by patching together the extensions from W 1,2(Q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) to W 1,2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) given by [9, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7] via partitions of unity (this is also the reason why the constant c above depends only on Q1 and Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The extensions in [9, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7] preserve equiintegrability, because they rely on the classical reflection procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The first application of the extension lemma is the following Poincaré inequality on periodic heterogeneous media (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) in [2] where, however, the proof is not provided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let Ω, Ω0 ε and Ω1 ε be as in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' There exists a constant c independent of ε, and such that for every y ∈ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) ∥y∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3) ≤ c � ε∥∇y∥L2(Ω0ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3×3) + ∥∇y∥L2(Ω1ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3×3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For ε fixed, we use the extension operator Tε from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 to obtain ∥y∥L2 ≤ ∥y − Tεy∥L2 + ∥Tεy∥L2 = ∥y − Tεy∥L2(Ω0ε) + ∥Tεy∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) Observe that Tεy ∈ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), as Tεy = y a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' in Ω1 ε and there exists a tubular neighborhood O of ∂Ω such that Ω1 ε ∩ O ≡ Ω ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, by the standard Poincaré’s inequality, ∥Tεy∥L2 ≤ c∥∇(Tεy)∥L2 ≤ c∥∇y∥L2(Ω1ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2) 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Observe that y − Tεy ∈ W 1,2 0 (Ω0 ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In view of the periodic structure of Ω0 ε and of Poincaré inequality on each cube, we infer ∥y − Tεy∥2 L2(Ω0ε) = � t∈Tε ∥y − Tεy∥2 L2(ε(t+D0)) = � t∈Tε ε3 ˆ D0 |y(ε(t + z)) − Tεy(ε(t + z))|2 dz ≤ c � t∈Tε ε5 ˆ D0 |∇(y − Tεy)(ε(t + z))|2 dz = cε2∥∇(y − Tεy)∥2 L2(Ω0ε), where c depends only on D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By applying again Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 we find ∥y − Tεy∥L2(Ω0ε) ≤ c � ε∥∇y∥L2(Ω0ε) + ∥∇y∥L2(Ω1ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2), yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Two-scale convergence and the unfolding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From a mathematical perspective, the high-contrast structure of the functional Jε results in the absence of uniform bounds in L2 for sequences with equibounded energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' indeed, only bounds on {ε∇yεP −1 ε } are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Such degenerate bounds are conveniently dealt with by means of two-scale convergence [2, 41], whose definition we recall next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hereafter, the subscript per denotes spaces of Q-periodic functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 per(R3) := {u ∈ W 1,2 loc (R3) : u(x + t) = u(x) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' for all t ∈ Z3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {εk} ⊂ (0, +∞) be infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A sequence {yk} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) weakly two- scale converges in L2 to a function y ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' L2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) if for every v ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cper(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) lim k→+∞ ˆ Ω yk(x) · v � x, x εk � dx = ˆ Ω ˆ Q y(x, z) · v(x, z) dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A sequence {yk} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) strongly two-scale converges in L2 to y ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' L2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) if yk 2⇀ y in L2 and ∥yk∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3) → ∥y∥L2(Ω×Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We use the notations yk 2⇀ y and yk 2→ y for the weak and strong two-scale convergence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Recalling that for i = 0, 1 χi k(x) = 1 if x ∈ Ωi k and χi k(x) = 0 otherwise, an example of strong two-scale convergence is provided by the sequences {χ0 k} and {χ1 k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Indeed, χi k 2→ χi strongly two-scale in L2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) where χi(x, z) := χQi(z) for all (x, z) ∈ Ω × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We collect in the next lemma some basic properties of two-scale convergence which we will resort to in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proofs and more details can be found in [2, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {εk} ⊂ (0, +∞) be infinitesimal and consider {yk} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (1) If {yk} is weakly two-scale convergent, then it is bounded in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' conversely, if {yk} is bounded in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), then it admits a weakly two-scale convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) If yk 2⇀ y weakly two-scale in L2, then yk ⇀ ´ Q y( · , z) dz weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3) If yk 2⇀ y weakly two-scale in L2 and if {uk} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) converges to u strongly two- scale in L2, then ykuk 2⇀ yu weakly two-scale in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 13 (4) Suppose that {yk} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and that {yk} and {εk∇yk} are bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, there exists y ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) such that, up to subsequences, yk 2⇀ y and εk∇yk 2⇀ ∇zy weakly two-scale in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Two-scale convergence in L2 can be related to L2 convergence by means of unfolding operator, which, for ε > 0, is the map Sε : L2(Ω) → L2(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' L2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) defined as Sεy(x, z) := ˆy � ε �x ε � + εz � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) where ˆy denotes the extension of y by 0 outside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If {yε} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) is bounded, the following hold: (1) yε 2⇀ y weakly two-scale in L2 if and only if Sεyε ⇀ y weakly in L2(R3 × Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) yε 2→ y strongly two-scale in L2 if and only if Sεyε → y strongly in L2(R3 × Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In addition, if {yε} is 2-equiintegrable, the family of unfoldings {Sεyε} is as well 2-equiintegrable on R3 × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lastly, if y ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), then Sε(ε∇y)(x, z) = ∇z(Sεy)(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For a proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 and for further reading on the unfolding operator we refer to [43, 44, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization of connected media in finite plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We present a variant of [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2] that is instrumental in dealing with the analysis of the stiff matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Its proof is an adaptation of the one in [24], the most substantial difference being the use of [9, Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1] instead of [9, Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We work in the space W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)×W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) endowed with the topology τ characterized by (yk, Pk) τ→ (y, P) if and only if \uf8f1 \uf8f2 \uf8f3 yk → y strongly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), Pk → P uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let E be an open and connected set that is Q-periodic and that has Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For every (y, P) ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K), let � W(x, F) := χE(x)W 1(F), �H(x, P) := χE(x)H(P), and define Fε(y, P) := ˆ Ω � W �x ε , ∇y(x)P −1(x) � dx + ˆ Ω �H �x ε , P(x) � dx + ˆ Ω |∇P(x)|q dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) which we extend by setting Fε(y, P) = +∞ on �L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) � \\ �W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If W 1 and H satisfy E1–E2 and H1–H2, respectively, then for all (y, P) ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)×Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) the Γ-limit F(y, P) := Γ(τ)- lim ε→0 Fε(y, P) exists and we have that F(y, P) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˆ Ω � � Whom(∇y(x), P(x)) + �Hhom(P(x)) + |∇P(x)|q� dx if (y, P) ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K), +∞ otherwise in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)), 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI where � Whom : R3×3 × K → [0, +∞) and �Hhom : K → [0, +∞) are defined as � Whom(F, G) := lim λ→+∞ 1 λ3 inf �ˆ (0,λ)3 � W �x, (F + ∇y(x))G−1� dx : y ∈ W 1,2 0 ((0, λ)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) � , �Hhom(F) := ˆ Q �H(z, F) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We observe that the theorem above is similar in spirit to homogenization results for perforated domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The case at stake is however different, in that later we will deal with functions defined on the nonperforated domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This makes the analysis simpler because it spares us the need of extending SL(3)-valued Sobolev maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1, we are able to refine the choice of recovery sequences for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This will come in handy in the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Under the same assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8, for any (y, P) ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) there exists a recovery sequence (yk, Pk) for F(y, P) satisfying the following: (1) yk ⇀ y weakly in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) yk = y in a neighborhood of ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3) {∇yk} is 2-equiintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {(wk, Pk)} be a recovery sequence for F(y, P) as provided by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 to {wk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We deduce the existence of sequences {kj} and {uj} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that the sequence defined by yk := � uj if k = kj for some j ∈ N, y otherwise satisfies properties (1)–(3) and (yk, Pk) τ→ (y, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Moreover lim j→+∞ L3(Nj) = 0, where Nj := {x ∈ Ω : wkj(x) ̸= uj(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We are left to prove that {(yk, Pk)} satisfies the upper limit inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Loosely speaking, this is a consequence of the fact that passing to a 2-equiintegrable sequence “does not increase the energy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Upon passing to a subsequence, which we do not relabel, we can assume that {Fk(yk, Pk)} is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We provisionally focus just on the elastic and hardening parts of the energy Fkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' It holds ˆ Ω � W � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ∇wkjP −1 kj � + H � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pkj �� dx = ˆ Nj � W � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ∇wkjP −1 kj � + H � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pkj �� dx + ˆ Ω\\Nj � W � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ∇ujP −1 kj � + H � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pkj �� dx ≥ ˆ Ω\\Nj � W � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ∇ujP −1 kj � + H � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pkj �� dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 15 so that lim sup j→+∞ ˆ Ω � W � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ∇wkjP −1 kj � + H � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pkj �� dx ≥ lim sup j→+∞ ˆ Ω\\Nj � W � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ∇ujP −1 kj � + H � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pkj �� dx = lim sup j→+∞ ˆ Ω � W � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ∇ujP −1 kj � + H � x εkj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pkj �� dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' where the equality follows from the growth condition E1 and from the 2-equiintegrability of {∇uj} (recall that supk∈N ∥P −1 k ∥∞ ≤ C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' together with the boundedness of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, coming back to the full functional Fkj, lim j→+∞ Fkj(wkj, Pkj) ≥ lim sup j→+∞ ˆ Ω � W � x εkj , ∇wkjP −1 kj � + H � x εkj , Pkj �� dx + lim inf j→+∞ ˆ Ω |∇Pkj|q dx ≥ lim sup j→+∞ ˆ Ω � W � x εkj , ∇ujP −1 kj � + H � x εkj , Pkj �� dx + lim inf j→+∞ ˆ Ω |∇Pkj|q dx ≥ lim j→+∞ Fkj(uj, Pkj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7) Recalling that {(wk, Pk)} is a recovery sequence, we find lim k→+∞ Fk(yk, Pk) = lim j→+∞ Fkj(uj, Pkj) ≤ lim j→+∞ Fkj(wkj, Pkj) = F(y, P), which in turn yields that {(yk, Pk)} is also a recovery sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finsler structure on SL(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In order to apply the results on homogenization of elasto- plastic media in [24] we endow SL(3) with a Finsler structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In doing so, we follow [38], whose approach is based on the notion of plastic dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Such line of thought links the geometry of SL(3) to the physics of the system under consideration, and allows to conveniently include dissipation effects in the model, see Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We start from the observation that SL(3) is a smooth manifold with respect to the topology induced by the inclusion in R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For every F ∈ SL(3) the tangent space at F is characterized as TFSL(3) = Fsl(3) := {FM ∈ R3×3 : trM = 0}, and, in particular, TISL(3) coincides with sl(3) := {M ∈ R3×3 : trM = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To the purpose of endowing SL(3) with a Finsler structure, we first consider a C2 function ∆I : sl(3) → [0, +∞), on which we make the following assumptions: D1: It is positively 1-homogeneous: ∆I(cM) = c∆I(M) for all c ≥ 0 and M ∈ sl(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' D2: It is 1-coercive and has at most linear growth: there exist 0 < c4 ≤ c5 such that for all M ∈ sl(3) c4|M| ≤ ∆I(M) ≤ c5|M|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' D3: It is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note that we consider more restrictive regularity assumptions than the ones in [38], because we appeal to results of differential geometry, where smoothness is customarily required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The drawback of this choice is that in our analysis we cannot encompass some models, such as single crystal plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' However, on the positive side, our assumptions cover Von Mises plasticity, see [33, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Let TSL(3) denote the tangent bundle to SL(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We can “translate” ∆I to the tangent spaces other than sl(3) by setting ∆: TSL(3) → [0, +∞) (F, M) �→ ∆I(F −1M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8) Then, it can be proved that (SL(3), ∆) is a C2 Finsler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For an introduction to Finsler geometry we refer to the monograph [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Next, we introduce the family C(F0, F1) of piecewise C2 curves Φ: [0, 1] → SL(3) such that Φ(0) = F0 and Φ(1) = F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We set D(F0, F1) := inf �ˆ 1 0 ∆ �Φ(t), ˙Φ(t) �dt : Φ ∈ C(F0, F1) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) where ˙Φ is the velocity of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The function D provides a non-symmetric distance on SL(3): it is positive, attains 0 if and only if it is evaluated on the diagonal of SL(3) × SL(3), and satisfies the triangular inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' in general, however, D(F0, F1) ̸= D(F1, F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By the direct method of the calculus of variations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [38, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1]) it can be proved that for every F0, F1 ∈ SL(3) there exists a curve Φ ∈ C1,1([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) such that Φ(0) = F0, Φ(1) = F1 and D(F0, F1) = ˆ 1 0 ∆ �Φ(t), ˙Φ(t) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10) We call such Φ a shortest path between F0 and F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We need the following local uniqueness result for shortest paths, which wraps up the content of [4, Exercises 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For any point F in the Finsler manifold SL(3) there exists a relatively compact neighborhood U of F such that for any F0, F1 ∈ U there exists a unique shortest path Φ joining F0 and F1, and such path depends smoothly on its endpoints F0 and F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10 we deduce the existence of a set K as in H1, but we first need to recall some terminology from differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A geodesic between F0 and F1 is a path that is a critical point of the length functional under variations that do not alter the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' When for any couple of points in a given subset S of a Finsler manifold there is a unique shortest path contained in S joining those points, we say that S is geodesically convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Assume that a C2 Finsler structure on SL(3) is assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, there exists a geodesically convex, compact neighborhood of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Owing to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10, there exists a relatively compact neighborhood U of I ∈ SL(3) such that for any F0, F1 ∈ U there is a unique shortest path Φ joining F0 and F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thanks to a Finsler variant of a theorem by Whitehead [4, Exercise 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3], there is an open neighborhood V of I that is compactly contained in U and geodesically convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us set K := ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since K ⊂ U, there is a unique shortest path Φ from F0 to F1 for any F0, F1 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The fact that K is geodesically convex as well may be proved by the same argument that proves that the closure of a convex set is still convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Compactness and splitting From now on we turn to the analysis of the high-contrast energy in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We investigate in this section the compactness properties of sequences with equibounded energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We will see that, as a consequence of the behavior of the hardening functional H, we can reduce the problem to the case of pure elasticity addressed by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cherdantsev & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cherednichenko [13], and we adapt their approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 17 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 (Compactness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {εk} be an infinitesimal sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We suppose that {(yk, Pk)}k∈N ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) satisfies ∥yk∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3) ≤ C, Jk(yk, Pk) ≤ C for some C ≥ 0, uniformly in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us denote by ˜yk the extension of yk in the sense of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, there exist subsequences of {εk}, {yk}, and {Pk}, which we do not relabel, as well as y ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)), y1 ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), v ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)), and P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) such that the following hold: y(x, z) = y1(x) + v(x, z) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (x, z) ∈ Ω × Q, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) yk 2⇀ y weakly two-scale in L2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2) εk∇yk 2⇀ ∇zv weakly two-scale in L2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) ˜yk ⇀ y1 weakly in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) Pk → P, P −1 k → P −1 weakly in W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) and uniformly in C(¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)), ∇˜ykP −1 k ⇀ ∇y1P −1 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From the definition of Jk, for all k ∈ N ∥∇Pk∥Lq ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) Besides, for all k, hypothesis E3, the definition of H and the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) imply ���εkχ0 k∇ykP −1 k ��� L2 + ���χ1 k∇ykP −1 k ��� L2 ≤ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7) ∥Pk∥L∞ + ���P −1 k ��� L∞ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8) Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5), from the first estimate we deduce ���εkχ0 k∇yk ��� L2 + ���χ1 k∇yk ��� L2 ≤ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) which is precisely formula (21) in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thus, for what concerns the sequence of deformations, the same bounds as the purely elastic case are retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' While referring to [13] for details, here we limit ourselves to sketch how (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) entails two-scale compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The boundedness of {yk} in L2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(4) yield the existence of a function y ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) such that, up to subsequences, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2) holds and εk∇yk 2⇀ ∇zy weakly two-scale in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10) Thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(3), we also infer that χ1 kyk 2⇀ χ1y, εkχ1 k∇yk 2⇀ χ1∇zy weakly two-scale in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Moreover, there exist y1 ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and v ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) such that the decomposi- tion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) and the convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We now turn to the sequence of plastic strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8), we see that {Pk} is bounded in W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since q > 3, Morrey’s embedding yields the uniform convergence of (a subsequence of) {Pk} to some P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, by definition of the inverse matrix P −1 k = (cofPk)T det Pk = (cofPk)T , we also deduce that P −1 k → P −1 uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Finally, we observe that, thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) and the uniform convergence of {P −1 k }, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) is also inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ It is well-known that Γ-limits are not additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In our case, however, we are able to show that the asymptotic behavior of the functionals Jε is given exactly by the sum of the Γ-limits of the soft and of the stiff contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Such splitting will enable us to treat the Γ-limits of J 0 ε and of J 1 ε separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We premise a simple lemma, which deals with the hardening part of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We recall that, for i = 0, 1, χi k is the characteristic function of Ωi k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Under assumptions H1–H2, for any sequence {Pk} ⊂ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) converging uniformly to P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) it holds lim k→+∞ ˆ Ω χi k(x)H �Pk(x) � dx = L3(Qi) ˆ Ω H �P(x) � dx for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us focus on the case i = 0 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We set E0 := � t∈Z3 (t + Q0) = R3 \\ E1, ˆΩ0 k := � t∈ ˆTk εk(t + Q0), where ˆTk := {t ∈ Z3 : εk(t + Q) ⊂ Ω} ⊂ Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) By definition of Ω0 k (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1)), we have εkE0 \\ Ω0 k ⊂ εkE0 \\ ˆΩ0 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note that Ω∩(εkE0 \\ ˆΩ0 k) is contained in the strip {x ∈ Ω : dist(x, ∂Ω) < √ 3εk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since {H(Pk)} is uniformly bounded by H1 and H2, we see that lim k→+∞ ˆ Ω χ0 k(x)H �Pk(x) � dx = lim k→+∞ ˆ Ω χεkE0(x)H �Pk(x) � dx − lim k→+∞ ˆ Ω �χεkE0(x) − χ0 k(x) �H �Pk(x) � dx = lim k→+∞ ˆ Ω χεkE0(x)H �Pk(x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, by the Lipschitz continuity of H on its domain, lim k→+∞ ˆ Ω χεkE0(x)H �Pk(x) � dx = lim k→+∞ ˆ Ω χεkE0(x)H �P(x) � dx = L3(Q0) ˆ Ω H �P(x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The case i = 1 follows from the previous one by the identities χ1 k = χΩ − χ0 k and L3(Q1) = 1 − L3(Q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ The splitting process is explained by the ensuing proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3 (Splitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {εk} be an infinitesimal sequence, and let {(yk, Pk)}k∈N ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) be a sequence satisfying ∥yk∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3) ≤ C, Jk(yk, Pk) ≤ C HOMOGENIZATION OF HIGH-CONTRAST MEDIA 19 for some C ≥ 0, uniformly in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let ˜yk be the extension of yk in the sense of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3, and let v ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) be as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, defining vk := yk − ˜yk, the following hold: {vk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), ∥vk∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3) ≤ C, εk∇vk 2⇀ ∇zv weakly two-scale in L2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12) lim inf k→+∞ J 0 k (vk, Pk) + lim inf k→+∞ J 1 k (˜yk, Pk) ≤ lim inf k→+∞ Jk(yk, Pk), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) lim sup k→+∞ Jk(yk, Pk) ≤ lim sup k→+∞ J 0 k (vk, Pk) + lim sup k→+∞ J 1 k (˜yk, Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14) Moreover, in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13), {vk} may be replaced with another sequence {wk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that {εk∇wk} is 2-equiintegrable and εk∇wk ⇀ 0 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We first prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12) – (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14) hold for the sequence {vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Afterwards, we will show how to recover the equiintegrability for the sequence of gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We split the functional Jk evaluated on (yk, Pk) as follows: Jk(yk, Pk) = J 0 k (yk, Pk) + J 1 k (yk, Pk) = J 0 k (vk, Pk) + J 1 k (˜yk, Pk) + Rk, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15) where J 0 k and J 1 k are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12), and Rk := J 0 k (yk, Pk) − J 0 k (vk, Pk) = ˆ Ω χ0 k � W 0 ε � εk∇ykP −1 k � − W 0 ε � εk∇vkP −1 k �� dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We next show that Rk is asymptotically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hypothesis E4 yields |Rk| ≤ c3 ˆ Ω χ0 k � 1 + ���εk∇ykP −1 k ��� + ���εk∇vkP −1 k ��� � ���εk∇˜ykP −1 k ��� dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='16) Since {(yk, Pk)} is equibounded in energy, the sequences {εkχ0 k∇ykP −1 k }, {χ1 k∇ykP −1 k }, and {P −1 k } are bounded in suitable Lebesgue spaces (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By the properties of the extension operator Tε in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2, we deduce that ˆ Ω ���∇˜ykP −1 k ��� 2 dx ≤ c ˆ Ω |∇˜yk|2 dx ≤ c ˆ Ω ���χ1 k∇yk ��� 2 dx ≤ c ˆ Ω ���χ1 k∇ykP −1 k ��� 2 dx ≤ C (recall estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' So, thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3), we deduce that εk∇vk = εk∇yk − εk∇˜yk 2⇀ ∇zv weakly two-scale in L2, In particular, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(1), {εkχ0 k∇vkP −1 k } is bounded in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By applying Hölder’s inequality to the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='16), we then find Rk = O(εk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Owing to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15) we conclude that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To complete the proof, we are only left to establish the existence of the sequence {wk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Upon extraction of a subsequence, which we do not relabel, we may assume that in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) the lower limit involving J 0 k is a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From the equiboundedness of the energy, by arguing as in the lines before (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9), we get ∥εk∇yk∥L2 ≤ C, ∥χ1 k∇yk∥L2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='17) 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Then, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) holds and, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(2), we obtain εk∇yk ⇀ 0 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 applied to the sequence {εk∇yk} yields two sequences, {kj} and {uj} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), such that {εkj∇uj} is 2-equiintegrable, εkj∇uj ⇀ 0 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='18) lim j→+∞L3(Nj) = 0, with Nj := {x ∈ Ω : ykj(x) ̸= uj(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Besides, we have εkjχ1 kj∇uj → 0 strongly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='19) Indeed, it holds ∥εkjχ1 kj∇uj∥L2 = ∥εkjχ1 kj∇uj∥L2(Nj) + ∥εkjχ1 kj∇ykj∥L2(Ω\\Nj) ≤ ∥εkj∇uj∥L2(Nj) + εkj∥χ1 kj∇ykj∥L2, and the conclusion follows by the 2-equiintegrability of {εkj∇uj} and from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We now define ˜uj := Tkjuj, with Tkj as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3 it follows that {εkj∇˜uj} is 2-equiintegrable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thus, the sequence defined by wk := � uj − ˜uj if k = kj for some j ∈ N, 0 otherwise has the properties that wk ∈ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and {εk∇wk} is 2-equiintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Moreover, εk∇wk ⇀ 0 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To see this, we write εkj∇wkj = εkj∇uj − εkj∇˜uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The first term converges to 0 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3), as stated in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Additionally, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 entails ∥εkj∇˜uj∥L2 ≤ c∥εkjχ1 kj∇uj∥L2, and the weak convergence of {εk∇wk} follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We are now ready to prove the validity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) when {εk∇vk} is replaced by the 2-equiintegrable sequence {εk∇wk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By the definition of the sequence at stake, we have εkj(∇vkj − ∇wkj) = εkj(∇ykj − ∇uj) − εkj(∇˜ykj − ∇˜uj) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='20) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 yields εkj∥∇˜ykj − ∇˜uj∥L2 = εkj∥∇ �Tkj(ykj − uj) �∥L2 ≤ cεkj∥χ1 kj∇(ykj − uj)∥L2 = cεkj∥χ1 kj(∇ykj − ∇uj)∥L2(Nj) ≤ c � εkj∥χ1 kj∇ykj∥L2 + ∥εkj∇uj∥L2(Nj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='17) and the 2-equiintegrability of {εkj∇uj} entail εkj � ∇˜ykj − ∇˜uj � → 0 strongly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='21) HOMOGENIZATION OF HIGH-CONTRAST MEDIA 21 Therefore, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='20) and the fact that the densities W 0 kj are bounded from below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' we have ˆ Ω χ0 kj(x)W 0 kj �εkj∇vkj(x)P −1 kj (x) � dx = ˆ Nj χ0 kj(x)W 0 kj �εkj∇vkj(x)P −1 kj (x) � dx + ˆ Ω\\Nj χ0 kj(x)W 0 kj ��εkj∇wkj(x) − εkj(∇˜ykj(x) − ∇˜uj(x)) �P −1 kj (x) � dx − ˆ Ω\\Nj χ0 kj(x)W 0 kj �εkj∇wkj(x)P −1 kj (x) � dx + ˆ Ω\\Nj χ0 kj(x)W 0 kj �εkj∇wkj(x)P −1 kj (x) � dx ≥ −c �ˆ Ω\\Nj |εkj(∇˜ykj(x) − ∇˜uj(x))|2 dx �1/2 + ˆ Ω\\Nj χ0 kj(x)W 0 kj �εkj∇wkj(x)P −1 kj (x) � dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' where the Lipschitz regularity E4 and Hölder’s inequality were employed to derive the last bound (recall that supk∈N ∥P −1 k ∥∞ ≤ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We now take the limit in the inequality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2, the hardening term has a limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, also the elastic contribution is convergent, and it satisfies lim k→+∞ J 0 k (vk, Pk) = lim j→+∞ ˆ Ω χ0 kj(x)W 0 kj �εkj∇vkj(x)P −1 kj (x) � dx + L3(Q0) ˆ Ω H �P(x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The strong converge (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='21) implies lim j→+∞ ˆ Ω χ0 kj(x)W 0 kj �εkj∇vkj(x)P −1 kj (x) � dx ≥ lim inf j→+∞ ˆ Ω\\Nj χ0 kj(x)W 0 kj �εkj∇wkj(x)P −1 kj (x) � dx = lim inf j→+∞ ˆ Ω χ0 kj(x)W 0 kj �εkj∇wkj(x)P −1 kj (x) � dx, where the equality follows from the growth condition E3 and from the equiintegrability of {εkj∇wkj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We thereby infer lim inf k→+∞ J 0 k (wk, Pk) ≤ lim inf j→+∞ J 0 kj(wkj, Pkj) ≤ lim k→+∞ J 0 k (vk, Pk), and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Γ-limit of the soft component We devote this section to the study of the asymptotics of the functional J 0 ε in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11), which encodes the energy of the inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' After some observations on the limiting functional J 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6), in the second and third subsections we deal respectively with the lower and with the upper limit inequality for the elastic part of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The other contributions will be taken into account in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4, where we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The limiting functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The definition of Q′W 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8), which encodes the limiting elastic contribution of the soft inclusions, may be regarded as a variant of the well known Dacorogna’s formula for the quasiconvex envelope [20, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As such, the infimum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8) does not depend on Q, and we may rewrite Q′W 0 as follows: Q′W 0(F, G) = inf � Q0 W 0��F + ∇v(z) �G � dz : v ∈ W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) 22 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Note that here quasiconvexification occurs just with respect to the first argument, since a very strong convergence is considered for the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The fact that different variables in a problem may call for different relaxation procedures has been already observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As an example, we mention the concept of cross-quasiconvexity introduced by Le Dret & Raoult [35] to deal with dimension reduction problems in elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For the sake of completeness, we explicitly mention some basic properties of Q′W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let W 0 : R3×3 → R, and assume there exist 0 < c1 ≤ c2 such that for all F ∈ R3×3 c1|F|2 ≤ W 0(F) ≤ c2 � |F|2 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let Q′W 0 be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (1) For all F, G ∈ R3×3 c1|FG|2 ≤ Q′W 0(F, G) ≤ c2 � |FG|2 + 1 � , and for all G ∈ R3×3 there exists c := c(G) > 0 such that for all F1, F2 ∈ R3×3 ���Q′W 0(F1, G) − Q′W 0(F2, G) ��� ≤ c (1 + |F1| + |F2|) |F1 − F2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Suppose further that there exists c3 > 0 such that for all F1, F2 ∈ R3×3 ���W 0(F1) − W 0(F2) ��� ≤ c3 (1 + |F1| + |F2|) |F1 − F2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2) (2) Then, Q′W 0(F, · ) is continuous for all F ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3) If {Pk} ⊂ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) converges weakly to P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)), then for any V ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3) lim k→+∞ ˆ Ω Q′W 0�V (x), P −1 k (x) � dx = ˆ Ω Q′W 0�V (x), P −1(x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The growth conditions on Q′W 0 are an immediate consequence of the ones on W 0 and of the definition of Q′W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For what concerns the 2-Lipschitz property, let us set W 0 G(F) := W 0(FG) for any fixed G ∈ R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, Q′W 0( · , G) coincides with the quasiconvex envelope of W 0 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By [20, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3(iii)] it follows that Q′W 0( · , G) is separately convex, and hence, in view of the growth assumptions on W 0, the proof of item (1) is concluded by [20, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As for point (2), let Gk → G in R3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In view of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2), for every δ > 0 there exists cδ > 0 such that Q′W 0(F, Gk) − Q′W 0(F, G) ≤ cδ|Gk − G| + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Similarly, for any k ∈ N there exists vk ∈ W 1,p 0 (Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3) such that Q′W 0(F, Gk) − Q′W 0(F, G) ≥ −c3|Gk − G| ˆ Q �1 + |(F + ∇vk)Gk| + |(F + ∇vk)G| �|F + ∇vk| dx − 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thanks to the coercivity of the integrand, it follows that {∇vk} is bounded in L2, whence Q′W 0(F, Gk) − Q′W 0(F, G) ≥ −c |Gk − G| − 1 k for a constant c independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The continuity of Q′W 0(F, · ) is then proved by letting first k → +∞ and then δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 23 Eventually, taking into account points (1) and (2), as well as the compact embedding of W 1,q(Ω) into C(¯Ω), we can employ the dominated convergence theorem to obtain the continuity property in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ We now exhibit an alternative expression for the soft limiting elastic energy, which is to be exploited in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For every couple (V, P) ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) we have ˆ Ω Q′W 0�V (x), P −1(x) � dx = inf �ˆ Ω Q0 W 0��V (x) + ∇zw(x, z) �P −1(x) � dz : w ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) The identity above rests on a measurable selection criterion that we recall next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10 in [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let S be a multifunction defined on the measurable space X and taking values in the collection of subsets of the separable metric space Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If S(x) is nonempty and open in Y for every x ∈ X, and if the set { x ∈ X : y ∈ S(x) } is measurable for every y ∈ Y , then S admits a measurable selection, that is, there exists a measurable function s: X → Y such that s(x) ∈ S(x) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The previous lemma is a variant of [12, Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6], and we refer to that monograph for a comprehensive treatment of measurable selection principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The argument follows the one proposed in [28, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us fix w ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)), so that, for almost every x ∈ Ω, w(x, · ) ∈ W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hence, according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1), we have Q′W 0�V (x), P −1(x) � ≤ Q0 W 0��V (x) + ∇zw(x, z) �P −1(x) � dz for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' x ∈ Ω, whence, after integration over Ω, we deduce that in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) the left-hand side is smaller that the righ-hand one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In order to establish the opposite inequality, we first observe that, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1), for every x ∈ Ω and every δ > 0 there exists vx,δ ∈ W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that Q0 W 0��V (x) + ∇vx,δ(z) �P −1(x) � dz − Q′W 0�V (x), P −1(x) � < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) We introduce the multifunction S defined for x ∈ Ω by S(x) := � v ∈ W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) : (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) holds for vx,δ = v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We show that it admits a measurable selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To this purpose, observe that, as a consequence of the growth assumptions on W 0 and of the dominated convergence theorem, S(x) is a nonempty, open subset of W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Second, for every v ∈ W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) the set {x ∈ Ω : v ∈ S(x)} is measurable, because it is the sublevel set of a measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thanks to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3, for every δ > 0 we retrieve a measurable function wδ : Ω → W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) that satisfies ˆ Ω Q0 W 0��V (x) + ∇zwδ(x, z) �P −1(x) � dz dx ≤ ˆ Ω Q′W 0�V (x), P −1(x) � + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI In particular, by the growth conditions on W 0, wδ must belong to L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' There- fore, since δ is arbitrary, we conclude that the left-hand side in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) bounds from above the right-hand one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lower bound for the elastic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The goal of this subsection is to prove the ensuing: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {W 0 k }k satisfy assumptions E3–E5, and let P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For every sequence {(vk, Pk)} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)×W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) such that {εk∇vk} is 2-equiintegrable and Pk → P uniformly, it holds L3(Q0) ˆ Ω Q′W 0�0, P −1(x) � dx ≤ lim inf k→+∞ ˆ Ω χ0 k(x)W 0 k �εk∇vk(x)P −1 k (x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) At a first glance, it may look bizarre that no convergence for the sequence {εk∇vk} is pre- scribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The statement becomes clearer once we recall that if Qf is the quasiconvex envelope of f : R3×3 → R, then Qf(0) ≤ Ω f �∇v(x) � dx for any v ∈ W 1,∞ 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In order to establish (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5), it is convenient to unfold the elastic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {W 0 k }k satisfy assumptions E3–E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For any (v, P) ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)×W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) it holds ˆ Ω χ0 k(x)W 0 k � εk∇v(x)P −1(x) � dx = � t∈Tk ˆ εk(t+Q) ˆ Q0 W 0 k � ∇zˆv(x, z) ˆP −1(x, z) � dz dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) where ˆv := Skv, ˆP := SkP and Sk := Sεk is the unfolding operator introduced in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' According to the definition of Ω0 k in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1), the left-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) equals ε3 k � t∈Tk ˆ Q0 W 0 k � εk∇v �εk(t + z) �P −1�εk(t + z) �� dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We use the unfolding operator to rewrite this quantity as a double integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Recalling Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7, we firstly observe that for every t ∈ Tk and z ∈ Q0 we have the identities Sk(εk∇v)(εkt, z) = εk∇v �εk(t + z) �, SkP −1(εkt, z) = P −1�εk(t + z) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, we also have Sk(εk∇v) = ∇z �Skv � = ∇zˆv, SkP −1 = (SkP)−1 = ˆP −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We obtain ˆ Ω χ0 k(x)W 0 k �εk∇v(x)P −1(x) � dx = ε3 k � t∈Tk ˆ Q0 W 0 k �Sk(εk∇v)(εkt, z)Sk(P −1)(εkt, z) � dz = � t∈Tk ˆ εk(t+Q) ˆ Q0 W 0 k � ∇zˆv � εk � x εk � , z � ˆP −1 � εk � x εk � , z �� dz dx, because ⌊x/εk⌋ = t for all x ∈ εk(t + Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since, in general, it holds Sku � εk � x εk � , z � = u � εk � x εk � + εkz � = Sku(x, z), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ HOMOGENIZATION OF HIGH-CONTRAST MEDIA 25 A crucial ingredient in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4 is a sort of lower semicontinuity result for the elastic contribution to the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {W 0 k }k satisfy assumptions E3–E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let also {wk} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) be such that {∇zwk} is 2-equiintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, for all P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)), L3(Q0) ˆ Ω Q′W 0�0, P −1(x) � dx ≤ lim inf k→+∞ ˆ Ω ˆ Q0 W 0 k �∇zwk(x, z)P −1 k (x) � dz dx, whenever Pk → P uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) it follows that for all k ∈ N L3(Q0) ˆ Ω Q′W 0�0, P −1 k (x) � dx ≤ ˆ Ω ˆ Q0 W 0�∇zwk(x, z)P −1 k (x) � dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7) Next, relying on the pointwise convergence of {W 0 k } to W 0, we adapt the argument in the proof of [21, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14] to pass from W 0 to W 0 k on the right-hand side (see also [26, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2] for a similar result in the context of A -quasiconvexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Fix δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If {∇zwk} is 2-equiintegrable, then so is {∇zwkP −1 k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, since the 2-growth assumptions on {W 0 k } transfer to the pointwise limit W 0, there exists r > 0 such that sup k∈N ˆ {(x,z)∈Ω×Q0:|∇zwk(x,z)P −1 k (x)|>r} W 0�∇zwk(x, z)P −1 k (x) � dz dx ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8) Owing to assumption E4 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2, we can find ρ > 0 such that for every F, G ∈ R3×3 contained in the open ball B(0, ρ) sup k∈N |W 0 k (F) − W 0 k (G)| + |W 0(F) − W 0(G)| ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) Let now F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' , Fn ∈ B(0, r) be such that B(0, r) ⊂ n � i=1 B (Fi, ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10) Due to the pointwise convergence of W 0 k to W 0, for any such Fi there exist ¯ki ∈ N such that |W 0 k (Fi)−W 0(Fi)| ≤ δ if k > ¯ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Letting ¯k := max{¯k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' , ¯kn}, it follows that for any i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' , n |W 0 k (Fi) − W 0(Fi)| ≤ δ if k > ¯k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10), for every G ∈ B(0, r) there exists i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' , n} such that G ∈ B(Fi, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For this particular i, the combination of the triangle inequality, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) yields |W 0 k (G) − W 0(G)| ≤ |W 0 k (G) − W 0 k (Fi)| + |W 0 k (Fi) − W 0(Fi)| + |W 0(G) − W 0(Fi)| ≤ 3δ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12) for every G ∈ B(0, r) and every k > ¯k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Thanks to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1(3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7) we deduce L3(Q0) ˆ Ω Q′W 0�0, P −1(x) � dx = L3(Q0) lim k→+∞ ˆ Ω Q′W 0�0, P −1 k (x) � dx ≤ lim inf k→+∞ ˆ Ω ˆ Q0 W 0�∇zwk(x, z)P −1(x) � dz dx ≤ lim inf k→+∞ ˆ {(x,z)∈Ω×Q0:|∇zwk(x,z)P −1 k (x)|≤r} W 0�∇zwk(x, z)P −1 k (x) � dz dx + δ ≤ lim inf k→+∞ ˆ Ω ˆ Q0 W 0 k �∇zwk(x, z)P −1 k (x) � dz dx + 3δL6(Ω × Q0) + δ, where the second inequality is due to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8), and the last one to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The arbitrariness of δ > 0 yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ We are now ready to prove the lower bound for the elastic contribution of the soft part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let ˆvk := Skvk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In view of the 2-equiintegrability of the sequence {εk∇vk} and of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7, {∇zˆvk} is 2-equiintegrable as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hence it is a fortiori bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5, restricting the summation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) to the set of translations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11), we deduce lim inf k→+∞ ˆ Ω χ0 k(x)W 0 k �εk∇vk(x)P −1 k (x) � dx ≥ lim inf k→+∞ ˆ ΩQ k ˆ Q0 W 0 k �∇zˆvk(x, z)P −1 k (x) � dz dx, where ΩQ k := � t∈ ˆTk εk(t + Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) We rewrite the right-hand side of the previous inequality as the difference between the integrals I′ k := ˆ Ω ˆ Q0 W 0 k �∇zˆvk(x, z)P −1 k (x) � dz dx, I′′ k := ˆ Ω\\ΩQ k ˆ Q0 W 0 k �∇zˆvk(x, z)P −1 k (x) � dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Being {∇zˆvk} 2-equiintegrable, the sequence {∇zˆvkP −1 k } is still 2-equiintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thus, by the growth condition E3, we obtain lim k→+∞ I′′ k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Taking into account Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6 we conclude lim inf k→+∞ ˆ Ω χ0 k(x)W 0 k �εk∇vk(x)P −1 k (x) � dx ≥ lim inf k→+∞ I′ k ≥ L3(Q0) ˆ Ω Q′W 0�0, P −1(x) � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ HOMOGENIZATION OF HIGH-CONTRAST MEDIA 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Upper bound for the elastic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In this subsection we address the proof of Γ-upper limit inequality for the elastic contribution of the soft component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Differently from the previous subsection, in order to establish the desired inequality we perform an analysis that is genuinely two-scale, in the sense that we interpret 0 as the average with respect to the periodic variable of the two-scale limit of the sequence {εk∇vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {W 0 k }k satisfy assumptions E3–E5, and let P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For all δ > 0 there exists a sequence {vk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that εk∇vk ⇀ 0 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3) and that lim sup k→+∞ ˆ Ω χ0 k(x)W 0 k � εk∇vk(x)P −1 k (x) � dx < L3(Q0) ˆ Ω Q′W 0�0, P −1(x) � dx + δ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14) whenever Pk → P uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We begin with a lemma that provides a strong two-scale approximation of any sufficiently regular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The result has already appeared in [13], where, however, the proof is just sketched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In order to keep the exposition self-contained, we include it in the Appendix, where we also compare our result with the one in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let w ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) ∩ C2(Ω × Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, there exists a sequence {vk} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that, letting ˆvk := Skvk, it holds ∇zˆvk → ∇zw strongly in L2(Ω × Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15) We are now ready to prove the Γ-limsup inequality for the soft inclusions functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2, for every δ > 0 there exists wδ ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) satisfying ˆ Ω ˆ Q0 W 0�∇zwδ(x, z)P −1(x) � dz dx < L(Q0) ˆ Ω Q′W 0�0, P −1(x) � dx + δ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='16) We would like to apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8 which, however, requires wδ ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3))∩C2(Ω× Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We therefore establish the inequality first for a sufficiently regular wδ, and we then extend the result by a density argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Case 1: wδ regular Let wδ ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) ∩ C2(Ω × Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We consider the recovery sequence {vk} coming from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(2) yield εk∇vk ⇀ 0 weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Assumption E4 and Hölder’s inequality entail � t∈Tk ˆ εk(t+Q) ˆ Q0 ���W 0 k � ∇zˆvk(x, z)P −1 k (x) � − W 0 k � ∇zwδ(x, z)P −1 k (x) ���� dz dx ≤ c � t∈Tk �ˆ εk(t+Q) ˆ Q0 |∇zˆvk(x, z) − ∇zwδ(x, z)|2 dz dx �1/2 , 28 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI where the constant c bounds ∥P −1 k ∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thanks to the strong convergence of {∇zˆvk}, we obtain that the term above is infinitesimal when k → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5 we then deduce lim sup k→+∞ ˆ Ω χ0 k(x)W 0 k � εk∇vk(x)P −1 k (x) � dx = lim sup k→+∞ � t∈Tk ˆ εk(t+Q) ˆ Q0 W 0 k � ∇zwδ(x, z)P −1 k (x) � dz dx = lim sup k→+∞ � t∈Tk ˆ εk(t+Q) ˆ Q0 W 0 k � ∇zwδ(x, z)P −1(x) � dz dx = lim sup k→+∞ � t∈Tk ˆ εk(t+Q) ˆ Q0 W 0 � ∇zwδ(x, z)P −1(x) � dz dx, where the second identity follows from E4 and the last one from E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note also that, by absolute continuity of the Lebesgue integral, lim sup k→+∞ � t∈Tk ˆ εk(t+Q) ˆ Q0 W 0 � ∇zwδ(x, z)P −1(x) � dz dx = ˆ Ω ˆ Q0 W 0 � ∇zwδ(x, z)P −1(x) � dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, by combining the equalities that we have just found with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='16), we achieve the conclusion in the case under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Case 2: wδ generic Let now wδ ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By mollification, we retrieve a function ˜wδ ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) ∩ C2(Ω × Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that ˆ Ω ˆ Q0 W 0�∇z ˜wδ(x, z)P −1(x) � dz ≤ ˆ Ω ˆ Q0 W 0�∇zwδ(x, z)P −1(x) � dz + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To achieve the conclusion, it only suffices to repeat the argument in Case 1 for ˜wδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We are eventually in a position to reap the fruits of the previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us start with the lower limit inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If the lower limit of J 0 k (vk, Pk) is not finite, there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Otherwise, recalling Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(4), we deduce that εk∇vk 2⇀ ∇z˜v weakly two-scale in L2 for some ˜v ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In particular, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6(2), it must be ∇v(x) = ˆ Q ∇z˜v(x, z) dz = 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' x ∈ Ω, whence, being Ω connected, must v be identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Statement (1) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10 then follows by combining Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We now turn to the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The only nontrivial case corresponds to v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Propo- sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 provides for all δ > 0 a sequence {vk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that εk∇vk ⇀ 0 = ∇v weakly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By the Rellich-Kondrachov theorem in W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3), it follows that εkvk → 0 strongly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) (up to subsequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We employ again Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2 to deduce that lim sup k→+∞ J 0 k (vk, Pk) < J 0(v, P) + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 29 This inequality is actually equivalent to the desired one (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [8, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2]), and the proof is therefore concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Conclusions and a variant We devote this final section to the proof of the homogenization result for high-contrast com- posites and to the discussion of a variant of the problem featuring plastic dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 and convergence of minimum problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As we outlined be- fore, the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 is achieved by combining the splitting procedure in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3 with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10, which account for the asymptotics of the stiff and the soft components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Once the homogenization theorem is on hand, the convergence of the minimum problems and of their minimizers will follow thanks to the compactness result in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {εk} be an infinitesimal sequence and let us fix y ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) and P ∈ Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We separate the proof of the lower and of the upper limit inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lower bound We consider a sequence {(yk, Pk)} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) such that yk → y in the sense of extensions and that Pk → P uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The only case to discuss is the one in which the lower limit of Jk(yk, Pk) is finite, and we may thus assume that {Jk(yk, Pk)} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Keeping in force the notation of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4, we let {˜yk} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) be a sequence such that yk = ˜yk in Ω1 k and ˜yk ⇀ y weakly in W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the light of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5, we may without loss of generality assume that ˜yk := Tkyk, with Tk as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We now apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3, which yields {vk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) and such that {vk} is bounded in L2, {εk∇vk} is 2-equiintegrable and εkvk → 0 strongly in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In particular, (εkvk, Pk) τ→ (0, P) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10 yields J 0(0, P) ≤ lim inf k→+∞ J 0 k (vk, Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' At this stage, recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13), the proof of the lower bound is concluded as soon as we show that J 1(y, P) ≤ lim inf k→+∞ J 1 k (˜yk, Pk) = lim inf k→+∞ J 1 k (yk, Pk) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) with J 1(y, P) given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This is what we prove next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us set � W 1(x, F) := χE1(x)W 1(F), �H(x, P) := χE1(x)H(P), � J 1 k (y, P) := ˆ Ω � � W 1 � x εk , ∇˜yP −1 � + �H � x εk , P � + |∇P|q � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2) It holds lim inf k→+∞ � J 1 k (˜yk, Pk) ≤ lim inf k→+∞ J 1 k (˜yk, Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since (˜yk, Pk) τ→ (y, P), by applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8 to the left-hand side of the previous inequality, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1) is deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Upper bound If P /∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) there is nothing to prove;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' let us then assume that P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 30 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI As we have already observed, { � J 1 k } satisfies the requirements of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In view of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9, for any (y, P) ∈ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) there exists a sequence {(uk, Pk)} ⊂ W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) such that {∇uk} is 2-equiintegrable, (uk, Pk) τ→ (y, P), and lim sup k→+∞ � J 1 k (uk, Pk) ≤ J 1(y, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note that 0 ≤ J 1 k (uk, Pk) − � J 1 k (uk, Pk) = ˆ Ω �χ1 k(x) − χεkE1(x) ��W 1(∇ukP −1 k ) + H(Pk) � dx ≤ c ˆ Ω �χ1 k(x) − χεkE1(x) ��|∇uk|2 + 1 � dx for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thanks to the 2-equiintegrability of {∇uk}, we deduce lim sup k→+∞ J 1 k (uk, Pk) = lim sup k→+∞ � J 1 k (uk, Pk) ≤ J 1(y, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) We focus now on the soft part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10 grants the existence of a sequence {vk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that εkvk → 0 strongly in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3) and that lim sup k→+∞ J 0 k (vk, Pk) ≤ J 0(0, P), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4) where {Pk} is as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Notice that if yk := uk + vk, then {Jk(yk, Pk)} is bounded and {yk} converges to y in the sense of extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Letting ˜yk := Tkyk, thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14) we conclude the proof of the upper limit inequality: lim sup k→+∞ Jk(yk, Pk) ≤ lim sup k→+∞ J 0 k (yk − ˜yk, Pk) + lim sup k→+∞ J 1 k (˜yk, Pk) = lim sup k→+∞ J 0 k (vk, Pk) + lim sup k→+∞ J 1 k (uk, Pk) ≤ J (y, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the previous lines, the equality is a consequence of the facts that {∇uk} and {∇˜yk} are bounded and that uk = yk on Ω1 k, whereas the last bound accounts for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ Finally, we are only left to establish the convergence of the minimum problems associated with the energy functionals Jε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' What we need is an adaptation of the Γ-convergence statement that we have just proved so as to make it comply with Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To this aim, as it is customary (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [9, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7]), we could employ the fundamental estimate derived in [24] on the functionals { � J 1 k } in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' indeed, boundary data concern only the stiff part, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the light of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9 we can adopt an alternative strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since {(yk, Pk)} is a sequence of almost-minimizers, there exists C such that Jk(yk, Pk) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The 2-growth condition from below, together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4, provides a bound on ∥yk∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1, there exists (y, P) ∈ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) such that, up to subsequences, yk → y in the sense of extensions and Pk → P uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 ensures that J (y, P) ≤ lim inf k→+∞ Jk(yk, Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We now prove the existence of a recovery sequence meeting the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As sug- gested by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6, we focus on the stiff part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let us consider again the functional � J 1 k in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since the sequence { � J 1 k } falls within the scopes of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8, for any (�y, �P) ∈ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × HOMOGENIZATION OF HIGH-CONTRAST MEDIA 31 W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9 provides a sequence {(uk, �Pk)} ⊂ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) such that {∇uk} is 2-equiintegrable, (uk, �Pk) τ→ (�y, �P) and lim sup k→+∞ � J 1 k (uk, �Pk) ≤ J 1(y, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By reasoning as in the proof of the upper bound in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 we retrieve a sequence {�yk, �Pk} ∈ W 1,2 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) × W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) such that �yk → �y in the sense of extensions, �Pk → �P uniformly and lim sup k→+∞ Jk(�yk, �Pk) ≤ J (�y, �P), whence lim sup k→+∞ (inf Jk) ≤ inf J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Recalling that {(yk, Pk)} is a sequence of almost minimizers, we conclude inf J ≤ J (y, P) ≤ lim inf k→+∞ Jk(yk, Pk) = lim inf k→+∞ inf Jk ≤ inf J , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A non degenerate upper bound for the soft component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We proved in Section 5 that the limiting behavior of the soft inclusions is described by a degenerate functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' However, under our assumptions, a non-degenerate upper bound may still be established, as we prove in the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The argument follows [13], where Cherdantsev & Cherednichenko derived the effective energy of high-contrast nonlinear elastic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Differently from us, the Γ-limit that they retrieve keeps track of both the macro- and the microscopic variable, and this roots in the choice of a stronger notion of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The drawback of such an approach is the lack of compactness for sequences with equibounded energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' It was shown in [26, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12] that, when weaker topologies are considered, the quasiconvex envelope does not provide the correct limiting energy density for the Γ-lower limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We start by proving a more detailed version of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lemma 22 in [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let w ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) ∩ C2(Ω × Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, there exists a sequence {wk} ⊂ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) such that ∇zwk → ∇zw strongly in L2(Ω × Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Besides, setting for x ∈ Ω vk(x) := wk � x, x εk � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) {vk} converges strongly two-scale to w in L2 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We extend w by setting it equal to 0 on Q\\Q0, so as to obtain a function in L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 per(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) which, by a slight abuse of notation, we denote again by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Keeping in mind the definition of ΩQ k (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13)), for (¯x, ¯z) ∈ Ω × R3 we define wk(¯x, ¯z) in terms of the averages of w( · , ¯z) on the cubes that form ΩQ k : wk(¯x, ¯z) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 εk(t+Q) w(x, ¯z) dx if ¯x ∈ εk(t + Q) for some t ∈ ˆTk, 0 for any other ¯x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) By definition, wk( · , z) is piecewise constant for all z ∈ ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Moreover, for almost every x ∈ Ω, wk(x, · ) is Q-periodic as well as weakly differentiable, and ∇zwk → ∇zw strongly in L2(Ω × 32 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Indeed, from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='6) and Jensen’s inequality, we have that ˆ Ω ˆ Q |∇zwk(x, z) − ∇zw(x, z)|2 dz dx = ˆ ΩQ k ˆ Q |∇zwk(x, z) − ∇zw(x, z)|2 dz dx + ˆ Ω\\ΩQ k ˆ Q |∇zw(x, z)|2 dz dx = � t∈ ˆTk ˆ εk(t+Q) ˆ Q |∇zwk(x, z) − ∇zw(x, z)|2 dz dx + o(1) ≤ � t∈ ˆTk ˆ εk(t+Q) ˆ Q εk(t+Q) ��∇zw(ξ, z) − ∇zw �x, z) ��2 dξ dz dx + o(1), and the last term is infinitesimal for k → +∞ (recall that w ∈ C2 and the mean value theorem applies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We now turn to the functions vk given by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' First of all, we point out that, thanks to the regularity of w, vk is measurable and vanishes on Ω1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Besides, it belongs to W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Secondly, we show that {vk} converges weakly two-scale to w in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' To this aim, let us fix φ ∈ C(¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cper(R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We find ˆ Ω vk(x) · φ � x, x εk � dx = ˆ Ω0 k wk � x, x εk � φ � x, x εk � dx = � t∈Tk ˆ εk(t+Q0) wk � x, x εk � φ � x, x εk � dx = ε3 k � t∈Tk ˆ Q0 wk �εk(t + z), z � · φ �εk(t + z), z � dz = � t∈ ˆTk ˆ Q0 ˆ εk(t+Q) w(x, z) · φ �εk(t + z), z � dx dz = ˆ ΩQ k ˆ Q0 w(x, z) · φk(x, z) dz dx, where φk(x, z) := φ(εk(t + z), z) if x ∈ εk(t + Q) with t ∈ ˆTk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By the dominated convergence theorem, we infer lim k→+∞ ˆ Ω vk(x) · φ � x, x εk � dx = ˆ Ω ˆ Q0 w(x, z) · φ(x, z) dz dx, that is, vk 2⇀ w weakly two-scale in L2 (recall that w(x, z) = 0 if z ∈ Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In order to prove that strong two-scale convergence actually holds, we study the limiting behavior of the L2 norm of {vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On one hand, the weak two-scale convergence yields ∥w∥L2(Ω×Q) ≤ lim inf k→+∞∥vk∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7) On the other hand, from the properties of {wk} and a change of variables we have the identities ˆ Ω |vk(x)|2 dx = ˆ Ω0 k ����wk � x, x εk ����� 2 dx = � t∈Tk ˆ εk(t+Q0) ����wk � x, x εk ����� 2 dx = � t∈Tk ε3 k ˆ Q0 ��wk �εk(t + z), z ���2 dz = � t∈ ˆTk ε3 k ˆ Q0 ����� εk(t+Q) w(x, z) dx ����� 2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 33 Thanks to Jensen’s inequality we deduce ˆ Ω |vk(x)|2 dx ≤ � t∈ ˆTk ε3 k ˆ Q0 εk(t+Q) |w(x, z)|2 dx dz = ˆ Q0 ˆ ΩQ k |w(x, z)|2 dx dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' This, combined with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7), ensures that lim k→+∞∥vk∥L2(Ω) = ∥w∥L2(Ω×Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In view of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5 the conclusion is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finally, the strong convergence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15) follows by observing that, if x ∈ εk(t + Q), it holds ∇zˆvk(x, z) = ∇zwk �εk(t + z), z �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ We are now in a position to prove a non-degenerate Γ-upper limit inequality that is the counterpart of the one in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 under the current stronger convergence assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let {W 0 k }k satisfy assumptions E3–E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For any (w, P) ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3))× W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' there exists a sequence {vk} ⊂ W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that: (1) vk 2→ w strongly two-scale in L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (2) εk∇vk 2⇀ ∇zw weakly two-scale in L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (3) whenever Pk → P uniformly, it holds lim sup k→+∞ ˆ Ω χ0 k(x)W 0 k �εk∇vk(x)P −1 k (x) � dx ≤ ˆ Ω ˆ Q0 Q′W 0�∇zw(x, z), P −1(x) � dz dx, where Q′W 0 is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The conclusion is not a straightforward consequence of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1, because along the sequence {vk} in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5) we would not end up with the correct limiting energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, the actual recovery sequence is obtained by adding a “correction” to vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The proof consists of several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' At first, to circumvent measura- bility issues, it is convenient to consider a sufficiently regular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Under such assumption, we are able to construct a recovery sequence of the form vk = ˜vk + ˜wk, where {˜vk} is provided by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 and { ˜wk} allows to pass from the densities W 0 k to Q′W 0 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The definition of ˜wk is given in Step 1, while Step 2 deals with the upper limit inequality in the regular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The general statement is eventually retrieved by approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Step 1: construction of ˜wk for a regular w Let us assume that w ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' W 1,2 0 (Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3)) ∩ C2(Ω × Q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We consider a cover of Q0 made of cubes whose edge length is εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We set ˆΣk := { s ∈ Z3 : εk(s + Q) ⊂ ¯Q0 } and, for all (t, s) ∈ ˆTk × ˆΣk, we introduce the averages Ak(t, s) := εk(t+Q) εk(s+Q) ∇zw(x, z) dz dx (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8) and the piecewise constant functions Ak(x, z) := \uf8f1 \uf8f2 \uf8f3 Ak(t, s) if (x, z) ∈ εk(t + Q) × εk(s + Q), (t, s) ∈ ˆTk × ˆΣk, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 34 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI We record here for later use that, by means of Lebesgue differentiation and dominated conver- gence theorems, it follows lim k→+∞∥Ak − ∇zw∥2 L2(Ω×Q) = lim k→+∞ � t∈ ˆTk � s∈ˆΣk ˆ εk(t+Q) ˆ εk(s+Q) |Ak(t, s) − ∇zw(x, z)|2 dz dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) By the definition of Q′W 0 k , for all k ∈ N there exists ψk ∈ W 1,2 0 (Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3) such that ˆ Q χ0(z)W 0 k ��Ak(t, s) + ∇ψk(z) �P −1 k (x) � dz ≤ Q′W 0 k �Ak(t, s), P −1(x) � + 1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10) Note that, due to the smoothness of w, the averages Ak are bounded uniformly in k, t and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In the light of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1, the values Q′W 0 k �Ak(t, s), P −1(x) � are uniformly bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, by combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10) with assumption E3, we deduce that {ψk} is bounded in W 1,2 0 (Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A change of variables in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10) yields ˆ εk(s+Q) χ0 � z εk − s � W 0 k �� Ak(t, s) + ∇ψk � z εk − s �� P −1(x) � dz ≤ ε3 k � Q′W 0 k �Ak(t, s), P −1(x) � + 1 k � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) and that suggests us to introduce the functions ˜ψk(x, z) := \uf8f1 \uf8f2 \uf8f3 εkψk � z εk − s � if (x, z) ∈ εk(t + Q) × εk(s + Q), (t, s) ∈ ˆTk × ˆΣk, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Note that, for each k and x ∈ Ω, ˜ψk(x, · ) admits a weak derivative with respect to z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' thus, by summing over (t, s) ∈ ˆTk × ˆΣk, from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11) we may write � (t,s)∈ ˆTk׈Σk ˆ εk(t+Q) ˆ εk(s+Q) χ0 � z εk − s � W 0 k ��Ak(x, z) + ∇z ˜ψk(x, z) �P −1(x) � dz dx ≤ � (t,s)∈ ˆTk׈Σk ˆ εk(t+Q) ε3 k � Q′W 0 k �Ak(t, s), P −1 k (x) � + 1 k � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12) We also observe that, since {ψk} is bounded, ˜ψk → 0 strongly in L2(Ω × Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, given that {∇z ˜ψ} is bounded L2(Ω × Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3), it must converge weakly in L2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' It follows that, if wk is as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 and if (x, z) ∈ εk(t + Q) × εk(s + Q) with (t, s) ∈ ˆTk × ˆΣk, ∇z(wk + ˜ψk) ⇀ ∇zw weakly in L2(Ω × Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13) We further notice that ˜wk(x) := ˜ψk � x, x εk � = � (t,s)∈ ˆTk׈Σk εkψk � x ε2 k − s � χεk(t+Q)(x)χεk(s+Q) � x εk � is a measurable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A quick application of the definition of weak derivative proves also that ˜wk belongs to W 1,2 0 (Ω0 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 35 Step 2: w regular We now turn to the proof of the limsup inequality along the sequence {vk} defined as vk := ˜vk + ˜wk, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='14) where ˜vk(x) := wk � x, x εk � with wk as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1, and where { ˜wk} was introduced in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We have ˆvk(x, z) := Skvk(x, z) = wk � εk � x εk � + εkz, z � + ˜ψk � εk � x εk � + εkz, z � , so that if (x, z) ∈ εk(t + Q) × εk(s + Q) ∇zˆvk(x, z) = ∇zwk �εk(t + z), z � + ∇ψk � z εk − s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15) Taking into account (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='13), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7(1), it follows that εk∇vk 2⇀ ∇zw weakly two-scale in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Recalling Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='5, we have that lim sup k→+∞ ˆ Ω χ0 k(x)W 0 k �εk∇vk(x)P −1 k (x) � dx = lim sup k→+∞ � t∈Tk ˆ εk(t+Q) ˆ Q0 W 0 k � ∇zˆvk(x, z)P −1 k (x) � dz dx = lim sup k→+∞ Ik, where Ik := � (t,s)∈ ˆTk׈Σk ˆ εk(t+Q) ˆ εk(s+Q) W 0 k �∇zˆvk(x, z)P −1 k (x) � dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Indeed, ˆvk vanishes if x ∈ Ω\\ΩQ k or if z ∈ Q0\\∪{εk(s+Q) : s ∈ ˆΣk}, and the sequence {W 0 k (0)} is bounded by virtue of E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Therefore, since the measure of Ω\\ΩQ k and of Q0\\∪{εk(s+Q) : s ∈ ˆΣk} vanish for k → +∞, the second equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Being the value of ∇zˆvk (x, z) expressed by formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='15), we introduce I′ k := � t,s ˆ εk(t+Q) ˆ εk(s+Q) W 0 k �� Ak(t, s) + ∇ψk � z εk − s �� P −1 k (x) � dz dx, where the summation runs over ˆTk × ˆΣk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' By exploiting assumption E4 and Hölder’s inequality, we obtain the estimate ��Ik − I′ k �� ≤ c � t,s ˆ εk(t+Q) ˆ εk(s+Q) ��� � ∇zwk �εk(t + z), z � − Ak(t, s) � P −1 k (x) ��� 2 dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In view of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) we deduce lim k→+∞ ��Ik − I′ k �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='16) Next, let us set I′′ k := ˆ ΩQ k ˆ Q0 Q′W 0 k �Ak(x, z), P −1 k (x) � dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 36 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI According to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='12), the difference between the integrands of I′ k and I′′ k is of order k−1: lim k→+∞ ��I′ k − I′′ k �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='17) Finally, we compare I′′ k and the limiting functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We have �����I′′ k − ˆ Ω ˆ Q0 Q′W 0�∇zw(x, z), P −1(x) � dz dx ����� ≤ ˆ ΩQ k ˆ Q0 ���Q′W 0 k �Ak(x, z), P −1 k (x) � − Q′W 0 k �∇zw(x, z), P −1 k (x) ���� dz dx + ˆ ΩQ k ˆ Q0 ���Q′W 0 k �∇zw(x, z), P −1 k (x) � − Q′W 0 k �∇zw(x, z), P −1(x) ���� dz dx + ˆ ΩQ k ˆ Q0 ���Q′W 0 k �∇zw(x, z), P −1(x) � − Q′W 0�∇zw(x, z), P −1(x) ���� dz dx + ˆ Ω\\ΩQ k ˆ Q0 Q′W 0�∇zw(x, z), P −1(x) � dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' All the terms on the right-hand side vanish as k → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Indeed, by using the Lipschitz continuity of Q′W 0 k (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1(1)) and the uniform bound on {Pk}, the first summand is controlled by the norm of Ak − ∇zv, which, according to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9), is infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For what concerns the second term, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1(2) and the uniform convergence of {Pk} imply that the integrand is infinitesimal for k → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The third quantity vanishes because {Q′W 0 k } pointwise converges to Q′W 0 (recall that they are just variants of the quasiconvex envelopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lastly, the fourth summand is negligible since L3(Ω \\ ΩQ k ) tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On the whole, taking into account (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='16) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='17), we conclude lim k→+∞ Ik = ˆ Ω ˆ Q0 Q′W 0�∇zw(x, z), P −1(x) � dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Step 3: w generic The argument follows the one of Case 2 in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A variant with plastic dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' With a view to applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='7 to time- dependent problems, it is useful to modify the functionals Jε by adding a term that encodes the plastic dissipation mechanism of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Precisely, we take into account the non-symmetric distance D: R3×3 × R3×3 → [0, +∞] in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) and we define the dissipation between P0, P1 : Ω → SL(3) as D(P0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' P1) := ˆ Ω D(P0, P1) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' From a physical viewpoint, if P0, P1 : Ω → SL(3) are admissible plastic strains, D(P0, P1) is interpreted as the minimum amount of energy that is dissipated when the system moves from a plastic configuration to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, assuming that ¯P ∈ W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SL(3)) represents a pre- existent plastic strain of the body, we set J diss ε (y, P) := Eε(y, P) + D( ¯P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' P) + ∥∇P∥q Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='R3×3×3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='18) HOMOGENIZATION OF HIGH-CONTRAST MEDIA 37 In the same spirit of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='10), we distinguish between the dissipation of the inclusions and the one of the matrix, respectively D0 ε( ¯P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' P) := ˆ Ω χ0 ε(x)D( ¯P, P) dx, D1 ε( ¯P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' P) := ˆ Ω χ1 ε(x)D( ¯P, P) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' For what concerns the compactness of sequences with equibounded energy, we notice that the presence of the dissipation D does not affect Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1: the same conclusions hold if the bound on Jk(yk, Pk) is replaced by a bound on J diss k (yk, Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Also our Γ-convergence results easily extend to the family {J diss ε }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The dissipation is indeed a continuous perturbation: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Let P, ¯P ∈ C(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' If {Pk} ⊂ C(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' K) converges uniformly to P, then lim k→+∞ Di k( ¯P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pk) = L3(Qi)D( ¯P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' P) for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' We firstly observe that if Pk → P pointwise, then D �Pk(x), P(x) � → 0, D �P(x), Pk(x) � → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='19) To see this, let γ be such that for all (t, F, G) ∈ [0, 1]×SL(3)×SL(3), γ(t, F, G) is the evaluation at t of the unique minimizing geodesic connecting F and G, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='9) and the definition of γ, D �Pk(x), P(x) � = ˆ 1 0 ∆ � γ �t, Pk(x), P(x) �, ˙γ �t, Pk(x), P(x) �� dt ≤ c ˆ 1 0 |˙γ �t, Pk(x), P(x) �| dt, where the inequality follows from the definition of ∆ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Since ˙γ is continuous and bounded, by dominated convergence we deduce that the last term vanishes as k → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In a similar fashion, we show that D(P, Pk) → 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' As second step, we notice that D � ¯P(x), Pk(x) � → D � ¯P(x), P(x) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='20) Indeed, the triangular inequality yields D � ¯P(x), P(x) � − D �Pk(x), P(x) � ≤ D � ¯P(x), Pk(x) � ≤ D � ¯P(x), P(x) � + D �P(x), Pk(x) �, and the assertion follows as a consequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finally, we observe that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='20) grants that lim k→+∞Di k( ¯P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pk) = lim k→+∞ ˆ Ω χi k(x)D � ¯P(x), P(x) �dx, and the conclusion is achieved by arguing as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' □ Acknowledgements We acknowledge support from the Austrian Science Fund (FWF) projects F65, V662, Y1292, from the FWF-GAČR project I 4052/19-29646L, and from the OeAD-WTZ project CZ04/2019 (MŠMTČR 8J19AT013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 38 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DAVOLI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' GAVIOLI, AND V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' PAGLIARI References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Acerbi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Buttazzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On the limits of periodic Riemannian metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 43 (1983), 183–201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Allaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization and two-scale convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 23(6) (1992), 1482–1518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Baía, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Fonseca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The limit behavior of a family of variational multiscale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 56(1) (2007), 1–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Bao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Chern, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' An Introduction to Riemann-Finsler Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Springer Science & Business Media, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Barchiesi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Toughening by Crack Deflection in the Homogenization of Brittle Composites with Soft Inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 227 (2018), 749–766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Barchiesi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lazzaroni, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Zeppieri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A bridging mechanism in the homogenization of brittle composites with soft inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 48 (2016), 1178–1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Bartels, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Carstensen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hackl, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hoppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Effective relaxation for microstructure simulations: algorithms and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 193 (2004), 5143–5175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Braides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Γ-convergence for beginners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Oxford University Press, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Braides, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Defranceschi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization of Multiple Integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Oxford University Press, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Braides, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Garroni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization of periodic nonlinear media with stiff and soft inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Models Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 5 (1995), 543–564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Carstensen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hackl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mielke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Non-convex potentials and microstructures in finite-strain plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 458 (2002), 299–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Castaing, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Valadier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Convex Analysis and Measurable Multifunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lecture Notes in Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 580, Springer-Verlag, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cherdantsev, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cherednichenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Two-scale Γ-convergence of integral functionals and its appli- cation to homogenisation of nonlinear high-contrast periodic composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 204(2) (2012), 445–478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Christowiak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Kreisbeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization of layered materials with rigid components in single-slip finite crystal plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Partial Differential Equations 56 (2018), 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Christowiak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Kreisbeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Asymptotic rigidity of layered structures and its application in homoge- nization theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 235 (2020), 51–98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cioranescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Damlamian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Griso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Periodic unfolding and homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Paris 335(1) (2002), 99–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Cioranescu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Damlamian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Griso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' The periodic unfolding method in homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 40(4) (2008), 1585–1620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Conti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Reina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Kinematic description of crystal plasticity in the finite kinematic framework: a mi- cromechanical understanding of F = FeFp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Solids 67 (2014), 40–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Conti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Reina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Schlömerkemper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Derivation of F = FeFp as the continuum limit of crystalline slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Solids 89 (2016), 231–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [20] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Dacorogna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Direct Methods in the Calculus of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Springer Science & Business Media, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [21] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Dal Maso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' An Introduction to Γ-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [22] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Davoli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Ferreira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Kreisbeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization in BV of a model for layered composites in finite crystal plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 14 (2021), 441–473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Davoli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Francfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A Critical Revisiting of Finite elastoplasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 47 (2015), 526–565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Davoli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Gavioli, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pagliari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A homogenization result in finite plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ArXiv preprint 2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='09084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Davoli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Kreisbeck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' On static and evolutionary homogenization in crystal plasticity for stratified composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In: Research in the Mathematics of Materials Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Springer AWM series, 2022, to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [26] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Davoli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Kružík, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pagliari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization of high-contrast composites under differential con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=', to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='1515/acv-2022-0009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Davoli, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Roubíček, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Stefanelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A note about hardening-free viscoelastic models in Maxwellian- type rheologies at large strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Solids 26 (2021), 1483–1497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [28] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Fonseca, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Krömer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Multiple integrals under differential constraints: two-scale convergence and homog- enization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 59(2) (2010), 427–457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [29] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Fonseca, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Müller, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pedregal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Analysis of concentration and oscillation effects generated by gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 29 (1998), 736–756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Gilbarg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Trudinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Elliptic Differential Equations of Second Order, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=', Springer-Verlag, Berlin, New York, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' HOMOGENIZATION OF HIGH-CONTRAST MEDIA 39 [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Grandi, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Stefanelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finite plasticity in P T P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Part I: constitutive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thermodyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 29 (2017), 97–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Grandi, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Stefanelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finite plasticity in P T P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Part II: quasi-static evolution and linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 49 (2017), 1356–1384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Hackl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mielke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mittenhuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Dissipation distances in multiplicative elastoplasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In: W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Wendland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Efendiev (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=') Analysis and Simulation of Multifield Problems, Springer, New York, 2003, 87–100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [34] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Kröner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 4 (1959), 273–334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [35] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Le Dret, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Raoult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Variational convergence for nonlinear shell models with directors and related semicontinuity and relaxation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 154 (2000), 101–134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [36] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Elastic-plastic deformation at finite strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 36 (1969), 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lubliner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Plasticity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Macmillan Publishing Company, New York, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mielke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Finite Elastoplasticity Lie Groups and Geodesics on SL(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' In: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Newton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Weinstein, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Holmes (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=') Geometry, Mechanics, and Dynamics, Springer-Verlag, New York, 2002, 61–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mielke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Energetic formulation of multiplicative elastoplasticity using dissipation distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Thermodyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 15 (2003), 351–382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [40] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Naghdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A critical review of the state of finite plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 41 (1990), 315–394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [41] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Nguetseng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' A general convergence result for a functional relate to the theory of homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 20(3) (1989), 608–623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [42] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pellet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Scardia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Zeppieri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Homogenization of high-contrast Mumford-Shah energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 51 (2019), 1696–1729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Visintin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Some properties of two-scale convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Atti Accad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Naz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lincei Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Natur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Lincei (9) Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 15(2) (2004), 93–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' [44] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Visintin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Towards a two-scale calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' ESAIM Control Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' 12(3) (2006), 371–397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Davoli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Gavioli, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Pagliari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content=' E-mails: elisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='davoli@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='at, chiara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='gavioli@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='at, valerio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='pagliari@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} +page_content='at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99A0T4oBgHgl3EQfO__U/content/2301.02170v1.pdf'} diff --git a/9dE3T4oBgHgl3EQfSQko/content/tmp_files/2301.04430v1.pdf.txt b/9dE3T4oBgHgl3EQfSQko/content/tmp_files/2301.04430v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da083f0595c155eaf148ef6f084a9af0dc57a9ff --- /dev/null +++ b/9dE3T4oBgHgl3EQfSQko/content/tmp_files/2301.04430v1.pdf.txt @@ -0,0 +1,5171 @@ +Network Adaptive Federated Learning: +Congestion and Lossy Compression +Parikshit Hegde +Electrical and Computer Engineering +The University of Texas at Austin +Austin, Texas, USA +hegde@utexas.edu +Gustavo de Veciana +Electrical and Computer Engineering +The University of Texas at Austin +Austin, Texas, USA +gustavo@ece.utexas.edu +Aryan Mokhtari +Electrical and Computer Engineering +The University of Texas at Austin +Austin, Texas, USA +mokhtari@austin.utexas.edu +Abstract—In order to achieve the dual goals of privacy +and learning across distributed data, Federated Learning (FL) +systems rely on frequent exchanges of large files (model updates) +between a set of clients and the server. As such FL systems +are exposed to, or indeed the cause of, congestion across a +wide set of network resources. Lossy compression can be used +to reduce the size of exchanged files and associated delays, +at the cost of adding noise to model updates. By judiciously +adapting clients’ compression to varying network congestion, an +FL application can reduce wall clock training time. To that end, +we propose a Network Adaptive Compression (NAC-FL) policy, +which dynamically varies the client’s lossy compression choices +to network congestion variations. We prove, under appropriate +assumptions, that NAC-FL is asymptotically optimal in terms +of directly minimizing the expected wall clock training time. +Further, we show via simulation that NAC-FL achieves robust +performance improvements with higher gains in settings with +positively correlated delays across time. +Index Terms—federated learning, rate adaptation, resilience +I. INTRODUCTION +Communication costs and delays of sending model updates +from clients to the server are a known bottleneck in training +Federated Learning (FL) systems [1]–[4]. Two common tech- +niques used to alleviate this issue are: 1) local computations +where clients perform several local steps before communicat- +ing with the server, and 2) (lossy) compression where clients +communicate quantized/compressed updates to the server. The +eventual end goal of these approaches is to minimize the wall +clock time for convergence of the training algorithm (hereon +referred to as FL algorithm) by reducing the amount of data +communicated from clients to the server. +To this end, several works have analyzed the relationship +between compression, local computations and the number +of rounds needed by FL algorithms to converge [5]–[12]. +However, these works ignore the impact of changing network +congestion, both across clients and across time, on the wall +This material is based upon work of Hegde and de Veciana supported +by the National Science Foundation (NSF) under grant No. 2148224 and is +supported in part by funds from OUSD R&E, NIST, and industry partners as +specified in the Resilient & Intelligent NextG Systems (RINGS) program and +the WNCG/6G@UT industrial affiliates. The work of Mokhtari is supported +in part by the NSF AI Institute for Future Edge Networks and Distributed +Intelligence (AI-EDGE) via NSF grant 2112471, the Machine Learning Lab +(MLL) at UT Austin, and the Wireless Networking and Communications +Group (WNCG) Industrial Affiliates Program. +clock time to converge. For instance, a client may choose +a high degree of compression when it sees high network +congestion, while a client seeing lower congestion may op- +portunistically choose not to compress as much. In this work, +we ask the following question: “Can we design a policy that +adapts the amount of compression across clients and time +according to changing network conditions in order to opti- +mize the wall clock time?” To answer this question, we first +characterize the impact that changing network congestion and +an adaptive compression policy have on the wall clock time. +Second, we propose the Network Adaptive Compression for +Federated Learning (NAC-FL) policy that judiciously chooses +compression levels based on network congestion to minimize +the wall clock time. Crucially, NAC-FL does not rely on the +prior knowledge of the distribution of network congestion. +Instead, it learns to optimize its compression decisions on- +the-fly based on the congestion seen by clients. +NAC-FL works in an opportunistic manner by adaptively +choosing high or low amounts of compression across clients +and across time based on low or high network congestion. +It further considers two effects that compression has on the +wall clock time. First, with increasing amount of compres- +sion, the FL algorithm would require more communication +rounds to converge, as the server receives “noisier”, and hence +inaccurate, model updates. Second, with higher degrees of +compression, the duration of each round would decrease as +a smaller model update is communicated. Since the wall +clock time is affected by both the number of rounds and +the duration of each round (it is effectively the product of +the two quantities), a policy for choosing compression levels +should consider these jointly. Fig. 1 provides an illustrative +visualization. Hence, NAC-FL aims to find the “sweet-spot” +compression levels over time varying network congestion. +Contributions. We propose a general framework to study how +to best adapt compression of client model updates. Assuming +a stationary Markov model for the underlying network conges- +tion state, we show that optimal policies are state dependent +and characterize the expected stopping time for convergence +to a predefined model accuracy. +This characterization provides the underlying insight for our +proposed NAC-FL policy. To our knowledge this is the first +1 +arXiv:2301.04430v1 [cs.LG] 11 Jan 2023 + +Fig. 1: Illustration of how compression level affects round +duration, number of rounds and wall clock time. +policy for compression that adapts to the stochastic variations +of the underlying network congestion process. Under appropri- +ate assumptions on the FL algorithm and underlying network +congestion and delays, we provide a proof of the asymptotic +optimality of NAC-FL in terms of minimizing the mean time +until the convergence criterion is met. To our knowledge this +is the first theoretical result of this type. +Finally we demonstrate via simulation the performance +gains and robustness of NAC-FL vs alternative fixed compres- +sion and/or fixed error per round policies. We explore a variety +of models for network congestion, finding that in particular +NAC-FL excels in the practically relevant setting where the +network sees positive correlations in the network congestion +accross time. +A. Related Work +Perhaps the most related papers to our work are [13]–[17] +which explored adaptive compression schemes for FL settings. +In [13]–[15] the authors propose adapting compression to +network congestion. In these works, the algorithm to select +compression has a per round budget, e.g., a budget on delay +(or compression error) per round, and possibly heterogeneous +compression levels are chosen across the clients based on the +current network congestion to minimize the compression error +(or delay) for the round. These works exploit the diversity of +network congestion across the clients, but not across time. +Meanwhile [16], [17] have observed that using a higher +amount of compression at the start and gradually reducing +compression through time may improve the wall clock time. +Our proposed policy is novel in that it learns how to best +exploit congestion variation across clients and across time to +optimize the wall clock time. +Another line of work that aims to reduce the overall commu- +nication cost is client sampling [18]–[21], where at each round, +only a subset of the clients are chosen to participate. The +authors of [21] propose a client sampling and power control +policy that adapts to time varying channels of clients sharing +a single base station and optimizes a proxy for wall clock +time. Overall we veiw lossy compression and client sampling +as alternative approaches geared at addressing communication +bottlenecks. A study of how to jointly adapt lossy compression +and client sampling to changing network congestion is left for +future work. +B. Paper Organization +In Section II, we introduce our system model. In Section +III, we propose our NAC-FL algorithm for lossy compression +and under appropriate assumptions prove it is asymptotically +optimal. Section IV is devoted to exploring the method for +several problem instances and in particular for various models +for the underlying network congestion in terms of correlation +across clients and time. In Section V, we comment on the +practical aspects of estimating the file transfer delay of clients +when deploying NAC-FL. Finally, in Section VI, we close the +paper with some concluding remarks. +Notation. Throughout this document, unless otherwise men- +tioned, quantities denoted with lowercase letters correspond +to constants, and uppercase letters correspond to random +variables. Bold symbols correspond to vectors, and regular +symbols indicate scalars. For example, x is a constant vector, +X is a random vector, x is a constant scalar, and X is a +random scalar/variable. Lowercase and uppercase forms of +the same letter correspond to constant and random variable +notions of the same quantity. A sequence indexed by n will be +denoted as (xn)n. +II. MODEL SETUP +In this paper, we focus on a federated architecture, where +a server aims to find a model that performs well with respect +to the data of a group of m clients, and in which nodes +exchange updates based on their local information with only +the server. More precisely, suppose the loss function associated +with client j is denoted by fj(w), where w represents the +weights of the model, e.g., the weights of a neural network. +The goal is to find the model that minimizes the average loss +across clients +f(w) = 1 +m +m +� +j=1 +fj(w). +The FL algorithm proceeds in rounds. Each round consists +of two stages: (i) a local stage in which each client updates the +most recent model received from the server via gradient-based +updates based on its local data and (ii), an aggregation stage in +which the server updates the global model by aggregating the +local updates received from clients. We shall let wn denote +the global model at the server at round n. Further, we let τ n +denote the total number of local steps (such as gradient steps) +that each client performs at round n, and let wτ n,n +j +denote the +resulting local model at node j. +In this paper, we are interested in the setting where each +client sends a compressed version ˜gn +Qj of its local model +wτ n,n +j +to the server using a lossy compression algorithm +(or, compressor) Q(·, ·). The compressor accepts a vector +x and a parameter q ∈ [0, qmax] indicating the amount of +compression with the maximum value being qmax, and outputs +ˆX = Q(x, q) which is an approximation of x, but has a +decreased file size as compared to x. ˆX is capitalized to +highlight that the compressor Q(·, ·) may use randomness +in its compression. We shall denote by qn +j the compression +2 + +Wall Clock Time +Round Duration +No. of Rounds +Compression Amount +Compression Amountparameter used by client j for round n, and denote by +qn ≜ (qn +j )m +j=1 the vector of parameters used by the clients +in round n. After receiving updates from all the clients, the +server aggregates the compressed local models and produces +the next global model wn+1. +Given a target tolerance ε > 0, the goal of FL is to +generate a sequence of global models until on some round +rε a prespecified stopping criterion is first met, e.g., the +norm of the global loss function gradient is at most ϵ, i.e., +∥∇f(wrε)∥ ≤ ε. Our goal is to find an adaptive compression +policy that dynamically adapts to the possibly time varying +network states such that the target accuracy is achieved with +a minimum overall wall clock time. +We formalize the overall wall clock time, denoted tε, +required to achieve the target accuracy as follows. The duration +d(τ n, qn, cn) of a round n depends on: +• τ n, the number of local computations performed by +clients which we will assume to be the same across +clients; +• qn, an m dimensional vector of clients’ compression +parameters ; +• cn, the network state which models network congestion +and is assumed to be an element of a finite set C. +This allows some flexibility, e.g., the round’s duration may +depend on the max delay to deliver the model update from +clients to server, or the sum of the delays if clients share a +single resource in TDMA (Time Division Multiple Access) +fashion. The total wall clock time is then given by +tε = +rε +� +n=1 +d (τ n, qn, cn) . +(1) +In our system model, the sequence of network states, (cn)n, +is assumed to be exogenous, i.e., not be controlled by the +server or the clients nor their choices of τ n and qn . The delays +associated with the server multicasting global models to clients +are assumed to be exogeneous i.e., can not be controlled by +the FL server/clients and are not compressed, whence are not +part of the model. Still, in this work, based on observing the +network state we will devise an approach to select the clients +compression parameters so as to minimize the wall clock +time. As discussed in Section V, in practice observation of +the network state may involve light weight in band estimation +by probing delays of message bits as they are delivered in a +given round. +A policy for choosing compression parameters is called a +state dependent stationary policy if it can be expressed as a +function π of the current network state, i.e., qn = π(cn) for +all rounds n ∈ N. Such a policy will be referred to simply as +policy π. Given a random sequence of network states, (Cn)n, +let Rπ +ε be the random variable denoting the minimum number +of rounds needed to converge to error tolerance ε under policy +π. Then, the corresponding wall clock time, denoted by T π +ε , +is expressed as, +T π +ε = +Rπ +ε +� +n=1 +d (τ n, π (Cn) , Cn) . +III. NETWORK ADAPTIVE COMPRESSION FOR FEDERATED +LEARNING (NAC-FL) +Our approach to designing a policy to adapt clients’ com- +pression parameters centers on recognizing that the expected +wall clock time can be broken up into a product of the expected +number of rounds rε needed to converge to an error tolerance +ε and the average duration of each round ˆd. We start by +characterizing the relationship between rε, ˆd, and the sequence +of selected quantization parameters (qn)n and network states +(cn)n for a given FL algorithm. +Below we state an assumption relating rε to (qn)n. To that +end we introduce a strictly increasing, continuous and bounded +scalar function hε : [0, qmax] → R+ of compression parameter +q and an associated vector function hε : [0, qmax]×m → Rm ++ +of a compression vector q where hε,j(q) = hε(qj). We let +h−1 +ε +denote the inverse of this vector function. +Assumption 1. For a given FL algorithm there exists a +strictly increasing, continuous and bounded function hε(q) +and norm ∥·∥ such that given a sequence of compression +parameters (qn)n, the FL algorithm has reached the desired +error tolerance ε by round r if and only if, +r > 1 +r +r +� +n=1 +∥hε (qn)∥ +for some norm. +The above assumption implies that the expected number of +rounds can be written as the average of an increasing function +of the sequence of selected quantization parameters. Roughly +speaking, given a lossy compression policy that generates a +stationary parameter sequence (Qn)n whose marginal distri- +bution is the same as the random vector Q, the above criterion +means that the expected number of rounds to converge to the +desired error tolerance is approximately E[∥hε (Q)∥]. +This is a general condition that is motivated by convergence +bounds of several FL algorithms with compression, including, +[5], [8], [11]. In particular in Appendix A, we illustrate this +motivation for an extension of the FedCOM algorithm [11], +when q indicates the normalized-variance introduced by the +compressor, the scalar function is hε(q) = O(√q + 1/ε) and +the norm is the L2 norm. +Assumption 2. For any sequence of compression parameters +(qn)n the minimum number of rounds rε needed to converge +to an error tolerance ε is such that rε = Θ(1/poly(ε)), where +poly(ε) denotes a polynomial of ε. +Assumption 2 is a natural assumption for gradient based +optimization algorithms. It requires the convergence guaran- +tees for the FL algorithm to be such that when we require a +more accurate solution, the number of required communication +rounds grows. This argument indeed holds even for the settings +that we do not exchange compressed signals. +We also make the following additional assumption about +the round duration function. +3 + +Fig. 2: Illustration of a round duration as a function of +compression parameter q for a fixed local computation τ and +network state c. +Assumption 3. Given a network state c, number of local +computations τ, and compression parameters q = h−1 +ε (r), +the round duration d (τ, q, c) = d +� +τ, h−1 +ε (r), c +� +is bounded, +convex in r and decreasing in every coordinate of r. +In Assumption 3, the round duration being decreasing in +r is reasonable, since we expect more rounds as well as +smaller file sizes with higher compression. The convexity is +motivated by the notion that we use a “good compressor” as +illustrated next. Consulting Fig. 2, for any two parameters +q1, q2 and 0 < α < 1, a new time-sharing compressor Q′ +may be derived which outputs Q(x, q1) with probability α +and outputs Q(x, q2) with probability (1−α). This compressor +has expected round duration αd(τ, q1, c) + (1 − α)d(τ, q2, c). +And, in certain cases, its compression parameter is qα = αq1+ +(1−α)q2 (such as when the stochastic quantizer parameterized +by its normalized variance [5] is used). If Q is a “good +compressor”, then its round duration, d(τ, qα, c), should be +lower compared to that of the simple time-shared compressor, +αd(τ, q1, c)+(1−α)d(τ, q2, c). Therefore, the convexity of the +round duration function is a reasonable assumption for “good +compressors” (considering hε(q) ∝ q for simplicity). +Assumption 4. The sequence of network states (Cn)n forms +an irreducible aperiodic stationary Markov Chain on a finite +state space C with invariant distribution µ. +Assumption 4 is a natural assumption made to facilitate the +analysis of algorithms (see e.g., [22]). +A. Expected Wall Clock Time Formulation +Given the above mentioned assumptions, we are now ready +to introduce the proposed framework. We begin by showing +that we need only consider state dependent stationary policies +for choosing compression parameters when optimizing the +overall wall clock time. +Lemma 1. Under Assumptions 1-4 there exists a state depen- +dent stationary policy to select compression parameters which +is asymptotically optimal in terms of minimizing the wall clock +time to reach a desired error tolerance of ε as ε → 0. +The proof of Lemma 1 depends on two critical observations. +First, since by Assumption 2 the number of rounds needed +to converge grows large as ε → 0, one can expect the +empirical distribution of the network states modelled by the +finite state Markov Chain to concentrate around the invariant +prior to the stopping time. Second, due to the convexity of the +round duration function in Assumption 3, given a sequence +of network states there exists a state dependent stationary +policy that is near optimal and depends solely on the empirical +distribution of the sequence. The proof is in Appendix C. +Here, we will focus on the setting where ε is small, hence by +Lemma 1, we only need to consider state dependent stationary +policies, qn = π(cn). +Lemma 2. Under Assumptions 1-4 and a fixed number of +local computations per round τ, for every δ > 0, there exists +an εth > 0 such that, for all ε < εth and any state-dependent +stationary policy π, the expected wall clock time is bounded +as, +1 − δ ≤ +E [T π +ε ] +E[∥hε (π(C))∥] E[d (τ, π(C), C)] ≤ 1 + δ, +(2) +where, C denotes a random variable whose distributions is µ +(see Assumption 3). +Lemma 2 is proved in Appendix D. Define, +ˆtπ +ε ≜ E[∥hε (π(C))∥] E[d (τ, π(C), C)]. +(3) +Due to Lemma 2, for small enough ε, ˆtπ +ε provides an accurate +approximation for E[T π +ε ]. Therefore, from here onwards, we +shall assume implicitly that that a small ε is considered and +focus on finding a policy to optimize ˆtπ +ε . +Suppose the distribution of C is known. Then, one could +compute expected wall clock time as given in (3) for any +state dependent stationary policy π. In this case, we could +determine an optimal policy π∗ by solving the optimization +problem, +min +π∈Qm|C| +ˆtπ +ε = E[∥hε (π(C))∥] E [d (τ, π(C), C)] , +(4) +where Qm|C| is the set of all state-dependent stationary poli- +cies. +Alas, in practice, we often cannot directly solve the above +problem, as the distribution of C is unknown. Hence, below, +we propose a stochastic approximation like algorithm that +achieves the optimal wall clock time of π∗ asymptotically. +B. NAC-FL: Informal Description +The idea underlying NAC-FL is to keep running estimates +for E [∥hε (Q)∥] and E [d(τ, Q, C)] i.e., +ˆrn +ε = 1 +n +n +� +k=1 +���hε +� +q(k)���� , +ˆdn = 1 +n +n +� +k=1 +d +� +τ, q(k), c(k)� +. +4 + +d(T, q, C) +q1 +q2 +qa +bGiven a network state of cn+1 at round n + 1, and, a possible +choice for compression parameters q, the running averages +would be updated as follows, +ˆrn+1 +ε += +n +n + 1 ˆrn +ε + +1 +n + 1 ∥hε (q)∥ , +ˆdn+1 = +n +n + 1 +ˆdn + +1 +n + 1d +� +τ, q, cn+1� +. +(5) +As seen in (3), to minimize the wall clock time one should +minimize ˆrn+1 +ε +ˆdn+1, which can be expanded as, +ˆrn+1 +ε +ˆdn+1 = +n +(n + 1)2 +� +rn +ε d +� +τ, q, cn+1� ++ ˆdn ∥hε(q)∥ +� ++ +n2 +(n + 1)2 rn +ε ˆdn + O +� +1 +(n + 1)2 +� +. +Given the fact that ˆrn +ε and ˆdn are constants, and neglecting +the term O +� +1/(n + 1)2� +, an optimal choice for qn+1 is +qn+1 = argmin +q +ˆrn +ε d +� +τ, q, cn+1� ++ ˆdn ∥hε (q)∥ . +(6) +The NAC-FL policy is summarized in Algorithm 1. To +retrieve policy informally described above the tunable param- +eters (βn)n and α should be set to βn = 1 +n and α = 1. +Consider two possible network states c and c′ at a round +n. If the delay under state c is higher compared to c′ for any +compression parameters, then NAC-FL would choose a higher +compression amount q for state c compared to compression +amount q′ for state c′, i.e., q > q′ elementwise. This may +be concluded from the selection policy of (6), and noting +that rε(q) is increasing in q (Assumption 1), and d(τ, q, c) +is decreasing in q (Assumption 3). +Observe that since the estimates ˆrn +ε and ˆdn will initially +change across rounds, NAC-FL may choose different compres- +sion parameters in two rounds for which the network was in +the same state, i.e., NAC-FL is not a state-dependent stationary +policy. Still, we will show NAC-FL is asymptotically near +optimal. To develop this result we shall next present NAC-FL +in a more formal manner. +Algorithm 1: NAC-FL +Input +: Initialization: ˆr(0) +ε , ˆd(0) ; step size +schedule {βn}∞ +n=1; parameter α. +1 for n = 1, . . . , until termination do +2 +Server observes network state cn ; +3 +qn = +argmin +q +αˆr(n−1) +ε +d (τ, q, cn) + ˆd(n−1) ∥hε (q)∥; +4 +ˆrn +ε = (1 − βn)ˆr(n−1) +ε ++ βn ∥hε (qn)∥ ; +5 +ˆdn = (1 − βn) ˆd(n−1) + βnd(τ, qn, cn); +6 end +C. NAC-FL: Formal Description +Our NAC-FL approach is also inspired by the Frank-Wolfe +Algorithm [23]. We start by reformulating the optimization +program in (4). Denote by set Vε all possible pairs of expec- +tations (ˆrε, ˆd), +Vε = +� +(ˆrε, ˆd) : ∃ π ∈ Qm|C| s.t. ˆrε = E [∥hε (π(C))∥] , +ˆd = E [d (τ, π(C), C)] +� +. +(7) +Using the set Vε, and denoting H(r, d) ≜ rd, we may write +the optimization (4) characterizing the optimal policy π∗ as +min +ˆrε, ˆd +{H(ˆrε, ˆd) : (ˆrε, ˆd) ∈ Vε}. +(8) +In this case, from a point (ˆrn +ε , ˆdn), the Frank-Wolfe update +would be given as, +(ˆrε, ˆd) = argmin +(r,d)∈Vε +∇H +� +ˆrn +ε , ˆdn�⊤ �r +d +� +, +(9) +ˆrn+1 +ε += (1 − β)ˆrn +ε + βˆrε, +ˆdn+1 = (1 − β) ˆdn + β ˆd. +The gradient ∇H(ˆrε, ˆd) is, ∇H(ˆrε, ˆd) = +� +ˆd +ˆrε +�⊤ +. Vε is a +set of feasible averages of ˆrε and ˆd. Therefore, at round (n+1), +not all the pairs (r, d) ∈ Vε may be achievable. Hence, NAC- +FL approximates equation (9) as, +qn+1 = argmin +q +ˆrn +ε d +� +τ, q, cn+1� ++ ˆdn ∥hε (q)∥ . +We have thus retrieved our proposed NAC-FL algorithm +based on the Frank-Wolfe update, with one difference. The +above derivation suggests the use of a fixed step-size β at all +rounds while the previously derived algorithm used a decaying +the step-size βn = 1/n. In our simulations, we will embrace +the latter. +The following assumption is required to show the asymp- +totic optimality of NAC-FL. A state dependent stationary +policy π maps from a domain of finite size |C|, to a range +positive-real vectors of dimension m. Therefore, the policy +may be represented by a positive-real vector, π, of dimension +m |C|. Further, a vector rπ may be obtained by applying +hε(·) elementwise to the policy vector π, rπ ≜ hε(π). This +representation is used in the following assumption. +Assumption 5. The objective function ˆtπ +ε of the optimization +problem in (4) is a strictly quasiconvex function in π in the +following sense, +rπ⊤ � +∇rπˆtπ +ε +� += 0 +=⇒ +rπ⊤ � +∇2 +rπˆtπ +ε +� +rπ > 0. +(10) +Assumption 5 ensures that there is a unique state dependent +stationary policy π∗ which optimizes (4). We have observed +that the considered network model, compression model and +the ∥hε (q)∥ function associated with the FedCOM algorithm +indeed satisfy this assumption. +Next we shall establish an optimality property for NAC- +FL. To that end we shall consider executing NAC-FL without +termination with βn = β for all n and let +� +Qn +β +� +n, ˆRn +ε,β and +5 + +ˆDn +β be the corresponding sequence of compression parameters +and the associated estimates. +Theorem 1. Let π∗ be the solution and ˆtπ∗ +ε +the minimum +of the optimization problem in (4). If Assumptions 1-5 hold, +then there exists a positive sequence (βi)∞ +i=1 with βi → 0 as +i → ∞, such that for every ρ > 0, there exists a thereshold +nth(ρ) such that, +lim +i→∞ +sup +n≥nth(ρ)/βi +P +������ +� ˆRn +ε,βi − E[∥hε (π∗(C))∥] +ˆDn +βi − E[d (τ, π∗(C), C)] +������ > ρ +� += 0, +The proof of Theorem 1 is included in Appendix B. +Remark 1. Theorem 1 should be interpreted with some +subtlety. Say the desired error-tolerance ε is very small such +that the number of rounds needed to converge under any +compression policy is such that rε ≫ nth(ρ)/β. Then, based +on Theorem 1, one can show that NAC-FL compression +choices will be near optimal after nth(ρ)/β rounds. Thereafter, +since rε is large, NAC-FL will make near optimal choices for +long enough leading to a near optimal expected wall clock +time. +We further remark on the meaning of the asymptotic result +in the context of minimizing the wall clock time. In appli- +cations that require a very low error-tolerance ε, one needs +to have a large number (i.e., in the asymptotic region) of +communication rounds rε for convergence. Therefore, even +though the wall clock time obtained by using NAC-FL may +be large in this setting, it is near-optimal compared to other +methods of choosing compression parameters. +IV. SIMULATION +In this section, we present our simulation results. We begin +by describing additional model details used in our simulations. +A. Additional Model Details +1) Compression Model: We shall use the stochastic quan- +tizer in [5] which we will denote as Qq(·, b). The quantizer +has a parameter b ∈ {1, . . . , 32} corresponding to the number +of bits used to represent each co-ordinate, in addition to the +bit used to denote signs. When input a vector x, it outputs, +Qq(x, b) = ∥x∥∞ sign(x)ζ(x, b) +(11) +where sign(x) is the element-wise sign operator and where the +function ζ(x, b) uniformly quantizes each co-ordinate amongst +2b − 1 levels between 0 and 1. That is, if xi/∥x∥∞ ∈ +� +l +2b−1, l+1 +2b−1 +� +, then it is quantized as, +ζi(x, b) = +� +l+1 +2b−1, +with prob. +|xi| +∥x∥∞ (2b − 1) − l, +l +2b−1, +otherwise. +When x is quantized to b-bits per co-ordinate, its file size is +given by the function, s(b) = ∥x∥0 (b + 1) + 32 bits. Here, +the zero-norm, ∥x∥0, gives the length of the vector, the 1 +indicates the bit used to denote the sign, and the 32 bits are +for a floating point number denoting the norm, ∥x∥∞. Finally, +if client j uses the parameter bj, then the vector of parameters +used by the clients is denoted as, b = (bj)m +j=1. +2) Network Congestion Model: For purposes of evaluating +the performance of various algorithms over different types +of network congestion we propose the following general, +albeit idealized, model. We let Cn be a m dimensional +random vector denoting the Bit Transmission Delay (BTD) +for clients during round n. We further let Cn = exp (Zn) +i.e., coordinate-wise exponentiation of an m dimensional first +order autoregressive process given by (Zi)∞ +i=0 where Z0 = 0, +where +Zn = A Z(n−1) +En, +∀n ≥ 1, +(12) +where A is an m×m deterministic matrix, and En ∼ N(µ, Σ) +are i.i.d., m dimensional normal random vectors. Different +correlations across time and clients may be modelled by +varying A, µ and Σ. The marginal distributions of Cn are +thus log-normal but can be correlated in different ways based +on the underlying autoregressive process. In particular: +Homogeneous Independent: the parameters are set to A = +0, µ = 1, and Σ = σ2I. This results in a process which +is independent and identically distributed across clients +and time. +Heterogeneous Independent: the parameters are set to A = +0, µi = 0 for i ∈ {1, . . . , 5} and µi = 2 for i ∈ +{6, . . . , 10}, and Σ = I. This results in a process which is +independent across clients and time, with the BTD being +lower for the first 5 clients compared to the rest. +Perfectly correlated: the parameters are set to A such that +Ai,j = +a +m where a ∈ (0, 1), µ = 0, and Σ such that +Σi,j = σ2 = 1. This results in a process where all clients +see the same positively correlated time-varying delays. +Partially correlated: the parameters are set to A such that +Ai,j = a +m, µ = 0, and Σ such that Σi,i = 1 and Σi,j = +1/2 for i ̸= j. This results in a process where delays are +positively correlated accross clients and time. +3) Model for Round Durations: We will model the duration +of a round as the maximum across clients’ delays, i.e., +d(τ, b, c) = max +j [θτ + cjs(bj)], +where θ represents the compute time per local computation, +and cjs(bj) the BTD of client j times the size of the client +j’s file capturing the time taken to communicate its update. +For simplicity we will set θ = 0. +4) Compression Level Choice Policies: We compare NAC- +FL to the following policies, +a) Fixed Bit: Here, a number b is fixed, and all the +clients use the stochastic quantizer Qq(x, b) from (11) with +the parameter b. We present results for b ∈ {1, 2, 3}, as we +didn’t notice a performance improvement for larger parameters +in our experiments. +b) Fixed Error: This method was suggested in [13] and is +parameterized by a number q. At each round n, the parameters +bn of the stochastic quantizers are such that the average +normalized-variance ¯qn (see equation (15)) is smaller than +6 + +q, and the duration of the round d(τ, qn, cn) is minimized. +We fix q = 5.25 in all our experiments after finding it to be +performing well across different settings. +5) Machine Learning Model: We consider m = 10 clients. +We consider the MNIST dataset [24] which may be distributed +homogeneously or heterogeneously amongst the clients. Since +data is heterogeneous across clients in most FL applications, +we consider the heterogenous data case. That is, each client +has data corresponding to 1 unique label. The MNIST dataset +has 60,000 training samples, 10,000 test samples and 10 labels. +The clients and the server aim to train a fully connected neural +network with the architecture (784, 250, 10) with the sigmoid +activation for the hidden layer. The learning rate is initialized +to η0 = 0.07, and is decayed by a factor 0.9 every 10 rounds. +The aggregation rate and local computations per round are +fixed throughout the training to γ = 1 and τ = 2 respectively. +As for the parameters of the NAC-FL policy, we set βn = 1 +n, +and α = 2. +We measure the performance of the global model using the +following, +a) Training Loss: The training loss of the global model +is the empirical cross entropy loss across the entire set of +training samples. +b) Test Accuracy: The test accuracy is measured over all +the test samples. Here, in some experiments, we run 20 simu- +lations with different random seeds, and report the mean, 90th +percentile and 10th percentile times to reach a test accuracy +of 90%. The 90th and 10th percentile scores are reported to +capture the variation in performance across the 20 simulations. +We also report a gain metric, which is sample mean of the +time gained to reach 90% accuracy by NAC-Fl compared to a +another policy reported in percentage. For instance, let xi, yi +be the times under NAC-FL and another policy for a random +seed i, then the gain is 100 ∗ +��20 +i=1 yi/xi − 1 +� +/20. +B. Simulation Results +1) Homogeneous Independent BTD: We simulated over +σ2 ∈ {1, 2, 3} in order to study the change in performance +over increasing variance. We observe that in all the cases, +NAC-FL and the Fixed Error policy have very similar perfor- +mance across all the considered statistics. This is because the +Fixed Error policy was designed to operate well in the i.i.d., +network delay case. However, both NAC-FL and Fixed Error +policy perform better than all the Fixed Bit policies according +to all the statistics across all the considered parameters. More- +over, we observed that the gap in the performance to Fixed +Bit policies increased with increasing variance. For instance, +the gain of the best Fixed Bit policy increased from 145% +to 250% when the variance was increased from 1 to 3, while +the gain of the worst fixed bit policy increased from 314% to +881%. This is as expected because both NAC-FL and Fixed +Error policy adapt to the heterogenous delay of clients at any +given time. Surprisingly, NAC-FL lagged behind Fixed Error +policy in some metrics, but it performed better in terms of the +gain metric in all the 3 cases, with the gain over Fixed Error +policy ranging from 1% to 8%. +σ2 +1 bit +2 bits +3 bits +Fixed Error +NAC-FL +1 +Mean +6.31 +3.82 +4.15 +1.58 +1.60 +90th +6.95 +4.72 +5.00 +1.86 +2.05 +10th +5.63 +3.20 +3.38 +1.20 +1.14 +Gain +314% +145% +168% +3% +- +2 +Mean +54.8 +32.5 +34.9 +12.5 +12.2 +90th +70.6 +44.7 +43.1 +19.0 +20.8 +10th +42.5 +19.2 +21.0 +6.26 +5.82 +Gain +522% +216% +240% +8% +- +3 +Mean +799 +430 +458 +165 +168 +90th +1430 +752 +665 +318 +320 +10th +418 +157 +148 +46.2 +57.9 +Gain +881% +270% +250% +1% +- +TABLE I: Performance comparison of policies with homoge- +neous independent BTD in terms of the mean, 90th percentile +and 10th percentile times to reach 90% test accuracy under +the different policies, and their average sample-path gain +compared to NAC-FL. All the numbers represented are in 107 +seconds. +2) Heterogeneous Independent BTD: We considered this +case since the first 5 clients would have consistently worse +delay, NAC-FL and the Fixed Error policy would consis- +tently compress the updates of those clients heavily. Since +the data distribution is heterogeneous, it may be possible +heavy compression of updates from specific clients throughout +the training may hurt the performance. On the other hand, +the Fixed Bit policies use the same amount of compression +across all clients equally irrespective of their delays. Still, we +observed that NAC-FL and the Fixed Error policy perform +better than the Fixed Bit policies as can be seen in Table +II. In fact, performance in terms of the gain metric is very +comparable to the i.i.d., network delay case with σ2 = 1 in +Table I. +1 bit +2 bits +3 bits +Fixed Error +NAC-FL +Mean +9.49 +5.85 +6.46 +2.49 +2.48 +90th +11.5 +7.16 +8.09 +3.48 +3.54 +10th +8.30 +4.37 +4.98 +1.74 +1.54 +Gain +319% +146% +173% +4% +- +TABLE II: Performance comparison of policies with heteroge- +nous independent BTD. The numbers shown are the mean, +90th percentile and 10th percentile times to reach 90% test +accuracy under the different policies, and their average sample- +path gain compared to NAC-FL. All the numbers represented +are in 108 seconds. +3) Perfectly Correlated BTD: We will demonstrate that +NAC-FL performs better than Fixed Error and Fixed Bit +policies under increasing correlated delay across time since +they are not designed to optimize the wall clock time under +this case. +To study the variation of network delay across rounds, +consider the marginal auto-regressive process of 1 client which +may be represented by the following scalar autoregressive +process, +Zn = a′Z(n−1) + En, +(13) +where En ∼ N(0, 1). We define metric called asymptotic +7 + +variance, denoted σ2 +∞, which is designed to capture the +variance, and long and short term correlations of a random +process, +σ2 +∞ ≜ lim +n→∞ +E +�� +Z(1) + · · · + Zn�2� +n +. +(14) +For the autoregressive process in (13), it may be computed to +be, σ2 +∞ = 1/(1 − a′)2. +Table III shows the performance of the different policies un- +der varying asymptotic variance of the marginals. We observe +that in addition to beating the baseline fixed bit policies on all +the metrics, the NAC-FL performs better than the Fixed Error +policy in most metrics as well. Considering the gain metric, +we observe gain of 13% over the Fixed Error policy for low +asymptotic variance of σ2 +∞ = 1.56, and is as large as 27% +for higher asymptotic variance of σ2 +∞ = 4. Notably, in terms +of the 10th percentile time to reach 90% accuracy, the Fixed +Error policy required 40%, 23% and 32% more time compared +to NAC-FL in the σ2 +∞=1.56, 4 and 16 cases respectively. +σ2 +∞ +1 bit +2 bits +3 bits +Fixed Error +NAC-FL +1.56 +Mean +5.14 +3.04 +3.47 +2.21 +2.11 +90th +5.94 +3.65 +4.43 +2.66 +3.32 +10th +3.88 +2.38 +2.18 +1.43 +1.02 +Gain +191% +58% +75% +13% +- +4 +Mean +5.82 +3.49 +4.03 +2.47 +2.23 +90th +7.43 +4.77 +6.28 +3.94 +4.00 +10th +3.88 +2.22 +1.98 +1.21 +0.981 +Gain +252% +82% +107% +27% +16 +Mean +8.42 +5.19 +6.15 +3.75 +3.36 +90th +12.8 +10.3 +13.4 +7.94 +7.2 +10th +4.34 +1.40 +1.67 +1.15 +0.87 +Gain +316% +72% +98% +21% +- +TABLE III: Performance comparison of policies with perfectly +correlated BTD in terms of the mean, 90th percentile and 10th +percentile times to reach 90% test accuracy under the different +policies, and their average sample-path gain compared to +NAC-FL. All the numbers represented are in 107 seconds. +4) Partially Correlated BTD: In Table IV, we show results +for the partially correlated BTD case with asymptotic variance +σ2 +∞ = 4. We consider this case to demonstrate that NAC-FL +is effective with positive (but, not 100%) correlation across +clients as well. Indeed, we observe NAC-FL performing better +compared to all the other policies across all the considered +metrics, with a gain of 10% over the Fixed Error policy, and +129% over the best fixed bit policy. Notably, in terms of the +10th percentile and 90th percentile metrics, NAC-FL outper- +formed Fixed Error policy by 30% and 15% respectively. +Figure 3 contains sample path plots of Training Loss and +Accuracy vs Wall Clock Time for the independent homo- +geneous (σ2 = 2), heterogeneous and perfectly correlated +(σ2 +∞ = 4) BTD cases. Both accuracy and loss plots for +NAC-FL and Fixed Error are overlapping in the independent +homogeneous and heterogeneous BTD cases, as expected. +However, in the perfectly correlated BTD case, NAC-FL +dominates the performance of Fixed Error policy. +In summary, we observe that NAC-FL’s performance is +robust under a range of network models considered. NAC-FL +1 bit +2 bits +3 bits +Fixed Error +NAC-FL +Mean +13.6 +8.33 +9.51 +4.22 +3.83 +90th +15.9 +10.5 +13.9 +6.24 +5.46 +10th +9.51 +5.47 +5.80 +2.64 +2.02 +Gain +307% +129% +159% +10% +- +TABLE IV: Performance comparison of policies with partially +correlated BTD in terms of the mean, 90th percentile and 10th +percentile times to reach 90% test accuracy under the different +policies, and their average sample-path gain compared to +NAC-FL. All the numbers represented are in 107 seconds. +vastly outperformed the baseline Fixed Bit policies in all the +network models. The performance of NAC-FL was observed to +be similar to that of Fixed Error policy in the independent BTD +setting, albeit, it outperformed Fixed Error policy in terms of +the gain metric under all the network models. Notably, the +gap between NAC-FL and Fixed Error policy was observed +to be noticeably high in the perfectly and paritally correlated +BTD settings, where NAC-FL was able to adapt to positive +correlations of BTD across time, whereas Fixed Error could +not. +V. NAC-FL IN PRACTICE +In this section we briefly comment on some practical aspects +underlying estimating model update delays. This involves +estimating the network’s current average BTD to each client. A +simple approach to doing so is to observe that for the stochastic +quantizer described in Section IV-A1, clients always send the +vector of signs of their updates, no matter what are the bits +per coordinate that will be chosen. So, as the clients send +their signs, the server may probe the delay characteristics to +estimate the BTD of clients without having to request vacuous +(non update related) bits to do so. It may then use these +estimates to perform the optimization in (6) for the round. +VI. CONCLUSION +Due to their distributed character FL algorithms are exposed +to congestion across a potentially large number of network +resources, whence one might say they are exposed to network +congestion and variability at scale. Building adaptive algo- +rithms that minimize the impact of time varying congestion +across clients presents a significant challenge, particularly +when the aim is to directly optimize the expected wall clock +time. NAC-FL exemplifies a new class of robust algorithms to +optimally adapt clients’ lossy compression. This paper further +provides the technical roadmap to formalizing and showing +asymptotic optimality for such algorithms. +REFERENCES +[1] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, +“Communication-efficient learning of deep networks from decentralized +data,” in Artificial intelligence and statistics. +PMLR, 2017, pp. 1273– +1282. +[2] T. Li, A. K. Sahu, A. Talwalkar, and V. Smith, “Federated learning: +Challenges, methods, and future directions,” IEEE Signal Processing +Magazine, vol. 37, no. 3, pp. 50–60, 2020. +8 + +(a) +(b) +(c) +(d) +(e) +(f) +Fig. 3: Plots of Training Loss and Test Accuracy vs Wall Clock time on different network models. Figures (a) and (d) correspond +to homogeneous independent BTD case (σ2 = 2), Figures (b) and (e) correspond to the heterogeneous independent BTD case, +and Figures (c) and (f) correspond to the perfectly correlated BTD case (σ2 +∞ = 4). +[3] P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. +Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, R. Cummings et al., +“Advances and open problems in federated learning,” Foundations and +Trends® in Machine Learning, vol. 14, no. 1–2, pp. 1–210, 2021. +[4] J. Koneˇcn`y, H. B. McMahan, F. X. Yu, P. Richt´arik, A. T. Suresh, and +D. Bacon, “Federated learning: Strategies for improving communication +efficiency,” arXiv preprint arXiv:1610.05492, 2016. +[5] D. Alistarh, D. Grubic, J. Li, R. Tomioka, and M. Vojnovic, “QSGD: +Communication-efficient SGD via gradient quantization and encoding,” +Advances in neural information processing systems, vol. 30, 2017. +[6] D. Basu, D. Data, C. Karakus, and S. Diggavi, “Qsparse-local-SGD: Dis- +tributed SGD with quantization, sparsification and local computations,” +Advances in Neural Information Processing Systems, vol. 32, 2019. +[7] S. U. Stich, “Local SGD converges fast and communicates little,” arXiv +preprint arXiv:1805.09767, 2018. +[8] S. U. Stich, J.-B. Cordonnier, and M. Jaggi, “Sparsified SGD with +memory,” Advances in Neural Information Processing Systems, vol. 31, +2018. +[9] S. Ghadimi and G. Lan, “Stochastic first-and zeroth-order methods for +nonconvex stochastic programming,” SIAM Journal on Optimization, +vol. 23, no. 4, pp. 2341–2368, 2013. +[10] A. Reisizadeh, A. Mokhtari, H. Hassani, A. Jadbabaie, and R. Pedarsani, +“Fedpaq: A communication-efficient federated learning method with +periodic averaging and quantization,” in International Conference on +Artificial Intelligence and Statistics. +PMLR, 2020, pp. 2021–2031. +[11] F. Haddadpour, M. M. Kamani, A. Mokhtari, and M. Mahdavi, “Feder- +ated learning with compression: Unified analysis and sharp guarantees,” +arXiv preprint arXiv:2007.01154, 2020. +[12] F. Sattler, S. Wiedemann, K.-R. M¨uller, and W. Samek, “Robust and +communication-efficient federated learning from non-iid data,” IEEE +transactions on neural networks and learning systems, vol. 31, no. 9, +pp. 3400–3413, 2019. +[13] P. S. Bouzinis, P. D. Diamantoulakis, and G. K. Karagiannidis, “Wireless +quantized federated learning: A joint computation and communication +design,” arXiv preprint arXiv:2203.05878, 2022. +[14] X. Zhang, X. Zhu, J. Wang, H. Yan, H. Chen, and W. Bao, “Federated +learning with adaptive communication compression under dynamic +bandwidth and unreliable networks,” Information Sciences, vol. 540, pp. +242–262, 2020. +[15] H. Sun, X. Ma, and R. Q. Hu, “Adaptive federated learning with gra- +dient compression in uplink NOMA,” IEEE Transactions on Vehicular +Technology, vol. 69, no. 12, pp. 16 325–16 329, 2020. +[16] D. Jhunjhunwala, A. Gadhikar, G. Joshi, and Y. C. Eldar, “Adaptive +quantization of model updates for communication-efficient federated +learning,” in ICASSP 2021-2021 IEEE International Conference on +Acoustics, Speech and Signal Processing (ICASSP). +IEEE, 2021, pp. +3110–3114. +[17] R. H¨onig, Y. Zhao, and R. Mullins, “DAdaQuant: Doubly-adaptive quan- +tization for communication-efficient federated learning,” in International +Conference on Machine Learning. +PMLR, 2022, pp. 8852–8866. +[18] W. Chen, S. Horvath, and P. Richtarik, “Optimal client sampling for +federated learning,” arXiv preprint arXiv:2010.13723, 2020. +[19] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, “A joint +learning and communications framework for federated learning over +wireless networks,” IEEE Transactions on Wireless Communications, +vol. 20, no. 1, pp. 269–283, 2020. +[20] M. Ribero and H. Vikalo, “Communication-efficient federated learning +via optimal client sampling,” arXiv preprint arXiv:2007.15197, 2020. +[21] J. Perazzone, S. Wang, M. Ji, and K. S. Chan, “Communication-efficient +device scheduling for federated learning using stochastic optimization,” +in IEEE INFOCOM 2022-IEEE Conference on Computer Communica- +tions. +IEEE, 2022, pp. 1449–1458. +[22] A. L. Stolyar, “On the asymptotic optimality of the gradient scheduling +algorithm for multiuser throughput allocation,” Operations research, +vol. 53, no. 1, pp. 12–25, 2005. +[23] M. Frank and P. Wolfe, “An algorithm for quadratic programming,” +Naval research logistics quarterly, vol. 3, no. 1-2, pp. 95–110, 1956. +[24] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning +applied to document recognition,” Proceedings of the IEEE, vol. 86, +no. 11, pp. 2278–2324, 1998. +9 + +101 +NAC-FL +b=1 +b=2 +b=3 +Training Loss +Fixed Error +100 +10-1 +10-2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Wall Clock Time +1e9101 +NAC-FL +b=1 +b=2 +b=3 +Training Loss +Fixed Error +100 +10-1 +10-2 +0 +1 +2 +3 +4 +5 +Wall Clock Time +1e8101 +Training Loss +100 +10-1 +NAC-FL +b=1 +b=2 +b=3 +Fixed Error +10-2 +0 +1 +2 +3 +4 +5 +6 +Wall Clock Time +1e70.9 +0.8 +Test Accuracy +0.7 +0.6 +0.5 +NAC-FL +0.4 +b=1 +b=2 +0.3 +b=3 +Fixed Error +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Wall Clock Time +1e90.9 +0.8 +Test Accuracy +0.7 +0.6 +0.5 +NAC-FL +0.4 +b=1 +b=2 +0.3 +b=3 +Fixed Error +0.2 +0 +1 +2 +3 +4 +5 +Wall Clock Time +1e80.9 +0.8 +Test Accuracy +0.7 +0.6 +0.5 +0.4 +NAC-FL +0.3 +b=1 +b=2 +0.2 +b=3 +Fixed Error +0 +L +2 +3 +4 +5 +6 +Wall Clock Time +1e7[25] G. +Sparling, +“Honors +calculus +notes.” +[Online]. +Available: +http://www.math.pitt.edu/ sparling/23014/23014notes6/node24.html +[26] P. Billingsley, Convergence of probability measures. +John Wiley & +Sons, 2013. +[27] S. N. Ethier and T. G. Kurtz, Markov processes: characterization and +convergence. +John Wiley & Sons, 2009. +[28] A. El Gamal and Y.-H. Kim, Network information theory. +Cambridge +university press, 2011. +[29] K.-M. Chung, H. Lam, Z. Liu, and M. Mitzenmacher, “Chernoff- +Hoeffding bounds for markov chains: Generalized and simplified,” arXiv +preprint arXiv:1201.0559, 2012. +[30] D. A. Levin and Y. Peres, Markov chains and mixing times. +American +Mathematical Soc., 2017, vol. 107. +10 + +APPENDIX A +FEDERATED LEARNING WITH ADAPTIVE COMPRESSION (FLAC) +In this section, we consider a variant of the FedCOM algorithm [11], which we will call FedCOM-V. FedCOM is based on +fixing a quantization parameter throughout run of the FL algorithm. On the other hand, FedCOM-V allows for an arbitrary +sequence of quantization parameters (qn)n, in order to account for adaptive compression policies such as NAC-FL. FedCOM-V +is presented in Algorithm 2. +Algorithm 2: FedCOM-V +Input +: number of local computations schedule (τn)∞ +n=1, local learning rate schedule (ηn)∞ +n=1, adaptively chosen +global learning rate schedule (γn)∞ +n=1, adaptively chosen number of rounds r, initial global model w1. +Result: wr+1: Final model +1 for n = 1, . . . , r do +2 +for each client j ∈ [m] do +3 +Set w1,n +j += wn ; +4 +for a = 1, . . . , τn do +5 +Sample a minibatch Za,n +j +and compute ˜ga,n +j +≜ ∇f(wa,n +j +; Za,n +j +) ; +6 +wa+1,n +j += wa,n +j +−ηn˜ga,n +j +; +7 +end +8 +Device sends ˜gn +Qj = Q((wn − wτn+1,n +j +)/ηn, qn +j ) back to the server; +9 +end +10 +Server computes, ˜gn +Q = 1 +m +�m +j=1 ˜gn +Qj ; +11 +Server computes wn+1 = wn −ηnγn˜gn +Q and broadcasts to all devices; +12 end +In order to study the convergence properties of FedCOM-V, we make the following standard assumptions. +Assumption 6 (Smoothness and Lower Boundedness). The objective function f(·) is differentiable and L-smooth. That is, +∥∇f(x) − ∇f(y)∥ ≤ L ∥x − y∥, for every x, y ∈ Rd. Moreover, the optimal value of f is lower bounded, f ∗ = minw f(w) > +−∞. +Assumption 7 (Bounded Variance). For all clients j ∈ [m] and rounds n and local step a, we can sample an independent mini- +batch Za,n +j +of size |Za,n +j +|= b and compute an unbiased stochastic gradient ˜ga,n +j += ∇f(w; Za,n +j +), i.e., EZa,n +j +[˜gj] = ∇f(wa,n +j +). +Moreover, the variance is bounded by a constant σ2, i.e., EZa,n +j +���˜ga,n +j +− ∇f +� +wa,n +j +���2� +≤ σ2. +Assumption 8 (Compression Model). The output of the compressor Q(x, q) is an unbiased estimator of x, i.e., E[Q(x, q)|x] = +x, and, its variance is bounded as, E[∥Q(x, q) − x∥2 |x] ≤ q ∥x∥2. +We denote the maximum normalized-variance by qmax and the average normalized-variance used at round n by +¯qn = 1 +m +m +� +j=1 +qn +j . +(15) +The following Theorem states the relationship between (qn)n, ε and rε and is proved in Appendix F. +Theorem 2. Let Algorithm 2 be run with a sequence of compressors such that the average normalized-variance at round n +is ¯Qn. Further, assume that the sequence +� ¯Qn� +n forms a stationary process with the stationary distribution represented by a +random variable Q. To obtain E[∥∇f(w)∥2] ≤ ε, we can choose, +rε = O +� +log(1/ε)E +�√Q + 1 +� +ε +� +, +τ n = O (n) . +The upper bound on rε in Theorem 2 provides a justification for Assumption 1 with hε(q) = O(√q + 1). Here, τ n is a +function of n, but for the purposes of NAC-FL we may use the average of τ (1) to τ (rε) in the expression of the duration +function. One may obtain a similar expression for other popular FL algorithms [5], [8]. +11 + +APPENDIX B +PROOF OF THEOREM 1 +In this section we show that NAC-FL converges to the optimal solution asymptotically as β ↓ 0. +In order to consider the effect of β ↓ 0 on NAC-FL estimates ˆRn +ε and ˆDn in (9), we shall denote these as ˆRn +ε,β and ˆDn +β +respectively. Let conv(Vε) be the convex hull of the set Vε defined in (7). Recall the positive sequence (βi)i with βi → 0 +from the statement of Theorem 1. Letting Xn +β ≜ ( ˆRn +ε,β ˆDn +β)⊤, and H(x) ≜ x1x2 over the domain R2 ++, we have the following +result. +Proposition B.1. Let the initialization X0 +β be equal to x0 ∈ R2 ++ almost surely for any 0 < β < 1, then, for any s > 0, +limi→∞ X⌊s/βi⌋ +βi +exists, is almost surely deterministic and denoted as x(s) ≜ limi→∞ X(⌊s/βi⌋) +βi +. Further, for any x0 ∈ R2 ++, +x(s) obeys the following differential equation, +x(0) = x0, +˙x(s) = v(s) − x(s), +s > 0, +v(s) = +argmin +v∈conv (Vε) +∇H (x(s))⊤ v, +s > 0. +(16) +The proof of Proposition B.1 is very similar to that of the main result of [22]. For the sake of completeness, we briefly +show the proof at the end of this section. From hereon, (16) will be referred to as the Fluid-Frank-Wolfe (FFW) process. +Proposition B.2. Under Assumption 5, the FFW process in (16) has a unique fixed point x∗ ∈ conv(Vε) such that, +x∗ = +argmin +x ∈conv (Vε) +∇H (x∗)⊤ x . +Moreover, x∗ ∈ Vε, and x∗ is the minimizer of H over the set Vε. +Proposition B.2 is proved in Appendix E. +Denote, G(x) = minv∈conv(Vε) ∇H(x)⊤(v − x). Since, ∇H is a continuous function of x, G(x) is a continuous function +of x as well. +As a consequence of Proposition B.2, there exists a unique point x∗ ∈ conv(V )ε such that G(x∗) = 0. For all other +x ∈ conv(Vε), G(x) < 0. We will, in fact, prove a stronger result that G(x) is bounded away from 0 for points that are a +distance away from x∗. +Claim 1: for any ω > 0, there exists a ξ > 0 such that if x ∈ conv(Vε) and ∥x − x∗∥ ≥ ω, then G(x) < −ξ. +Proof. We prove this claim by contradiction. Suppose there exists an ω > 0 such that for all ξ > 0, the set, +X ξ ≜ +� +xξ : xξ ∈ conv(Vε), +��xξ − x∗�� ≥ ω and G(xξ) ≥ −ξ +� +, += conv(Vε) +� � +xξ : +��xξ − x∗�� ≥ ω +� � � +xξ : G(xξ) ≥ −ξ +� +, +is non-empty. +conv(Vε) is a compact set because it is the convex hull of a compact set, Vε. Further, the sets +� +xξ : +��xξ − x∗�� ≥ ω +� +and +� +xξ : G(xξ) ≥ −ξ +� +are also closed because they are the pre-image of continuous functions over closed sets. Therefore, X ξ is +a closed set since it is the intersection of three closed sets. Further, it is also bounded because conv(Vε) is bounded. Therefore, +X ξ is a compact set. +Consider ξ1 > ξ2 > 0. Since, G(x) ≥ −ξ2 implies that G(x) ≥ −ξ1, we have that X ξ1 ⊃ X ξ2. Consider a decreasing +sequence (ξi)i∈N with limi→∞ ξi = 0. Then, (X ξi)i∈N is a decreasing sequence of compact and non-empty sets. We know +that a decreasing sequence of non empty compact sets has a limit, and the limit is non-empty [25]. Therefore, +X 0 ≜ +∞ +� +i=1 +X ξi, +exists and is non-empty. Since ξi ↓ 0, this means that any x ∈ X 0 satisfies G(x) = 0. Since ∥x − x∗∥ ≥ ω and x ∈ conv(Vε) +for any x ∈ X 0, this is a contradiction to the fact that x∗ is a unique point in conv(Vε) with G(x) = 0. Therefore, there must +exist some ξ > 0 for which X ξ is empty. +Next we proceed to study the asymptotic convergence of the process x(·). Note that since Vε is apriori unknown, the +initialization x0 may not be in the set Vε. Nevertheless, the FFW process x(·) eventually reaches the set conv(Vε). In order +to formalize this, let convζ(Vε) denote the ζ-thickening of the set conv(Vε), +convζ(Vε) = {y : ∃ x ∈ conv(Vε) such that ∥y − x∥2 ≤ ζ}. +12 + +Proposition B.3. Consider the FFW process defined in (16). For every ζ > 0, there exists an sζ > 0 such that, x(s) ∈ convζ(Vε) +for all s > sζ. +The proof is the same as that of Corollary 2 in [22]. +Claim 2: x(s) → x∗ as s → ∞. +Proof. First we prove that lim inf +s→∞ +∥x(s) − x∗∥ = 0 by contradiction. As a contradiction assume that there exists an ω > 0 +and sω > 0 such that ∥x(s) − x∗∥ > ω for all s > sω. +Let ξ > 0 be the constant according to Claim 1 which ensures that G(x) < −ξ for all x in conv(Vε) satisfying ∥x − x∗∥ > ω. +Moreover, due to continuity of G(·), there exists a ξ′ > 0 and a small enough ζ > 0 such that G(x) < −ξ′ for all x in +convζ(Vε) that satisfy ∥x − x∗∥ ≥ ω. Due to Proposition B.3, there exists a constant sζ > 0 such that x(s) ∈ convζ(Vε) for +all s > sζ. +Define, sω +∗ = sω + sζ + (H(x(sζ + sω)) + 1)/ξ′. Then, +H (x (sω +∗ )) = H(x(sζ + sω)) + +� sω +∗ +sζ+sω dH(x(s)), += H(x(sζ + sω)) + +� sω +∗ +sζ+sω ∇H(x(s))⊤ ˙x(s)ds, += H(x(sζ + sω)) + +� sω +∗ +sζ+sω ∇H(x(s))⊤(v(s) − x(s))ds, += H(x(sζ + sω)) + +� sω +∗ +sζ+sω G(x(s))ds, +< H(x(sζ + sω)) + +� sω +∗ +sζ+sω −ξ′ds, += H(x(sζ + sω)) − H(x(sζ + sω)) − 1 < 0. +Since, H is a positive function, this is a contradiction. Therefore, there exists a time s > sω + sζ such that ∥x(s) − x∗∥ < ω. +Since this is true for every ω > 0 and sω > 0, we have proved that lim inf +s→∞ +∥x(s) − x∗∥ = 0. +Next we prove that lims→∞ x(s) = x∗. Define, +Hω = +max +x∈convζ(Vε) +∥x − x∗∥≤ω +H(x). +Since lim inf +s→∞ ∥x(s) − x∗∥ = 0, there exists a constant sω +th > sζ such that ∥x(sω +th) − x∗∥ ≤ ω. Due to Proposition B.3, for all +s > sω +th, we have x(s) ∈ convζ(Vε). Therefore, if for any s > sω +th, H(x(s)) > Hω is true, then x(s) satisfies x(s) ∈ convζ(Vε) +and ∥x(s) − x∗∥ > ω. Therefore, due to Claim 1 at all such points, the gradient satisfies, dH(x(s))/ds = G(x(s)) < 0. This +implies that H(x(s)) ≤ Hω for all s > sω +th. +Moreover, by the continuity of H(·), Hω → H(x∗) as ω ↓ 0. And, by definition of the minimum x∗, H(x(s)) ≥ H(x∗) +for any s > 0. Therefore, by the Sandwich Theorem, lims→∞ H(x(s)) = H(x∗). +Further, by the continuity of H(·) and the uniqueness of the minimum x∗, lims→∞ H(x(s)) = H(x∗) implies that +lims→∞ x(s) = x∗. +Claim 2 proves that the Fluid-Frank-Wolfe process converges to the optimal solution x∗ asymptotically. In particular, for +any ρ > 0, there exists an nth(ρ) > 0 such that, +sup +s>nth(ρ) +∥x(s) − x∗∥ ≤ ρ. +Denote, xβ(s) = X⌊s/β⌋ +β +. Then, since the functions converge as follows, (xβi) → x as i → ∞, from the Continous Mapping +Theorem [26, Theorem 2.7], we have, +lim +i→∞ +sup +s>nth(ρ) +P (∥xβi(s) − x∗∥ > ρ) = 0. +The above implies the Theorem statement, +lim +i→∞ +sup +n>nth(ρ)/βi +P +���Xn +βi − x∗�� > ρ +� += 0. +13 + +A. Proof of Proposition B.1 +Define the “scaled process” as, xβ(s) ≜ X⌊s/β⌋ +β +. Denote DR2[0, ∞) as the set of functions with domain [0, ∞), range R2, +and which are right continuous with left limits. Observe that xβ has sample paths in DR2[0, ∞) for any 0 < β < 1. +Denote, V n +β ≜ +�∥hε(qn)∥ +d(τ, qn, c) +� +, which is the action taken by the NAC-FL algorithm (Algorithm 1) at round n, and vβ(s) ≜ +V ⌊s/β⌋ +β +. Defining, +K = max +� +x0, +max +q∈[0,qmax],C∈C +���� +� ∥hε(q)∥ +d(τ, q, C) +����� +� +, +by the update rule of NAC-FL, xβ(s) = (1 − β) xβ(s − β) + βvβ(s − β), we have xβ(s) ≤ K for any 0 < β < 1 and s > 0. +Further, rearranging the NAC-FL update rule as, +xβ(s) − xβ(s − β) = β (vβ(s − β) − xβ(s − β)) +we obtain, ∥xβ(s) − xβ(s − β)∥ ≤ 2βK. More generally, for any s1, s2 > 0, we have, +∥xβ(s1) − xβ(s2)∥ ≤ 2K max(β, |s1 − s2|). +This implies the “asymptotic Lipschitz” property, +lim +β→0 ∥xβ(s1) − xβ(s2)∥ ≤ 2K|s1 − s2|, +∀s1, s2 > 0. +Then, by Corollary 7.4 in Chapter 3 of [27], the set of stochastic processes {xβ(·)}0<β<1 is relatively compact. Therefore, +there exists a sequence (βi)i with βi → 0 as i → ∞ such that xβi(·) → x(·) as i → ∞ for some stochastic process x(·) with +sample paths in DR2[0, ∞). +Next, we need to prove that x(·) behaves according to (16). To do so, observe that due to the “continuity property” (i.e., +���Xn +β − Xn−1 +β +��� ≤ 2Kβ), for any δ > 0, there exists a small enough β > 0 and ∆ > 0 such that, for any integer n in the +range [s/β, (s + ∆)/β], we have, +| +� +∇H(Xn +β) +�⊤ V n +β − Y ∗ +Cn| ≤ δ, +where, +Y ∗ +C ≜ +min +q∈[0,qmax] (∇H(xβ(s)))⊤ +� ∥hε(q)∥ +d(τ, q, C) +� +, +C ∈ C. +The above equations say that the optimal action at any round in the considered range is very close to the optimal action at the +start of the range, for an appropriate selection of parameters. Summing across n in the range [s/β, (s + ∆)/β] we obtain, +������ +� +s/β≤n≤(s+∆)/β +� +∇H(Xn +β) +�⊤ V n +β − +� +s/β≤n≤(s+∆)/β +Y ∗ +Cn +������ +≤ δ∆/β. +Multiplying by β on both sides, from the definition of the scaled process, we have, +������ +� s+∆ +s +(∇H(xβ(ξ)))⊤ vβ(ξ)dξ − +� +s/β≤n≤(s+∆)/β +βY ∗ +Cn +������ +≤ δ∆. +From the Law of Large Numbers for Markov Chains, we have limβ→0 +� +s/β≤n≤(s+∆)/β βY ∗ +Cn = ∆ � +C∈C µ(C)Y ∗ +C. Similar +to the convergence of xβ shown above, one can prove convergence of vβ to a process v. Therefore, taking limit i → ∞ along +the sequence (βi)i, we get, +����� +� s+∆ +s +(∇H(x(ξ)))⊤ v(ξ)dξ − ∆ +� +C∈C +µ(C)Y ∗ +C +����� ≤ δ∆. +Observe that � +C∈C µ(C)Y ∗ +C = minv∈conv(Vε) (∇H(x(s)))⊤ v. Therefore, by choosing a ∆ small enough, we get, +����(∇H(x(s)))⊤ v(s) − +min +v∈conv(Vε) (∇H(x(s)))⊤ v +���� ≤ δ. +Since δ can also be chosen arbitrarily small, we have, +(∇H(x(s)))⊤ v(s) = +min +v∈conv(Vε) (∇H(x(s)))⊤ v. +14 + +APPENDIX C +PROOF OF LEMMA 1 +In this section we show that a state-dependent stationary policy asymptotically optimizes the wall clock time. To do so, we +first define the notion of a type for sequences of network states and compression parameters. Then, we show that for a given +network state sequence, a policy for choosing compression parameters which depends on the sequence type optimizes the +wall clock type. Finally, because the type asymptotically concentrates for markov processes, we show that a state-dependent +stationary policy asymptotically optimizes the wall clock time. +We start by defining the notion of an empirical distribution, called type, and its associated expectation and conditional +expectations. +Definition 1 (Type). The type of a finite sequence, x[r] ≜ (xn)r +n=1 with elements in domain X, is a function, ˆp +� +· ; x[r]� +: +X → [0, 1], defined as, +ˆp +� +x ; x[r]� += +�r +n=1 1 (xn = x) +r +, +∀x ∈ X, +where 1(x = y) = 1 if x = y, and 0 otherwise. +Similarly, the conditional type and the joint type are defined as follows. +Definition 2 (Joint Type and Conditional Type). The joint type of two finite sequences, x[r] and y[r] with domains X and Y +respectively, is a function, ˆp +� +· ; x[r], y[r]� +: X × Y → [0, 1], defined as, +ˆp +� +x, y ; x[r], y[r]� += +�r +n=1 1 (xn = x , yn = y) +r +, +∀x ∈ X, y ∈ Y. +The conditional type ˆp +� +·|· ; x[r], y[r]� +: X × Y → [0, 1] is defined as, +ˆp +� +x|y ; x[r], y[r]� += +�r +n=1 1 (xn = x , yn = y) +�r +n=1 1 (yn = y) +, +∀x ∈ X, y ∈ Y such that ˆp(y; y[r]) > 0, += ˆp +� +x, y; x[r], y[r]� +ˆp +� +y; y[r]� +. +Then, the expectation and conditional expectation with respect to the type may be defined as follows. +Definition 3 (Expectation and Conditional Expectation). The expectation of a non-negative function g : X → R+ with respect +to type ˆp(·; x[r]) is defined as1, +ˆE +� +g(X); x[r]� +≜ +� +x∈X +g(x)ˆp(x; x[r]), +where X denotes a random variable with distribution ˆp(x; x[r]). Similarly, the conditional expectation of a non-negative function +l : X → R+ with respect to the type ˆp(·|·; x[r], y[r]) is defined as, +ˆE +� +l(X)|Y = y; x[r], y[r]� +≜ +� +x∈X +l(x)ˆp(x|y; x[r], y[r]), +∀y ∈ Y such that ˆp(y; y[r]) > 0, +where the random variable pair (X, Y ) has joint distribution ˆp(x, y; x[r], y[r]). +Proposition C.1. Suppose Assumptions 1 and 3 hold, and let (cn)n denote an observed sequence of network states and (qn) +denote a sequence of compression parameters. Then, for any positive integer r and positive ε with the associated function +hε(·) defined in Assumption 1, there exists a sequence dependent, state dependent stationary policy π such that, +r +� +n=1 +∥hε (qn)∥ ≥ +r +� +n=1 +∥hε (π (cn))∥ , +(17) +and, +r +� +n=1 +d (τ, qn, cn) ≥ +r +� +n=1 +d (τ, π (cn) , cn) . +(18) +1If X is uncountably infinite, then, � +x∈X g(x) ≜ sup +�� +x∈F g(x) : F ⊂ X, F is finite +� +. +15 + +Proof of Proposition C.1. Given the sequence (qn, cn)r +n=1, we obtain the joint type p(·, ·; q[r], c[r]). Thus, one may interpret the +sequence as given an observed network state c, the policy plays the compression parameters q with probability ˆp(q|c ; q[r], c[r]). +Define the state-dependent stationary policy π as playing the conditional mean (w.r.t., the function hε) given any network state +c. That is, +π(c) = h−1 +ε +� +ˆE +� +hε(Q)| C = c; q[r], c[r]�� +, +∀ c ∈ C such that ˆp(c; c[r]) > 0. +(19) +Such a choice for π(c) always exists because hε(·) is continuous, bounded and strictly increasing coordinate-wise applied +function which implies that the inverse operator of hε(·) is well-defined. Note that hε(π(c)) = ˆE +� +hε(Q)| C = c; q[r], c[r]� +. +So, due to the convexity of ∥·∥, +∥hε (π(c))∥ ≤ ˆE +� +∥hε (Q)∥ |C = c ; q[r], c[r]� +, +∀ c ∈ C. +(20) +Then, +r +� +n=1 +∥hε (π(cn))∥ = rˆE +� +∥hε(π(C))∥ ; c[r]� +, +(a) +≤ rˆE +� +ˆE +� +∥hε(Q)∥ |C = c ; q[r], c[r]� +; c[r]� +, +(b) += rˆE +� +∥hε(Q)∥ ; q[r]� +, += +r +� +n=1 +∥hε (qn))∥ , +(21) +where (a) follows from (20), and (b) follows from the Tower-rule of expectations. +Next we bound d(τ, π(c), c). By the definition of π in (19), for all c ∈ C, +d (τ, π(c), c) = d +� +τ, h−1 +ε +(hε(π(c))) , c +� +, +(a) += d +� +τ, h−1 +ε +� +ˆE +� +hε(Q)| C = c; q[r], c[r]�� +, c +� +, +(b) +≤ ˆE +� +d +� +τ, h−1 +ε +(hε(Q)) , c +� +|C = c ; q[r], c[r]� +, += ˆE +� +d (τ, Q, c) |C = c ; q[r], c[r]� +, +(22) +where (a) follows from the definition of policy π, and (b) follows from the convexity of d(τ, h−1 +ε (·), c) (Assumption 3). (22) +is analogous to (20). Therefore, we may repeat the same calculation in (21) for the delay, +r +� +n=1 +d(τ, π(cn), cn) = rˆE +� +d(τ, π(C), C) ; c[r]� +, +≤ rˆE +� +ˆE +� +d(τ, Q, C)|C = c ; q[r], c[r]� +; c[r]� +, += rˆE +� +d(τ, Q, C) ; q[r], c[r]� +, += +r +� +n=1 +d(τ, qn, cn), +Equation (17) in Proposition C.1 states that if the FL algorithm with a sequence of compression parameters (qn)n has +reached an error tolerance of ε by round r, then, under Assumption 1, it has also reached error tolerance ε under sequence +of compression parameters (π(cn))n by around r. Moreover, (18) states that (π(cn))n takes lesser amount of time up to +round r compared to (qn)n. However, this construction of π is sequence dependent. More specifically, it is dependent on the +type ˆp(· ; c[r]) of the network state sequence observed. In order to prove Lemma 1, we need to construct a state-dependent +but sequence-independent stationary policy that is near-optimal in minimizing the expected wall clock time. Therefore, in the +following, we first define the notion of a typical set and show in Proposition C.2 that the type of an observed network state +sequence concentrates around its mean with high probability. +Definition 4 (Typical Set). For a distribution p on network sets, a typical set with parameters (r, ν), is defined as, +T r +ν (p) ≜ {cr : |ˆp(c|cr) − p(c)| ≤ νp(c), for all c ∈ C} . +16 + +We will use the following result called the Typical Averaging Lemma for typical sets [28, Section 2.4]. +Lemma C.1. Let cr ∈ T r +ν (p). Then, for any non-negative function g : C → R+, +(1 − ν) E [g(C)] ≤ 1 +r +r +� +n=1 +g (cn) ≤ (1 + ν) E [g(C)] , +where C is a random variable with distribution p. +Next, due to ergodicity of stationary Markov chains, we have the following proposition which is proved at the end of this +section. +Proposition C.2. Let Assumption 4 be true. Then, there exist positive constants κ1 and κ2 such that, for every ν > 0, and +r ∈ N, +P +� +∃r′ ≥ r such that Cr′ ̸∈ T r′ +ν (µ) +� +≤ κ1 exp +� +−κ2ν2r +� +. +Lemma C.1 will be used to argue that if two network-state sequences have similar types (they belong to T r +ν (µ)), then a state +dependent stationary policy π will have a similar expected wall clock to converge under both sequences. Then, Proposition +C.2 will be used to argue that one observes a typical network state sequence with high probability. We proceed to prove this +formally in the following proof of Lemma 1. +Proof of Lemma 1. Denote hmin +ε +and hmax +ε +as the minimum and maximum of the bounded function hε(·). Then, Assumption +1 implies that the number of rounds needed to converge to an error tolerance ε under any sequence of compression parameters +is bounded between hmin +ε +and hmax +ε +. +For a positive ν, let ε be small enough such that hmin +ε +> 2(1 + ν)/ν. First, we will consider network state sequences which +are typical with repsect to ν. Specifically, we consider a sequence (cn)n such that cr belongs to T r +ν (p) for every r > hmin +ε +. +Due to Proposition C.1, there exists a state-dependent stationary policy that optimizes the wall clock time to reach error +tolerance ε with respect to the sequence (cn)n. Let π′ represent this policy, and rπ′ +ε +be the minimum number of rounds taken +to converge by π′. Then, the wall clock time for π′ can be lower bounded as, +rπ′ +ε +� +n=1 +d (τ, π′ (cn) , cn) = +� +� 1 +rπ′ +ε +rπ′ +ε +� +n=1 +d (τ, π′ (cn) , cn) +� +� rπ′ +ε , +(a) +≥ (1 − ν) E [d(τ, π′(C), C)] rπ′ +ε , +(b) +≥ (1 − ν) E [d(τ, π′(C), C)] +� +� 1 +rπ′ +ε +rπ′ +ε +� +n=1 +∥hε (π (cn))∥ +� +� , +(c) +≥ (1 − ν)2 E [d(τ, π′(C), C)] E [∥hε (π′(C))∥] , +(d) +≥ (1 − ν)2 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] . +(23) +(a) and (c) follow from Lemma C.1, (b) follows from Assumption 1, and (d) follows by the following definition, +π∗ = arg min +π ∈Π +E [d(τ, π(C)), C)] E [∥hε (π(C))∥] . +Performing a similar calculation for π∗, +rπ∗ +ε� +n=1 +d (τ, π∗ (cn) , cn) = +� +� 1 +rπ∗ +ε +rπ∗ +ε� +n=1 +d (τ, π∗ (cn) , cn) +� +� rπ∗ +ε , +(a) +≤ (1 + ν) E [d(τ, π∗(C), C)] rπ∗ +ε , +(b) +≤ (1 + ν)2 E [d(τ, π∗(C), C)] +� +� 1 +rπ∗ +ε +rπ∗ +ε� +n=1 +∥hε (π∗ (cn))∥ +� +� , +(c) +≤ (1 + ν)3 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] . +(24) +17 + +(a) and (c) follow from Lemma C.1 and (b) follows from Proposition C.3 proved at the end of this section. +The expected wall clock time to reach error tolerance ε under the state-dependent stationary policy π∗ can be upper bounded +as, +E +� +T π∗ +ε +� (a) +≤ P +� +CRπ∗ +ε +∈ T Rπ∗ +ε +ν +(µ) +� +(1 + ν)3 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] + P +� +CRπ∗ +ε +̸∈ T Rπ∗ +ε +ν +(µ) +� +hmax +ε +dmax, +(b) +≤ (1 + ν)3 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] + P +� +∃r′ ≥ hmin +ε +, Cr′ ̸∈ T r′ +ν (µ) +� +hmax +ε +dmax, +(c) +≤ (1 + ν)3 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] + κ1 exp +� +−κ2ν2hmin +ε +� +hmax +ε +dmax, +(25) +where, (a) follows by using (25) for typical sequences, and upper bounding Rπ∗ +ε +by hmax +ε +and the round duration +by dmax for non-typical sequences. (b) follows by upper bounding P +� +CRπ∗ +ε +∈ T Rπ∗ +ε +ν +(µ) +� +by 1, and upper bounding +P +� +CRπ∗ +ε +̸∈ T Rπ∗ +ε +ν +(µ) +� +by P +� +∃r′ ≥ hmin +ε +, Cr′ ̸∈ T r′ +ν (µ) +� +because, almost surely, Rπ∗ +ε +≥ hmin +ε +. (c) follows from Proposition +C.2. +Let T ∗ +ε be the random variable representing the wall-clock time to reach error tolerance ε when one uses the optimal sample- +path sequence dependent policy on random network-state sequence (Cn)n. Then, denoting R∗ +ε as the corresponding random +variable denoting the number of rounds needed to reach error tolerance ε, we have, +E [T ∗ +ε ] +(a) +≥ P +� +CR∗ +ε ∈ T R∗ +ε +ν +(µ) +� +(1 − ν)2 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] , +(b) +≥ P +� +∀r′ ≥ hmin +ε +, Cr′ ∈ T r′ +ν (µ) +� +(1 − ν)2 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] , +(c) +≥ (1 − ν)2 � +1 − κ1 exp +� +−κ2ν2hmin +ε +�� +E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] . +(26) +(a) follows due to (23), and (b) follows since, almost surely, R∗ +ε ≥ hmin +ε +. (c) follows from Proposition C.2. +Due to Assumption 2, both hmin +ε +and hmax +ε +are Θ(1/poly(ε)). And, ν can be made as small as desired. Therefore, from +(25) and (26), +E +� +T π∗ +ε +� +→ E [T ∗ +ε ] +as +ε → 0. +Proposition C.3. Under Assumptions 1 and 2, for ν > 0, if ε is small enough such that hmin +ε +> 2(1 + ν)/ν, then, for any +state-dependent stationary policy π, the minimum number of rounds rπ +ε to reach an error tolerance ε is such that, +rπ +ε ≤ (1 + ν) 1 +rπ +ε +rπ +ε +� +n=1 +∥hε (qn)∥ . +Proof. Due to Assumption 1, rπ +ε satisfies, +rπ +ε > 1 +rπ +ε +rπ +ε +� +n=1 +∥hε (qn)∥ . +Moreover, since rπ +ε is the earliest round at which error tolerance ε is reached, due to Assumption 1, round rπ +ε − 1 satisfies, +rπ +ε − 1 ≤ +1 +rπ +ε − 1 +rπ +ε −1 +� +n=1 +∥hε (qn)∥ , +≤ +1 +rπ +ε − 1 +rπ +ε +� +n=1 +∥hε (qn)∥ , += +� +rπ +ε +rπ +ε − 1 +� 1 +rπ +ε +rπ +ε +� +n=1 +∥hε (qn)∥ , +=⇒ rπ +ε ≤ +� +rπ +ε +rπ +ε − 1 +�2 1 +rπ +ε +rπ +ε +� +n=1 +∥hε (qn)∥ . +(27) +18 + +Now, we upper bound rπ +ε /(rπ +ε − 1), +rπ +ε +rπ +ε − 1 = 1 + +1 +rπ +ε − 1, +(a) +< 1 + +ν +2 + ν , += +1 + ν +1 + ν/2, +(b) +≤ +√ +1 + ν. +(28) +(a) follows by rearranging the assumption rπ +ε > 2(1+ν)/ν to obtain rπ +ε −1 > (2+ν)/ν. (b) follows since 1+ν/2 ≥ √1 + ν +for ν > 0. Substituting (28) in (27) completes the result. +In order to prove Proposition C.2, we use Theorem 3.1 from [29] which we re-state below for clarity. +Theorem 3. Let Assumption 4 hold. Define rmix as the 1/8 mixing time2 of the Markov chain (Cn)n and f : C → [0, 1] be a +function. Let, µf ≜ E +� +f(C(1)) +� +. Then, there exists a constant κc such that for every 0 ≤ ν ≤ 1 and r ∈ N, +P +������ +1 +r +r +� +n=1 +f(Cn) − µf +����� ≥ νµf +� +≤ κc exp +� +−ν2µfr +72rmix +� +. +Proof of Proposition C.2. For some c ∈ C, define f(c′) = 1(c′ = c). Then, due to Theorem 3, there exist a constant κc such +that, +P +� +� +������ +1 +r′ +r′ +� +n=1 +1(Cn = c) − µ(c) +������ +≥ νµ(c) +� +� ≤ κc exp +� +−ν2µ(c)r′ +72rmix +� +. +(29) +Denote, κ ≜ � +c∈C κc and µmin = minc∈C µ(c). Since the Markov chain is irreducible, µmin > 0. Then, using (29) and taking +a union bound over all c ∈ C, we obtain, +P +� +Cr′ ̸∈ T r′ +ν (µ) +� +≤ κ exp +� +−ν2µminr′ +72rmix +� +. +Define κ2 ≜ µmin/(72rmix). Taking a further union bound over all r′ ≥ r, +P +� +∃r′ ≥ r such that Cr′ ̸∈ T r′ +ν (µ) +� +≤ +κ +1 − exp (−κ2ν2) exp +� +−κ2ν2r +� +Defining κ1 ≜ κ/(1 − exp(−κ2ν2)) completes the proof. +2Denoting M as the transition-matrix of the Markov chain, and µ as its stationary distribution, rmix ≜ maxψ +� +r : ∥Mrψ − µ∥T V ≤ 1/8 +� +, where ∥·∥T V +denotes the TV-norm. Due to Theorem 4.9 of [30], rmix is finite for an aperiodic and irreducible Markov chain. +19 + +APPENDIX D +PROOF OF LEMMA 2 +Here we prove Lemma 2 which states that the expected wall clock is approximately equal to the product of the expected +number of rounds and the expected round duration asymptotically. +The proof is very similar to the proof of Lemma 1 in Appendix C. As such, we will use the notation introduced in Appendix +C. +Denote hmin +ε +and hmax +ε +as the minimum and maximum of the bounded function hε(·). And, let dmin and dmax denote the +minimum and maximum of the positive, bounded function d(·, ·, ·). +Similar to the proof of Lemma 1, we consider network state sequences which are typical. In order to define the parameters +for the typical set, for any given δ > 0, we choose a small enough εth > 0 and δ′ > 0 such that, +1) εth is small enough such that hmin +εth > 2(1 + δ′)/δ′. +2) δ′ > 0 is such that for all 0 < ε ≤ εth, +(1 − δ) < (1 − δ′)2 � +1 − κ1 exp +� +−κ2(δ′2hmin +ε +) +�� +, +and, +max +� +� +�(1 + δ′)3, 1 + +κ1 exp +� +−κ2δ′2hmin +ε +� +hmax +ε +dmax +hmin +ε +dmin +� +� +� < (1 + δ/2). +Such a choice of δ′ is possible because hmin +ε += Θ(1/poly(ε)), and exp(−κ2δ′2hmin +ε +)hmax +ε +/hmin +ε += exp(−Ω(δ′2/poly(ε))). +We consider a sequence (cn)n such that cr ∈ T r +δ′(µ) for every r ≥ hmin +εth . +For a policy π, let rπ +ε denote the minimum number of rounds needed to converge to error tolerance ε < εth given network +state sequence (cn)n. Then, the wall clock time for π can be lower bounded as, +rπ +ε +� +n=1 +d (τ, π (cn) , cn) = +� +� 1 +rπ +ε +rπ +ε +� +n=1 +d (τ, π (cn) , cn) +� +� rπ +ε , +(a) +≥ (1 − δ′) E [d(τ, π(C), C)] rπ +ε , +(b) +≥ (1 − δ′) E [d(τ, π(C), C)] +� +� 1 +rπ +ε +rπ +ε +� +n=1 +∥hε (π (cn))∥ +� +� , +(c) +≥ (1 − δ′)2 E [d(τ, π(C), C)] E [∥hε (π(C))∥] , +(30) +(a) and (c) follow from Lemma C.1 since rπ +ε > hmin +εth , and (b) follows from Assumption 1. +Performing a similar calculation for the upper bound, +rπ +ε +� +n=1 +d (τ, π (cn) , cn) = +� +� 1 +rπ +ε +rπ +ε +� +n=1 +d (τ, π (cn) , cn) +� +� rπ +ε , +(a) +≤ (1 + δ′) E [d(τ, π(C), C)] rπ +ε , +(b) +≤ (1 + δ′)2 E [d(τ, π(C), C)] +� +� 1 +rπ +ε +rπ +ε +� +n=1 +∥hε (π (cn))∥ +� +� , +(c) +≤ (1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] . +(31) +(a) and (c) follow from Lemma C.1, and (b) follows from Proposition C.3. +20 + +Let Rπ +ε be the random variable denoting the minimum number number of rounds needed to converge to an error tolerance +ε < εth when policy π is used on the sequence (Cn)n. The expected wall clock times of π can be lower bounded as, +E [T π +ε ] ≥ P +� +CRπ +ε ∈ T Rπ +ε +δ′ +(µ) +� +(1 − δ′)2 E [d(τ, π(C)), C)] E [∥hε (π(C))∥] , +(a) +≥ P +� +∀r′ ≥ hmin +ε +, Cr′ ∈ T r′ +ν (µ) +� +(1 − δ′)2 E [d(τ, π(C), C)] E [∥hε (π(C))∥] +(b) +≥ (1 − δ′)2 � +1 − κ1 exp +� +−κ2δ′2hmin +ε +�� +E [d(τ, π(C), C)] E [∥hε (π(C))∥] +(c) +≥ (1 − δ) E [d(τ, π(C), C)] E [∥hε (π(C))∥] . +(32) +(a) follows since Rπ +ε ≥ hmin +ε +almost surely, (b) follows from Proposition C.2 and (c) follows from the choice of δ′. +The expected wall clock times of π can be upper bounded as, +E [T π +ε ] +(a) +≤ P +� +CRπ +ε ∈ T Rπ +ε +δ′ +(µ) +� +(1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] + P +� +CRπ +ε ̸∈ T Rπ +ε +δ′ +(µ) +� +hmax +ε +dmax, +(b) +≤ (1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] + P +� +∃r′ ≥ hmin +ε +Cr′ ̸∈ T r′ +δ′ (µ) +� +hmax +ε +dmax, +(c) +≤ (1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] + κ1 exp +� +−κ2δ′2hmin +ε +� +hmax +ε +dmax, +(d) +≤ (1 + δ) E [d(τ, π(C), C)] E [∥hε (π(C))∥] . +(33) +(a) follows by using (31) to upper bound the wall clock time for typical sequences, and using the worst-case upper bound +hmax +ε +dmax for non-typical sequences. (b) follows because, almost surely, Rπ +ε ≥ hmin +ε +. (c) follows from Proposition C.2. (d) +follows from the choice of δ′. +(32) and (33), jointly, conclude the proof. +21 + +APPENDIX E +PROOF OF PROPOSITION B.2 +In this section, we prove Proposition B.2 which states that the Fluid-Frank-Wolfe (FFW) process has a unique stationary point +in the set conv(Vε). It further states that the stationary point lies in the set Vε. In order to demonstrate the main arguments of +the proof, we will first consider the case with a single client (m = 1) and a single network state, C = {c}. Later, in Subsection +B, we will generalize these arguments to complete the proof of Proposition B.2. +A. Warmup: 1 Client and 1 Network State (1C1NS) Case +Recall that the set Vε is the set of pairs of achievable expected number of rounds ˆrε and expected round duration ˆd of +state-dependent stationary policy. Here, we observe that conv(Vε) may be interpreted as the corresponding feasibility set for +(possibly random) stationary policies. We start by making this observation precise. In the 1C1NS case, a stationary policy may +be represented by a one-dimensional random variable Π that denotes the possibly randomly selected compression parameter. +The space of possible stationary policies is denoted by the set of distributions Q1, +Q1 = {fΠ : fΠ is a distribution over [0, qmax]}. +Under a policy corresponding to Π and a small enough error tolerance ε, due to Lemma 2, the expression for the expected +wall clock time for stationary policies is given by, +E[T Π +ε ] ≈ ˆtΠ +ε = E [hε(Π)] E [d(τ, Π, c)] , +where T Π +ε is the wall clock time to reach error tolerance ε under compression parameter Π. Then, the feasible set conv(Vε) +for stationary policies may be written for this case as, +conv(Vε) = +� +(ˆrε, ˆd) : ∃fΠ ∈ Q1 s.t., Π ∼ fΠ satisfies ˆrε = E [hε(Π)] , +ˆd = E [d(τ, Π, c)] +� +. +Here, stationary policies may be separated into two categories, +• deterministic policies: here, Π is a constant. +• stochastic policies: here, Π is non-deterministic random variable. +Our aim is to prove that there exists a unique fixed-point of the FFW update in conv(Vε). As we will see, this will prove +Proposition B.2 for the special case of 1 client and 1 network state. Before delving into the proof of this statement, we state +a couple of properties of conv(Vε) which are useful in proving the existence of a unique fixed-point of the FFW update. +Proposition E.4. For any hmin +ε +≤ h ≤ hmax +ε +, there exists a deterministic policy that minimizes ˆtΠ +ε given a constraint E[hε(Π)] = +h. +Proof. As a contradiction, assume that there is no deterministic policy that minimizes the expected wall clock time given +a constraint E [hε(Π)] = h. Let Π∗ be a stochastic policy that minimizes the expected wall clock time with the constraint +E[hε(Π)] = h. Then, consider an alternate policy with deterministic compression parameter π chosen as, hε(π) = E [hε(Π∗)]. +Such a π exists due to the Intermediate Value Theorem since hε(·) is a continuous function. In this case, by the strict convexity +of the duration function assumed in Assumption 3, we have, +E [d(τ, π, c)] < E [d (τ, Π∗, c)] . +Therefore, the relation between their expected wall clock times is, +ˆtπ +ε = E [hε(π)] E [d(τ, π, c)] < E [hε(Π∗)] E [d (τ, Π∗, c)] = ˆtΠ∗ +ε . +This is a contradiction to the assumption that a stochastic policy minimizes the expected wall clock time given the constraint. +For notational brevity, denote, +¯d(ˆrε) = d(τ, h−1 +ε (ˆrε), c). +Recall that we may denote points (ˆrε, ˆd) ∈ conv(Vε) by a two-dimensional vector x = (ˆrε ˆd)⊤. Also, recall the function +H(x) = x1x2. The following proposition states several equivalent ways of describing a point x in the set Vε. +Proposition E.5. The following statements are equivalent +I. x ∈ conv(Vε) is of the form (ˆrε, ¯d(ˆrε)) for some hmin +ε +≤ ˆrε ≤ hmax +ε +. +II. x ∈ conv(Vε) is such that α x ̸∈ conv(Vε) for any 0 < α < 1. +III. x = (ˆrε, ˆd) ∈ conv(Vε) is such that, ˆd = min{d′ : (ˆrε, d′) ∈ conv(Vε)}. +IV. x = (ˆrε, ˆd) ∈ conv(Vε) is such that, ˆrε ˆd = min{ˆrεd′ : (ˆrε, d′) ∈ conv(Vε)}. +22 + +Fig. 4: Feasibility set for stationary policies conv(Vε). In the 1C1NS case, a point (ˆrε, ˆd) ∈ conv(Vε) corresponds to a policy +Π such that, ˆrε = E[hε(Π)] and ˆd = E[d(τ, Π, c)]. The function ¯d(ˆrε) is represented by the blue curve. +V. x is an extreme point3 of conv(Vε). +Proof Sketch. The equivalences may be inferred from the structure of the feasibility set conv(Vε) as shown in Fig. 4. We +briefly describe the arguments required to prove the equivalences. I ⇐⇒ III because (ˆrε, ¯dε) corresponds to a deterministic +policy by definition, and, due to part b, a deterministic policy minimizes the wall clock time ˆrε ˆd amongst the set of policies +{fΠ ∈ Q1 : s.t., Π ∼ fΠ satisfies E[hε(Π)] = ˆrε}. I, III ⇐⇒ II because ¯d(ˆrε) is a strictly decreasing function. I ⇐⇒ V +because ¯d is a strictly convex function. III and IV are trivially equivalent. +From here on, we will call points of the form described in Proposition E.5 as extreme points, and all other points in conv(Vε) +as non-extreme points. Note that in the 1C1NS case, the set of extreme points is equal to the set Vε. However, we refrain from +using this fact here because in the general case of multiple clients and multiple network states, this is no longer true. +Next, we prove the existence of a unique fixed point of the FFW update in two steps. First, we show that a non-extreme +point cannot be a fixed point of the FFW update. +Proposition E.6. If a point x ∈ conv(Vε) is not an extreme point of conv(Vε), then it is not a fixed-point of the FFW update. +Proof. Due Proposition E.5, x being a non-extreme point implies that there exists a constant 0 < α < 1 such that α x ∈ +conv(Vε). +Observe that ∇H(x) = (x2 x1)⊤. Since, hε() and d() are non-negative functions, we have that ∇H(x) has non-negative +entries for any x ∈ Vε. +Recall that x is a fixed-point of the FFW update if and only if x = +arg min +x′ ∈conv (Vε) +∇H(x)⊤ x′. Due to the elementwise +non-negativity of ∇H, ∇H(x)⊤(α x) < ∇H(x)⊤ x. Since α x ∈ conv(Vε), x is not a fixed point of the FFW-update. +Due to Proposition E.6, we focus on only extreme points in the next step. Before proving the existence of a unique fixed +point of the FFW update amongst the set of extreme points, we state a result about an equivalent description of a fixed point. +For this purpose, define ˆt(ˆrε) = ˆrε ¯d(ˆrε). +Proposition E.7. Under Assumption 5, an extreme-point x = +� +ˆrε, ¯d(ˆrε) +�⊤ with hmin +ε +< ˆrε < ˆhmax +ε +is a fixed point for the +FFW-update if and only if ˆt′(ˆrε) = 0. +Proof. First, as a contradiction, assume that there exists an ˆrε with ˆt′(ˆrε) = 0, but x = +� +ˆrε ¯d(ˆrε) +�⊤ is not a fixed point of +the FFW update. Recall that x is a fixed point of the FFW update if x = +arg min +x′ ∈conv (Vε) +∇H(x)⊤ x′. Therefore, this implies +that there exists a different point z = (ˆr′ ¯d(ˆr′))⊤ in conv(Vε) such that, +∇H(x)⊤(z − x) ≤ 0. +Refer to Fig. 5 for an illustration of the following argument. Due to strict convexity of the curve ¯d(·), there exists a point +w = (ˆr ¯d(ˆr′))⊤ with ˆr being in between ˆrε and ˆr′ such that ∇H(x)⊤(w − x) = −ξ < 0. Then, for any 0 < θ < 1, denote +3A point x ∈ conv(Vε) is called an extreme point if it cannot be written as the convex combination of two other points in conv(Vε). +23 + + 0. +Conversely, consider for contradiction that x = +� +ˆrε, ¯d(ˆrε) +�⊤ is a fixed point of the FFW update, but ˆt′(ˆrε) ̸= 0. Define +ˆrδ = ˆrε + δ, and zδ = (ˆrδ, ¯d(ˆrδ))⊤. Then, for small enough δ > 0, since ˆt′(ˆrε) ̸= 0, we have that +ˆt(ˆrδ) < ˆt(ˆrε) − Ω(δ), +(OR) +ˆt(ˆr−δ) < ˆt(ˆrε) − Ω(δ). +(35) +Again, by Taylor series expansion, +H(zδ) = H(x) + ∇H(x)(zδ − x) + O(δ2), +⇐⇒ ˆt(ˆrδ) = ˆt(ˆrε) + ∇H(x)(zδ − x) + O(δ2). +(36) +Comparing (35) and (36), we have that ∇H(x)(zδ − x) < 0 or ∇H(x)(z−δ − x) < 0. This is a contradiction to the premise +that x is a fixed point for the FFW update. +Remark 2. x = +� +hmin +ε +, ¯d(hmin +ε +) +� +is a fixed point of the FFW update if t′(hmin +ε +) ≥ 0. Similarly, x = +� +hmax +ε +, ¯d(hmax +ε +) +� +is a fixed +point of the FFW update if t′(hmax +ε +) ≤ 0. The proof of these statements is very similar to that of Proposition E.5 We don’t +prove these statements for the 1C1NS case for succinctness. But, we prove it rigorously in the multiple client and multiple +network state case. +Proposition E.8. For the 1C1NS case, there exists a unique fixed point x of the FFW update in conv(Vε). +Proof. Due to Proposition E.6, a non-extreme point of conv(Vε) cannot be a fixed point of the FFW update. +Proposition E.7 and Remark 2 jointly characterize the fixed point of the FFW update amongst the set of extreme points in +terms of ˆt(·). In particular, they imply that a point x = (ˆrε, ¯d(ˆrε)) is a fixed-point if and only if ˆrε is a local-minimum of ˆt(·) +in the domain hmin +ε +≤ ˆrε ≤ hmax +ε +(since ˆt(·) is quasiconvex, it does not have any local maximum in the interior of its domain). +Since ˆt is a strictly quasiconvex function (Assumption 5) on a bounded domain, it has a unique local minimum. Therefore, +the FFW update has a unique fixed point among the set of extreme points of conv(Vε). +B. Multiple Clients and Multiple Network States (MCMNS) Case +This section contains the complete proof of Proposition B.2. The argument is a generalization of what we saw in the previous +subsection to the case with multiple clients and multiple network states. We start by introducing notation to describe policies +in this general setting. +24 + +We may denote the policy π as a function, π : C → [0, qmax]m, or as a vector π of dimension m|C|. In specific, enumerating +the elements of C as, C = {c1, c2, . . . , c|C|}, the vector π is represented as, +π = +� +π1(c1), . . . , πm(c1), π1(c2), . . . , πm(c2), · · · · · · · · · , π1(c|C|), . . . , πm(c|C|) +�⊤ , +where πi(c) indicates the ith entry of m-dimensional vector π(c). +Next, we explain how to define the quantities, E [∥hε(π(C))∥] and E [d (τ, π(C), C)] in terms of the vector representation +of π. Denote ei as an m|C| dimensional vector with, +ei +j = +� +1 +, if (i − 1)|C| ≤ j < i|C|, +0 +, otherwise. +Recall that µ denotes the stationary distribution of the Markov chain of network states. Then, for a deterministic policy π, we +may write, +˜rε(π) ≜ E [∥hε(π(C))∥] = +|C| +� +i=1 +µ(ci) +��hε(π) ⊙ ei�� , +(37) +˜d(π) ≜ E [d(τ, π(C), C)] = +|C| +� +i=1 +µ(ci)d +� +τ, πi|C|−1 +(i−1)|C|, ci +� +, +(38) +where ⊙ denotes the elementwise product of two vectors, and πk +j denotes the vector +� +πj, πj+1, · · · , πk +�⊤. +Similar to the 1C1NS case, conv(Vε) may be interpreted as the set of achievable pairs of expected rounds and expected +round duration of (possibly, stochastic) stationary policies. Therefore, a stationary policy may be represented by a random +vector, Π. Then, we will denote the set of all (deterministic and stochastic) stationary policies as, +Qm|C| = +� +fΠ : fΠ is a m|C| dimensional distribution over [0, qmax]m|C|� +. +The feasible set conv(Vε) may be defined as, +conv(Vε) = +� +(ˆrε, ˆd) : ∃fΠ ∈ Qm|C| s.t., Π ∼ fΠ satisfies ˆrε = E [˜rε(Π)] , +ˆd = E +� +˜d(Π) +�� +. +Next, we prove an analogous result of Proposition E.4. +Proposition E.9. For any hmin +ε +≤ h ≤ hmax +ε +, there exists a deterministic policy π that minimizes E +� +˜d(Π) +� +over the constrained +domain +� +Π ∈ Qm|C| : E [˜rε(Π)] = h +� +. +Proof. The proof is similar to that of Proposition E.4. +As a contradiction, assume that there is no deterministic policy that minimizes the expected wall clock time given a constraint +E [˜rε(Π)] = h. Let Π∗ be a stochastic policy that minimizes the expected wall clock time with the constraint E[˜rε(Π)] = h. +Then, consider an alternate policy with deterministic compression parameters π chosen as, hε(π) = E [hε(Π∗)]. Such a π +exists due to the Intermediate Value Theorem since hε(·) is a continuous function. In this case, by the strict convexity of the +duration function assumed in Assumption 3, we have, +E +� +˜d(π) +� +< E +� +˜d (Π∗) +� +. +By the convexity of the norm operator, ˜rε(π) ≤ h. Notice that ˜rε(·) is increasing in every co-ordinate, whereas ˜d(·) is +decreasing in every co-ordinate. Therefore, for any π′ ≥ π (elementwise inequality) with ˜rε(π′) = h, we have, +E +� +˜d(π′) +� +≤ E +� +˜d(π) +� +< E +� +˜d (Π∗) +� +. +This is a contradiction to the assumption that a stochastic policy minimizes the expected wall clock time given the constraint. +At this point in the proof of the 1C1NS case, we defined a function ¯d(ˆrε) which was the round duration of the deterministic +policy whose number of rounds for convergence was ˆrε. In the MCMNS case, since there could be multiple deterministic +policies corresponding to a rounds for convergence ˆrε, we define ¯d(ˆrε) with respect to the policy that minimizes the round +duration. +¯d(rε) = +min +π:˜rε(π)=rε +˜d(π). +(39) +25 + +In the rest of the proof, we need to use the fact that ¯d(ˆrε) is strictly convex, and that ˆt(ˆrε) ≜ ˆrε ¯d(ˆrε) is strictly quasiconvex. +In the 1C1NS case, these facts were a direct consequence of Assumptions 3 and 5 because ¯d(ˆrε) was simply d(τ, h−1 +ε (ˆrε), c). +For the MCMNS case we prove these results in the following two propositions. +Proposition E.10. Under Assumptions 1 and 3, ¯d(rε) is decreasing and strictly convex in rε. +Proof. From Assumption 1, recall that since hε(·) is a strictly increasing, continuous function, it has an inverse h−1 +ε (·). Denote +by, h−1 +ε +: +� +hmin +ε +, hmax +ε +�m|C| → [0, qmax]m|C|, the function that outputs a vector obtained by applying h−1 +ε (·) elementwise to +the input vector. +¯d(rε) from (39) may be redefined as, +¯d(rε) = +min +r:r∈[hmin +ε +,hmax +ε +] +m|C| +˜rε(h−1 +ε +(r))=rε +˜d +� +h−1 +ε (r) +� +. +Due to Assumption 1, ˜rε +� +h−1 +ε (r) +� +is an increasing function in every element of r, and, due to Assumption 3, ˜d +� +h−1 +ε (r) +� +is +a decreasing function in every element of r. Therefore, we can further reformulate ¯d(rε) as, +¯d(rε) = +min +r:r∈[hmin +ε +,hmax +ε +] +m|C| +˜rε(h−1 +ε +(r))≤rε +˜d +� +h−1 +ε (r) +� +. +(40) +¯d(rε) is decreasing because the feasibility set in the minimization problem in (40) is an monotonically increasing set with +increasing ˆrε. +Consider two points, rε,1, rε,2 ∈ +� +hmin +ε +, hmax +ε +� +, and let rε,1 and rε,2 be their corresponding minimizers according to (40), +rε,1 ≜ +arg min +r : r ∈[hmin +ε +,hmax +ε +] +m|C| +˜rε(h−1 +ε +( r ))≤rε,1 +˜d +� +h−1 +ε (r) +� +, +rε,2 ≜ +arg min +r : r ∈[hmin +ε +,hmax +ε +] +m|C| +˜rε(h−1 +ε +( r ))≤rε,2 +˜d +� +h−1 +ε (r) +� +. +(41) +Let 0 < θ < 1, rε,θ = θrε,1 +(1−θ)rε,2 and rε,θ = θ rε,1 +(1−θ) rε,2. By the strict convexity of ˜d +� +h−1 +ε (·) +� +as considered +in Assumption 3, +˜d +� +h−1 +ε (rε,θ) +� +< θ ˜d +� +h−1 +ε (rε,1) +� ++ (1 − θ) ˜d +� +h−1 +ε (rε,2) +� +. +(42) +From the definition of ˜rε(·) in (37), +˜r +� +h−1 +ε +(rε,θ) +� += +|C| +� +i=1 +µ(ci) +���rε,θ ⊙e(i)��� , +(a) +≤ θ +|C| +� +i=1 +µ(ci) +���rε,1 ⊙e(i)��� + (1 − θ) +|C| +� +i=1 +µ(ci) +���rε,2 ⊙e(i)��� , +(b) += θ˜rε +� +h−1 +ε +(rε,1) +� ++ (1 − θ)˜rε +� +h−1 +ε +(rε,2) +� +, +(c) +≤ θrε,1 + (1 − θ)rε,2, +(d) += rε,θ. +(43) +(a) follows from the convexity of the norm operator, and (b), (c) and (d) follow by definition. +Due to (43), rε,θ is a feasible point in the constraint set of the minimization problem in (40) evaluated at rε,θ. Therefore, +¯d(rε,θ) ≤ ˜d +� +h−1 +ε (rθ) +� +, +(a) +< θ ˜d +� +h−1 +ε (r1) +� ++ (1 − θ) ˜d +� +h−1 +ε (r2) +� +, +(b) += θ ¯d(rε,1) + (1 − θ) ¯d(rε,2). +(a) follows from (42), and (b) follows by definition. Since this is true for any rε,1, rε,2 ∈ +� +hmin +ε +, hmax +ε +� +, and 0 < θ < 1, ¯d(·) +is strictly convex. +26 + +However, unlike the 1 client and 1 network state case, ¯d(rε) may not be differentiable. But, since it is convex, it has +left-derivative and right-derivative functions denoted as ¯d′ +L(rε) and ¯d′ +R(rε) respectively. +¯d′ +L(rε) = +� +limδ↓0 +¯d(rε)− ¯d(rε−δ) +δ +, +rε > hmin(ε) +−∞, +rε = hmin(ε), +¯d′ +R(rε) = +� +limδ↓0 +¯d(rε+δ)− ¯d(rε) +δ +, +rε < hmax(ε) +0, +rε = hmax(ε). +Similar to the case of 1 client and 1 network state, given a constraint, ˜rε(π) = rε, the optimal expected wall clock time +may be expressed as ˆt(rε) = rε ¯d(rε). Since ¯d(rε) has left and right derivatives everywhere, so does ˆt(rε). Denote them by +ˆt′ +L(rε) and ˆt′ +R(rε) respectively. +Proposition E.11. ˆt(rε) is strictly quasiconvex. That is, for any rε,1, rε,2 ∈ +� +hmin +ε +, hmax +ε +� +, and 0 < θ < 1, +ˆt(θrε,1 + (1 − θ)rε,2) < max{ˆt(rε,1), ˆt(rε,2)}. +Proof. Here, we reuse the definitions introduced in the proof of Proposition E.10. +Consider two points rε,1, rε,2 ∈ +� +hmin +ε +, hmax +ε +� +. Let rε,1 and rε,2 be defined as in (41). Consider, 0 < θ < 1. Since ˜rε +� +h−1 +ε (r) +� +is continuous in r, by the Intermediate Value Theorem, there exists a 0 < δ < 1 such that rδ = δ rε,1 +(1 − δ) rε,2 has +˜rε(rδ) = θrε,1 + (1 − θ)rε,2. +From the strict quasiconvexity of the wall clock time as in Assumption 5, we have, +˜rε +� +h−1 +ε (rδ) +� ˜d +� +h−1 +ε (rδ) +� +< max +�ˆt(rε,1), ˆt(rε,2) +� +. +By definition, +ˆt(θrε,1 + (1 − θ)rε,2) ≤ ˜rε +� +h−1 +ε (rδ) +� ˆd +� +h−1 +ε (rδ) +� +. +Therefore, +ˆt(θrε,1 + (1 − θ)rε,2) < max +�ˆt(rε,1), ˆt(rε,2) +� +. +At this point in the proof of the 1C1NS case, we stated some equivalent descriptions of a point x ∈ Vε. The description is +the same for the MCMNS case as well, which we restate here for clarity. +Proposition E.12. The following statements are equivalent +I. x ∈ conv(Vε) is of the form (ˆrε, ¯d(ˆrε)) for some hmin +ε +≤ ˆrε ≤ hmax +ε +. +II. x ∈ conv(Vε) is such that α x ̸∈ conv(Vε) for any 0 < α < 1. +III. x = (ˆrε, ˆd) ∈ conv(Vε) is such that, ˆd = min{d′ : (ˆrε, d′) ∈ conv(Vε)}. +IV. x = (ˆrε, ˆd) ∈ conv(Vε) is such that, ˆrε ˆd = min{ˆrεd′ : (ˆrε, d′) ∈ conv(Vε)}. +V. x is an extreme point of conv(Vε). +Proof Sketch. The only difference in the proof from that of Proposition E.5 is the equivalence I ⇐⇒ III. Here, I ⇐⇒ III +follows from the definition of ¯d(·) in (39). +At this point in the 1C1NS case, we showed that a non-extreme point of conv(Vε) cannot be a fixed point of the FFW +update. The result is the same in the MCMNS case. We restate the result for clarity, but skip the proof as it is the same as +that of Proposition E.6. +Proposition E.13. If a point x ∈ conv(Vε) is not an extreme point of conv(Vε), then it is not a fixed-point of the FFW update. +Next, similar to the 1C1NS case, we show necessary and sufficient condition for an extreme-point of conv(Vε) to be a +fixed-point of the FFW update. +Proposition E.14. A point x = +� +rε, ¯d(rε) +�⊤ is a fixed point for the FFW update if and only if ˆt′ +L(rε) ≤ 0 and ˆt′ +R(rε) ≥ 0. +Proof. The proof is very similar to the proof of Proposition E.7, but with some extra care because, here, ˆt(·) may not be +differentiable. +27 + +Consider a point x = +� +rε, ¯d(rε) +�⊤ which is not a fixed point of the FFW update. This implies that there exists another +point, z = (r, ¯d(r))⊤, such that, +∇H(x)⊤(z − x) ≤ 0. +Refer to Fig. 5 for an illustration of the following argument. Due to strict convexity of the curve ¯d(·)(Proposition E.10), +there exists a point w = (ˆr′ ¯d(ˆr′))⊤ with ˆr being in between ˆrε and ˆr′ such that ∇H(x)⊤(w − x) = −ξ < 0. Then, for +any 0 < θ < 1, consider wc +θ = (1 − θ) x +θ w and wθ = +� +(1 − θ)ˆrε + θˆr, ¯d((1 − θ)ˆrε + θˆr) +� +. It is easily verified that +∇H(x)⊤(wc +θ − x) = −ξθ. And, due the the convexity of ¯d(·), ∇H(x)⊤(wθ − x) ≤ ∇H(x)⊤(wc +θ − x). That is, +∇H(x)⊤(wθ − x) ≤ −ξθ, +(44) +for some positive constant ξ. +By a Taylor Series expansion of H, we have ˆt((1 − θ)ˆrε + θˆr) = ˆt(ˆrε) + ∇H(x)⊤(wθ − x) + O(θ2). Therefore, due to +(44), for small enough θ, we have ˆt((1 − θ)ˆrε + θˆr) < ˆt(ˆrε) − ξ′θ, where ξ′ is some positive constant. This, in turn, implies +that either ˆt′ +L(ˆrε) > 0 or ˆt′ +R(ˆrε) < 0. +Conversely, consider for contradiction that x = +� +rε, ¯d(rε) +�⊤ is a fixed point of the FFW update, but ˆt′ +L(rε) > 0 or +ˆt′ +R(rε) < 0. Define rδ = rε + δ, and zδ = (rδ, ¯d(rδ))⊤. Then, for small enough δ > 0, we have that +ˆt(rδ) < ˆt(rε) − Ω(δ), +(OR) +ˆt(r−δ) < ˆt(rε) − Ω(δ). +(45) +Again, by Taylor series expansion, +ˆt(rδ) = ˆt(rε) + ∇H(x)(zδ − x) + O(δ2). +(46) +Comparing (45) and (46), we have that ∇H(x)(zδ − x) < 0 or ∇H(x)(z−δ − x) < 0. This is a contradiction to the premise +that x is a fixed point for the FFW update. +Next, similar to the 1C1NS case, we use the equivalence derived in Proposition E.12 to prove that the fixed point of the +FFW update is unique. +Proposition E.15. There exists a unique point x = +� +r, ¯d(r) +�⊤ with hmin +ε +≤ r ≤ hmax +ε +such that ˆt′ +L(r) ≤ 0 and ˆt′ +R(r) ≥ 0. +Proof. Denote, +r(rε) = +arg min +r : r ∈[hmin +ε +,hmax +ε +] +m|C| +˜rε(h−1 +ε +( r ))=rε +˜d +� +h−1 +ε (r) +� +. +Also, denote, +˜tε(r) = ˜rε(h−1 +ε (r)) ˜d(h−1 +ε (r)) +We use the following claim which is proved at the end of this section. +Claim E.1. If ˆt′ +L(rε) ≤ 0 and ˆt′ +R(rε) ≥ 0, then, +∇r +�˜tε(r(rε)) +� += 0, +or, the left and right derivatives of r(·) evaluated at rε, r′ +L(rε) and r′ +R(rε), exist, and, +r′ +L(rε)⊤∇r +�˜tε(r(rε)) +� +≤ 0, +(AND) +r′ +R(rε)⊤∇r +�˜tε(r(rε)) +� +≥ 0. +In either case, Assumption 5 ensures that for all small enough δ > 0, we have ˆt(rε − δ) > ˆt(rε) and ˆt(rε + δ) > ˆt(rε). +That is, if ˆt′ +L(rε) ≤ 0, then for all small enough δ > 0, ˆt(rε − δ) > ˆt(rε) (whenever rε − δ is in [hmin +ε +, hmax +ε +]). Similarly, if +ˆt′ +R(rε) ≥ 0, then for small enough δ > 0, ˆt(rε + δ) > t(rε) (whenever rε + δ is in [hmin +ε +, hmax +ε +]). Therefore, rε is a strict local +minimum of of ˆt. +Since ˆt(·) is strictly quasiconvex (Proposition E.11), there can be at most one point rε which is a strict local minimum of +ˆt(·). +28 + +Moreover, since ˆt is a continuous function over a closed and bounded set, it attains a local minimum over its domain. And, +ˆt′ +L(rε) ≤ 0 and ˆt′ +R(rε) ≥ 0 is necessary condition for a local minimum. Therefore, there exists at least one point rε in the +domain such that ˆt′ +L(rε) ≤ 0 and ˆt′ +R(rε) ≥ 0. +Proof of Claim E.1. Recall the notation, ˜tε(r) = ˜rε(h−1 +ε (r)) ˜d(h−1 +ε (r)). +a) Case 1: : First, consider the case that r(rε) is in the interior of the domain [hmin +ε +, hmax +ε +]m|C|. In this case we will +show that, ∇˜tε(r(rε)) = 0. As a contradiction, assume that ∇˜tε(r(rε)) ̸= 0. +Let L(rε) ≜ {r : ˜r(r) = rε}. Since, r(rε) is the minimizer of ˜tε(r) over the set L(rε), ∇˜tε(r(rε)) has to be normal to +L(rε) at the point r(rε). Let n denote the normal to the set L at point rε. Then, one of the following is true, +˜tε(r(rε) + δn) = ˜tε(r(rε)) − Θ(δ), +(OR) +˜tε(r(rε) − δn) = ˜tε(r(rε)) − Θ(δ). +Then, by the definition of t(·), one of the following is true, +t(rε + δ) = t(rε) − Ω(δ), +(OR) +t(rε − δ) = t(rε) − Ω(δ). +This implies that either t′ +L(rε) > 0 or t′ +R(rε) > 0. This is a contradiction. Therefore, ∇˜tε(r(rε)) = 0, for all r(rε) in the +interior of [hmin +ε +, hmax +ε +]m|C|. +b) Case 2: : Consider the case that r(rε) is on the boundary of +� +hmin +ε +, hmax +ε +�m|C| and ∇˜tε(r(rε)) ̸= 0. +Here, the left derivative exists and its direction is expressed as, +r′ +L(rε) ∝ +arg max +s:r (rε)+s∈[hmin +ε +,hmax +ε +]m|C| +s⊤∇˜tε(r(rε)) +∥s∥2 +. +Since, t′ +L(rε) ≤ 0, we obtain r′ +L(rε)⊤∇˜tε(r(rε)) ≤ 0. +A similar argument holds for the right derivative as well. +Finally, we summarize how the results in this section prove Proposition B.2. +Proof Summary of Proposition B.2. Due to Proposition E.13, a non-extreme point of conv(Vε) is not a fixed-point of the FFW +update. Further, Proposition E.12 shows that an equivalent condition for an extreme point x = (ˆrε, ¯d(ˆrε)) of conv(Vε) being a +fixed point of the FFW update is that t′ +L(ˆrε) ≤ 0 and t′ +R(ˆrε) ≥ 0. Then, in Proposition E.15 we showed that there is a unique +point which satisfies t′ +L(ˆrε) ≤ 0 and t′ +R(ˆrε) ≥ 0. Therefore, this proves that there is a unique fixed point x∗ for the FFW +update in conv(Vε). +Further, since x∗ is an extreme point of conv(Vε), and conv(Vε) is the convex hull of the set Vε, x∗ lies in the set Vε. +Finally, to prove that x∗ is the minimizer of H(·) over the set Vε, we make the following observation which is a consequence +of IV in Proposition E.12, +min +x∈Vε H(x) = +min +r∈[hmin +ε +,hmax +ε +] +ˆt(r). +Recall from Proposition E.12 that x∗ = (r∗, ¯d(r∗)) is such that t′ +L(r∗) ≤ 0 and t′ +R(r∗) ≥ 0. Further, since ˆt is strictly +quasiconvex (Proposition E.11), r∗ is the global minimizer of ˆt. Therefore, x∗ is the minimizer of H(·) over the set Vε. +29 + +APPENDIX F +PROOF OF THEOREM 2 +In this section, we prove Theorem 2 which bounds the convergence of the FedCOM-V Algorithm shown in Algorithm 2. +Moreover, we explicitly state the choice of local learning rate schedule (ηn), and global learning rate schedule (γn) required +to achieve the convergence rate stated in Theorem 2. +In the rest of the section, we violate our convention by sometimes using lower case letters instead of capital letters to denote +random variables/vectors in order to stay consistent with the notation used in FL literature. +We start by defining sigma-algebras and filtrations associated with the probability space. +Remark 3. Let σ(X) denote the sigma-algebra generated by the random variable X. Let σ(X1, . . . , Xn) denote the sigma- +algebra generated by the set of random variables X1 to Xn. Similarly, let σ(F1, F2, . . . , Fn) denote the smallest sigma-algebra +containing the sigma-algebras F1 to Fn. +First, we describe the sigma-algebra associated with the network state process. Recalling that Cn denotes the network state +at round n, denote, +FC ≜ σ (Cn : n ≥ 1) . +We remark that although the compression parameters (qn)n may be random vectors dependent across various rounds n, their +randomness only depends on the network states under the NAC-FL policy as well as under other baseline policies we consider +in this paper. More precisely, qn is measurable in FC for all rounds n, and is therefore not dependent on the updates of the +FedCOM-V algorithm. Moreover, the aim in this proof is to study the convergence of the FedCOM-V algorithm for arbitrary +choices of (qn)n. So, in this section, all expectations will be conditioned on FC, and therefore, qn +j ’s will be treated as arbitrary, +but known, constants in this Section. +Next, we describe the sigma-algebras associated with the stochastic gradients. Recall that at round n, by local-step b, client +j has sampled mini-batches +� +Za,n +j +�b +a=1 to compute the stochastic-gradients. So, we denote the associated sigma-algebra across +all clients as, +Db,n ≜ σ +� +Za,n +j +: j ∈ [m], a ∈ [b] +� +, +b ∈ [τn], +The sigma-algebra associated with the compressors used in round n is denoted as, +Qn ≜ σ +� +Q(·, qn +j ) : j ∈ [m] +� +. +Finally, we describe the filtration for the entire system. Since, in this Section, we condition all events on the knowledge of +the network states, the filtration is initialized as, +F0 = FC . +At the bth local step of round n, the filtration includes the knowledge of all previous rounds and the stochastic-gradients up to +the bth local step, +Fb,n = σ (Fn−1, Db,n) , +, b ∈ [τn], n ≥ 1, +And, finally, at the end of round n, the filtration includes the knowledge of all previous rounds, the stochastic gradients and +compressors of round n, +Fn = σ (Fn−1, Dτn,n, Qn) , +n ≥ 1. +Next we recap the FedCOM-V Algorithm and introduce further notation used in the proof. +At the start of round n, client j recieves the global model wn from the server, which it initializes as w1,n +j +. Then, at local- +step b, it samples a mini-batch Zb,n +j +and computes the stochastic gradient, ˜gb,n +j +≜ ∇f(wb,n +j +, Zb,n +j +), while performing the local +model update as, wb+1,n +j += wb,n +j +−ηn˜gb,n +j +. At this point, we remark that there are two sources of randomness involved in the +evaluation of a stochastic gradient at a local step b of round n. One is from the model wb,n +j +at which the gradient is evaluated, +which is itself obtained by stochastic gradient and compressed aggregation updates of previous rounds and local steps (i.e., +wb,n +j +is measurable in Fb−1,n). The second source of randomness is from the mini-batch, Zb,n +j +, used to compute the stochastic +gradient ˜gb,n +j +. So, ˜gb,n +j +is measurable in Fb,n. +Additionally, for the analysis, we will define the “true-gradient” at the local model wb,n +j +as, gb,n +j +≜ ∇f(wb,n +j +). Observe that +the true gradient at local step b of round n is itself a random vector, as it is evaluated at wb,n +j +. But, it is independent of the +mini-batch, Zb,n +j +, sampled at that step. Therefore, gb,n +j +is measurable in Fb−1,n. Refer to Figure 6 for an illustration of this +process. +After τn local computations, the client computes its “pre-compressed” update, ˜gn +j ≜ �τn +b=1 ˜gb,n +j +, which can also be expressed +as, ˜gn +j = (wn − wτn+1,n +j +)/ηn. Next, the client sends the compressed message, ˜gn +Qj = Q(˜gn +j , qn +j ), to the server. +30 + +Fig. 6: Illustration of the local steps at a client. +The server aggregates the compressed messages received from the clients as, ˜gn +Q = 1/m �m +j=1 ˜gn +Qj, and performs the update +wn+1 = wn −ηnγn˜gn +Q. +For the purpose of analysis, define ˜gn ≜ 1/m �m +j=1 ˜gn +j , which may be interpreted as the message aggregated at the server +had the clients not used any compression. Further, define the “true-gradient” analogies of ˜gn +j and ˜gn as, gn +j = �τn +b=1 gb,n +j +and gn = 1/m �m +j=1 gn +j . gn and gn +j are random variables since their components, gb,n +j +’s, are evaluated at models which are +obtained by stochastic gradient updates. +Remark 4. The presence of a tilde and the subscript Q, such as in ˜gn +Q, will indicate that vector is both compressed and has +stochastic gradient components. The presence of just a tilde, such as in ˜gn +j , will indicate that vector (or its components) has +two sources of randomness: one from the model at which it (or its components) is evaluated, and second from the mini-batch +using which it (or its components) is evaluated. The absence of both the tilde and subscript Q, such as in gn +j , will indicate +that the vector (or its components) has one source of randomness, which is from the model at which it (or its components) is +evaluated. +Also, denote the average noise-variance across clients per round as ¯qn: +¯qn = 1 +m +m +� +j=1 +qn +j . +We start by stating some results which will assist in proving Theorem 2. First is a lemma that bounds the distance between +the sum of stochastic gradients at a client in a round to the sum of the true gradients across the local steps at the client in the +round. +Lemma F.2. E +���˜gn +j − gn +j +��2 ���Fn−1 +� +≤ τnσ2. +31 + +.n +Tn,n +9 +Tn,n +w2,n +1.n +2,n +1,n +gj +In,nProof. +E +���˜gn +j − gn +j +��2 ���Fn−1 +� (a) += E +� +� +����� +τn +� +b=1 +� +˜gb,n +j +− gb,n +j +������ +2 �����Fn−1 +� +� , +(b) += E +� +�E +� +� +����� +τn +� +b=1 +� +˜gb,n +j +− gb,n +j +������ +2 ���Fτn−1,n +� +� +�����Fn−1 +� +� , +(c) += E +� +E +� +� +����� +τn−1 +� +b=1 +� +˜gb,n +j +− gb,n +j +������ +2 ���Fτn−1,n +� +� + E +���˜gτn,n +j +− gτn,n +j +��2 ���Fτn−1,n +� ++ 2 E +� +�˜gτn,n +j +− gτn,n +j +�⊤ +τn−1 +� +b=1 +� +˜gb,n +j +− gb,n +j +� +|Fτn−1,n +� �����Fn−1 +� +, +(d) += E +� ����� +τn−1 +� +b=1 +� +˜gb,n +j +− gb,n +j +������ +2 ++ E +���˜gτn,n +j +− gτn,n +j +��2 ���Fτn−1,n +� ++ 2 E +��˜gτn,n +j +− gτn,n +j +� ���Fτn−1,n +�⊤ τn−1 +� +b=1 +� +˜gb,n +j +− gb,n +j +� �����Fn−1 +� +, +(e) += E +� +� +����� +τn−1 +� +b=1 +� +˜gb,n +j +− gb,n +j +������ +2 ++ E +���˜gτn,n +j +− gτn,n +j +��2 ���Fτn−1,n +� �����Fn−1 +� +� , +(f) +≤ E +� +� +����� +τn−1 +� +b=1 +� +˜gb,n +j +− gb,n +j +������ +2 �����Fn−1 +� +� + σ2, +... +≤ τnσ2 +(a) follows by definition. (b) follows by law of iterated expectations. (c) follows by partially expanding the summation. (d) +follows because all random vectors except ˜gτn,n +j +are measurable in Fτn−1,n. (e) follows because ˜gτn,n +j +is an unbiased estimate +of gτn,n +j +(Assumption 7). (f) follows by Assumption 7 which bounds the variance of the stochastic gradients. Repeating steps +(b)-(f) (τn − 1) more times by taking internal conditional expectations w.r.t., Fτn−2,n, ..., F1,n, Fn−1 gives us the result. +Next is a lemma comparing the expected norm of the (compressed and stochastic) gradient received by the server to the +true gradients at all the local steps. +Lemma F.3. Under Assumption 7 and 8, +E +���˜gn +Q +��2 ���Fn−1 +� +≤ 2τn +m +�qmax +m ++ 1 +� m +� +j=1 +τn +� +b=1 +E +����gb,n +j +��� +2 ���Fn−1 +� ++ (¯qn + 1) 2τnσ2 +m +. +32 + +Proof. +E +���˜gn +Q +��2 ���Fn−1 +� +(a) += E +� +��E +� +�� +������ +1 +m +m +� +j=1 +˜gn +Q,j +������ +2 ���Fτn,n +� +�� +���Fn−1 +� +�� , +(b) += E +� +��E +� +�� +������ +1 +m +m +� +j=1 +˜gn +Q,j − 1 +m +m +� +j=1 +E +� +˜gn +Q,j +���Fτn,n +� +������ +2 ���Fn−1 +� +�� + +������ +E +� +� 1 +m +m +� +j=1 +˜gn +Q,j +���Fτn,n +� +� +������ +2 ���Fn−1 +� +�� , +(c) += E +� +�� 1 +m2 +m +� +j=1 +� +E +���˜gn +Q,j − ˜gn +j +��2 ���Fτn,n +�� ++ +������ +1 +m +m +� +j=1 +˜gn +j +������ +2 ���Fn−1 +� +�� , +(d) +≤ E +� +� +m +� +j=1 +qn +j +m2 +��˜gn +j +��2 + ∥˜gn∥2 ���Fn−1 +� +� . +(47) +(a) follows by Law of Iterated Expectations. (b) follows by the identity, E[∥X∥2] = E[∥X − E[X]∥2] + ∥E[X]∥2 (this is the +E X2 = var(X) + (E X)2 identity applied to vectors). The first summation term of (c) is obtained by expanding out the first +squared norm from (b) and observing that ˜gn +Q,j − ˜gn +j is a zero mean random vector, and independent across different j’s. The +second term in (c) follows from the linearity of expectation and the unbiased property of the compressor (Assumption 8). (d) +follows from Assumption 8, which bounds the noise introduced by the compressor. +Let’s bound E +� +∥˜gn∥2 ���Fn−1 +� +, +E +� +∥˜gn∥2 ���Fn−1 +� (a) +≤ 2 E +� +∥˜gn − gn∥2 ���Fn−1 +� ++ 2 E +� +∥gn∥2 Fn−1 +� +, +(b) += 2 E +� +�� +������ +1 +m +m +� +j=1 +�˜gn +j − gn +j +� +������ +2 ���Fn−1 +� +�� + 2 E +� +�� +������ +1 +m +m +� +j=1 +gn +j +������ +2 ���Fn−1 +� +�� , +(c) += +2 +m2 +m +� +j=1 +E +���˜gn +j − gn +j +��2 ���Fn−1 +� ++ 2 E +� +�� +������ +1 +m +m +� +j=1 +gn +j +������ +2 +Fn−1 +� +�� , +(d) +≤ +2 +m2 +m +� +j=1 +E +���˜gn +j − gn +j +��2 ���Fn−1 +� ++ 2 +m +m +� +j=1 +E +���gn +j +��2 ���Fn−1 +� +, +(e) +≤ 2τnσ2 +m ++ 2 +m +m +� +j=1 +E +���gn +j +��2 ���Fn−1 +� +. +(48) +Above, (a) follows from the identity ∥x∥2 ≤ 2 ∥x − y∥2 +2 ∥y∥2 (to prove the identity, observe that ∥·∥2 is a convex function, +and use Jensen’s inequality, ∥x/2∥2 ≤ 1/2 ∥x − y∥2 + 1/2 ∥y∥2), and (b) follows by definition. (c) is true because, given +Fn−1, (˜gn +j − gn +j ) is independent of (˜gn +k − gn +k) for k ̸= j. (d) follows from Jensen’s Inequality, and (e) is true by Lemma F.2. +Performing a similar calculation again, +E +���˜gn +j +��2 ���Fn−1 +� +≤ 2 E +���˜gn +j − gn +j +��2 ���Fn−1 +� ++ 2 E +���gn +j +��2 ���Fn−1 +� +, +≤ 2τnσ2 + 2 E +���gn +j +��2 ���Fn−1 +� +. +(49) +Using Jensen’s inequality, we get, +��gn +j +��2 ≤ τn +τn +� +b=1 +���gb,n +j +��� +2 +. +(50) +Substituting (48), (49) and (50) in (47), we get the result, +E +���˜gn +Q +��2 ���Fn−1 +� +≤ 2τn +m +m +� +j=1 +τn +� +b=1 +�qn +j +m + 1 +� +E +����gb,n +j +��� +2 ���Fn−1 +� ++ +� +� +m +� +j=1 +qn +j +m + 1 +� +� 2τnσ2 +m +. +(51) +33 + +The following lemma bounds the inner product between the true gradient evaluated at the global model at the start of a +round and the approximate gradient received by the server. +Lemma F.4. Under Assumption 6, the FedCOM-V updates follow, +− E +�� +∇f(wn), ˜gn +Q +� ���Fn−1 +� +≤ +1 +2m +m +� +j=1 +τn +� +b=1 +� +− ∥∇f(wn)∥2 − E +����gb,n +j +��� +2 ���Fn−1 +� ++ L2 E +����wn − wb,n +j +��� +2 ���Fn−1 +� � +. +Proof. +− E +�� +∇f(wn), ˜gn +Q +� ���Fn−1 +� (a) += − E +�� +∇f(wn), E +� +˜gn +Q +���Fτn,n +�� ���Fn−1 +� +, +(b) += − E +� +⟨∇f(wn), ˜gn⟩ +���Fn−1 +� +, +(c) += − E +� +� +� +∇f(wn), 1 +m +m +� +j=1 +τn +� +b=1 +˜gb,n +j +� ���Fn−1 +� +� , +(d) += − E +� +� 1 +m +m +� +j=1 +τn +� +b=1 +� +∇f(wn), E +� +˜gb,n +j +���Fb−1,n +�� ���Fn−1 +� +� , +(e) += − E +� +� 1 +m +m +� +j=1 +τn +� +b=1 +� +∇f(wn), gb,n +j +� ���Fn−1 +� +� , +(f) += E +� +� 1 +2m +m +� +j=1 +τn +� +b=1 +� +− ∥∇f(wn)∥2 − +���gb,n +j +��� +2 ++ +���∇f(wn) − gb,n +j +��� +2� ���Fn−1 +� +� , +(g) +≤ E +� +� 1 +2m +m +� +j=1 +τn +� +b=1 +� +− ∥∇f(wn)∥2 − +���gb,n +j +��� +2 ++ L2 ���wn − wb,n +j +��� +2� ���Fn−1 +� +� , +(a) follows by Law of Iterated Expectations since ∇f(wn) is measurable in Fτn,n. (b) follows since ˜gn +Q is an unbiased +estimate of ˜gn by Assumption 8. (c) follows from the definition of ˜gn. (d) follows from the Law of Iterated Expectations. +(e) follows from the unbiased property of the stochastic gradients as stated in Assumption 7. (f) follows from the relation +2⟨x, y⟩ = ∥x∥2 + ∥y∥2 − ∥x − y∥2. (g) follows from L-smoothness of Assumption 6 since gb,n +j += ∇f(wb,n +j +). +The following Lemma bounds the distance between the global model at the start of a round to the local model at a local +step of a client. +Lemma F.5. Under Assumption 7, FedCOM-V updates follow, +E +����wn − wb,n +j +��� +2 ���Fn−1 +� +≤ 2η2τn +τn +� +b=1 +E +����gb,n +j +��� +2 ���Fn−1 +� ++ 2η2 +nτnσ2. +34 + +Proof. +E +����wn − wb,n +j +��� +2 ���Fn−1 +� +(a) += E +� +� +�����ηn +b−1 +� +a=1 +˜ga,n +j +����� +2 ���Fn−1 +� +� , +(b) +≤ 2η2 +n E +� +� +����� +b−1 +� +a=1 +�˜ga,n +j +− ga,n +j +� +����� +2 ���Fn−1 +� +� + 2η2 +n E +� +� +����� +b−1 +� +a=1 +ga,n +j +����� +2 ���Fn−1 +� +� , +(c) +≤ 2η2 +n(b − 1)σ2 + 2η2 +n E +� +� +����� +b−1 +� +a=1 +ga,n +j +����� +2 ���Fn−1 +� +� , +(d) +≤ 2η2 +n(b − 1)σ2 + 2η2 +n(b − 1) +b−1 +� +a=1 +E +���ga,n +j +��2 ���Fn−1 +� +, +≤ 2η2 +nτnσ2 + 2η2 +nτn +τn +� +a=1 +E +���ga,n +j +��2 ���Fn−1 +� +. +(a) follows from the local update rule of Algorithm 2. (b) follows from the identity, ∥x∥2 ≤ 2 ∥x − y∥2 + 2 ∥y∥2. (c) follows +from a similar calculation as in the proof of Lemma F.2. (d) follows from Jensen’s inequality. +We prove Theorem 2 in two steps. First, in Theorem 4, we bound the convergence rate of FedCOM-V for a general choice +of learning rates and local computations. Then, in Theorem 5, we prove a more explicit form of Theorem 2 for a specific +choice of learning rates and local computations. +Theorem 4. Under Assumptions 6 to 8, if the local learning rates (ηn), local computations (τn) and global learning rates +(γn) satisfy, +1 ≥ 2τ 2 +nL2η2 +n + 2 +�qmax +m ++ 1 +� +ηnγnLτn, +∀n, +then, the FedCOM-V updates satisfy, +�r +n=1 ηnτnγn E +� +∥∇f(wn)∥2 ���FC� +�r−1 +n=0 ηnτnγn +≤ 2 +� +f(w(0)) − f(w∗) +� +�r−1 +n=0 ηnτnγn ++ 2Lσ2 +m +�r−1 +n=0 η2 +nτnγ2 +n(¯qn + 1) +�r−1 +n=0 ηnτnγn ++ 2L2σ2 +�r−1 +n=0 η3 +nτ 2 +nγn +�r−1 +n=0 ηnτnγn +. +Proof. Recall the global update rule, wn+1 = wn −ηnγn˜gn +Q. From the L-smoothness of f(·), we can write, +f(wn+1) − f(wn) ≤ −ηnγn⟨∇f(wn), ˜gn +Q⟩ + η2 +nγ2 +nL +2 +��˜gn +Q +��2 . +35 + +Now, we bound the conditional expectation, +E +� +f(wn+1) − f(wn)|Fn−1 +� +≤ −ηnγn E +� +⟨∇f(wn), ˜gn +Q⟩ +���Fn−1 +� ++ η2 +nγ2 +nL +2 +E +���˜gn +Q +��2 ���Fn−1 +� +, +(a) +≤ ηnγn +2m +m +� +j=1 +τn +� +b=1 +� +− ∥∇f(wn)∥2 − E +����gb,n +j +��� +2 ���Fn−1 +� ++ L2 E +����wn − wb,n +j +��� +2 ���Fn−1 +� � ++ 2τnLη2 +nγ2 +n +2m +�qmax +m ++ 1 +� m +� +j=1 +τn +� +b=1 +E +����gb,n +j +��� +2 ���Fn−1 +� ++ (¯qn + 1) 2τnLη2 +nγ2 +nσ2 +2m +, +(52) +(b) +≤ ηnγn +2m +m +� +j=1 +τn +� +b=1 +� +− ∥∇f(wn)∥2 − E +����gb,n +j +��� +2 ���Fn−1 +� ++ 2L2η2 +nτn +τn +� +b′=1 +E +����gb′,n +j +��� +2 ���Fn−1 +� ++ 2L2η2 +nτnσ2� ++ τnLη2 +nγ2 +n +m +�qmax +m ++ 1 +� m +� +j=1 +τn +� +b=1 +E +����gb,n +j +��� +2 ���Fn−1 +� ++ (¯qn + 1) τnLη2 +nγ2 +nσ2 +m +, +(53) +(c) += −ηnγnτn +2 +∥∇f(wn)∥2 + Lτnη2 +nγn +m +(mLτnηn + γn(¯qn + 1))σ2 +− ηnγn +2m +� +1 − 2L2η2 +nτ 2 +n − 2Lτnηnγn +�qmax +m ++ 1 +�� m +� +j=1 +τn +� +b=1 +E +����gb,n +j +��� +2 ���Fn−1 +� +, +(d) +≤ −ηnγnτn +2 +∥∇f(wn)∥2 + Lτnη2 +nγn +m +(mLτnηn + γn(¯qn + 1))σ2, +(54) +where, (a) is obtained by using Lemmas F.3 and F.4, and (b) is obtained by using Lemma F.5. (c) is a rearrangement of terms, +and (d) follows from the premise of the theorem, +1 ≥ 2τ 2 +nL2η2 +n + 2 +�qmax +m ++ 1 +� +ηnγnLτn. +Taking an expectation, rearranging terms and summing up equation (54) for all the rounds r, we have a telescopic cancellation +to get, +�r +n=1 ηnτnγn E +� +∥∇f(wn)∥2 ���FC� +�r−1 +n=0 ηnτnγn += 2 +� +f(w(0)) − f(wr) +� +�r−1 +n=0 ηnτnγn ++ 2Lσ2 +m +�r−1 +n=0 η2 +nτnγ2 +n(¯qn + 1) +�r−1 +n=0 ηnτnγn ++ 2L2σ2 +�r−1 +n=0 η3 +nτ 2 +nγn +�r−1 +n=0 ηnτnγn +, +≤ 2 +� +f(w(0)) − f(w∗) +� +�r−1 +n=0 ηnτnγn ++ 2Lσ2 +m +�r−1 +n=0 η2 +nτnγ2 +n(¯qn + 1) +�r−1 +n=0 ηnτnγn ++ 2L2σ2 +�r−1 +n=0 η3 +nτ 2 +nγn +�r−1 +n=0 ηnτnγn +. +(55) +Theorem 5. If we choose the learning rates for round n, ηn and γn, and the number of local computations τn as, +ηn = cη +Ln, +γn = +cγ +√¯qn + 1, +τn = +n +2cη +, +where, +cη = 2 +�L∆f +√m +σ +�qmax +m ++ 1 +��2 +, +cγ = +1 +2 +� qmax +m ++ 1 +�, +where, ∆f ≜ +� +2(f(w0) − f(w∗))/L, and, if FedCOM-V is run for r communication rounds such that, +r +1 + log r ≥ max +��qmax +m ++ 1 +�2 4L2∆2 +fm√qmax + 1 +ε +, +�qmax +m ++ 1 +� 12L2∆2 +fσ +ε +�r +n=1 +√¯qn + 1 +r +� +, +(56) +then we have, +�r +n=1 ηnγnτn E +� +∥f(wn)∥2 ���FC� +�r +n=1 ηnτnγn +≤ ε. +(57) +36 + +Proof of Theorem 5. This proof will use the result of Theorem 4. Therefore, we first check that the premise of Theorem 4 is +satisfied. Due to the choice of ηn, τn and γn, +2τ 2 +nL2η2 +n + 2 +�qmax +m ++ 1 +� +ηnγnLτn = 1/2 + +�qmax +m ++ 1 +� +cγ, += 1. +Therefore, the following result of Theorem 4 holds true. +�r +n=1 ηnτnγn E +� +∥∇f(wn)∥2 ���FC� +�r +n=1 ηnτnγn +≤ 2 +� +f(w0) − f(w∗) +� +�r +n=1 ηnτnγn ++ 2Lσ2 +m +�r +n=1 η2 +nτnγ2 +n(¯qn + 1) +�r +n=1 ηnτnγn +� +�� +� +I ++ 2L2σ2 +�r +n=1 η3 +nτ 2 +nγn +�r +n=1 ηnτnγn +� +�� +� +II +. +Consider I, +I +(a) += 4L(f(w(0)) − f(w∗)) +cγ +�r +n=1 1/√¯qn + 1 + 2σ2cηcγ +m +�r +n=1 1/n +�r +n=1 1/√¯qn + 1, +(b) +≤ 4L(f(w(0)) − f(w∗)) +cγ +�r +n=1 1/√¯qn + 1 + 2σ2cηcγ +m +1 + log r +�r +n=1 1/√¯qn + 1, +(c) +≤ +2(1 + log r) +�r +n=1 1/√¯qn + 1 +� +L2∆2 +f +cγ ++ σ2cηcγ +m +� +, +(d) += +1 + log r +�r +n=1 1/√¯qn + 16L2∆2 +f +�qmax +m ++ 1 +� +, +(e) +≤ 6L2∆2 +f +�qmax +m ++ 1 +� 1 + log r +r +�r +n=1 +√¯qn + 1 +r +. +(58) +(a) is obtained by substituting the expressions for ηn, γn and τn. (b) is obtained by the bound, �r +n=1 1/n ≤ 1 + log r. (c) is +a rearrangement of terms and the bound 1 ≤ 1 + log r. (d) is obtained by substituting the expressions for cη and cγ. (e) is +obtained using the result that the harmonic mean is smaller than the arithmetic mean. +Consider II, +II = 2L2σ2 +�r +n=1 η3 +nτ 2 +nγn +�r +n=1 ηnτnγn +, += σ2cη +�r +n=1 1/(n√¯qn + 1) +�r +n=1, 1/√¯qn + 1 , +≤ σ2cη +� +qmax + 1 +�r +n=1 1/n +r +, +≤ σ2cη +� +qmax + 11 + log r +r +, += 2 +�qmax +m ++ 1 +�2 +L2∆2 +fm +� +qmax + 11 + log r +r +. +(59) +Therefore, if we chose the total number of communication rounds r such that, +6L2∆2 +f +�qmax +m ++ 1 +� 1 + log r +r +�r +n=1 +√¯qn + 1 +r +≤ ε +2, +(60) +and, +2 +�qmax +m ++ 1 +�2 +L2∆2 +fm +� +qmax + 11 + log r +r +≤ ε +2, +(61) +then (57) is satisfied. (60) and (61) are simply a restatement of (56). This completes the proof. +If the compression parameters (Qn)n formed a stationary process with a stationary distribution according to a random +variable Q, then �r +n=1 +� ¯Qn + 1/r → E[ +� ¯Q + 1]. Therefore, Theorem 5 proves Theorem 2. +37 + diff --git a/9dE3T4oBgHgl3EQfSQko/content/tmp_files/load_file.txt b/9dE3T4oBgHgl3EQfSQko/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c507ef47584e1776021741fd49051f24453ede3c --- /dev/null +++ b/9dE3T4oBgHgl3EQfSQko/content/tmp_files/load_file.txt @@ -0,0 +1,1695 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf,len=1694 +page_content='Network Adaptive Federated Learning: Congestion and Lossy Compression Parikshit Hegde Electrical and Computer Engineering The University of Texas at Austin Austin, Texas, USA hegde@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='edu Gustavo de Veciana Electrical and Computer Engineering The University of Texas at Austin Austin, Texas, USA gustavo@ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='edu Aryan Mokhtari Electrical and Computer Engineering The University of Texas at Austin Austin, Texas, USA mokhtari@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='edu Abstract—In order to achieve the dual goals of privacy and learning across distributed data, Federated Learning (FL) systems rely on frequent exchanges of large files (model updates) between a set of clients and the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As such FL systems are exposed to, or indeed the cause of, congestion across a wide set of network resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lossy compression can be used to reduce the size of exchanged files and associated delays, at the cost of adding noise to model updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' By judiciously adapting clients’ compression to varying network congestion, an FL application can reduce wall clock training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To that end, we propose a Network Adaptive Compression (NAC-FL) policy, which dynamically varies the client’s lossy compression choices to network congestion variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We prove, under appropriate assumptions, that NAC-FL is asymptotically optimal in terms of directly minimizing the expected wall clock training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, we show via simulation that NAC-FL achieves robust performance improvements with higher gains in settings with positively correlated delays across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Index Terms—federated learning, rate adaptation, resilience I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' INTRODUCTION Communication costs and delays of sending model updates from clients to the server are a known bottleneck in training Federated Learning (FL) systems [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Two common tech- niques used to alleviate this issue are: 1) local computations where clients perform several local steps before communicat- ing with the server, and 2) (lossy) compression where clients communicate quantized/compressed updates to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The eventual end goal of these approaches is to minimize the wall clock time for convergence of the training algorithm (hereon referred to as FL algorithm) by reducing the amount of data communicated from clients to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To this end, several works have analyzed the relationship between compression, local computations and the number of rounds needed by FL algorithms to converge [5]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' However, these works ignore the impact of changing network congestion, both across clients and across time, on the wall This material is based upon work of Hegde and de Veciana supported by the National Science Foundation (NSF) under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2148224 and is supported in part by funds from OUSD R&E, NIST, and industry partners as specified in the Resilient & Intelligent NextG Systems (RINGS) program and the WNCG/6G@UT industrial affiliates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The work of Mokhtari is supported in part by the NSF AI Institute for Future Edge Networks and Distributed Intelligence (AI-EDGE) via NSF grant 2112471, the Machine Learning Lab (MLL) at UT Austin, and the Wireless Networking and Communications Group (WNCG) Industrial Affiliates Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' clock time to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For instance, a client may choose a high degree of compression when it sees high network congestion, while a client seeing lower congestion may op- portunistically choose not to compress as much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In this work, we ask the following question: “Can we design a policy that adapts the amount of compression across clients and time according to changing network conditions in order to opti- mize the wall clock time?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To answer this question, we first characterize the impact that changing network congestion and an adaptive compression policy have on the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Second, we propose the Network Adaptive Compression for Federated Learning (NAC-FL) policy that judiciously chooses compression levels based on network congestion to minimize the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Crucially, NAC-FL does not rely on the prior knowledge of the distribution of network congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Instead, it learns to optimize its compression decisions on- the-fly based on the congestion seen by clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' NAC-FL works in an opportunistic manner by adaptively choosing high or low amounts of compression across clients and across time based on low or high network congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' It further considers two effects that compression has on the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' First, with increasing amount of compres- sion, the FL algorithm would require more communication rounds to converge, as the server receives “noisier”, and hence inaccurate, model updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Second, with higher degrees of compression, the duration of each round would decrease as a smaller model update is communicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since the wall clock time is affected by both the number of rounds and the duration of each round (it is effectively the product of the two quantities), a policy for choosing compression levels should consider these jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1 provides an illustrative visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Hence, NAC-FL aims to find the “sweet-spot” compression levels over time varying network congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We propose a general framework to study how to best adapt compression of client model updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assuming a stationary Markov model for the underlying network conges- tion state, we show that optimal policies are state dependent and characterize the expected stopping time for convergence to a predefined model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This characterization provides the underlying insight for our proposed NAC-FL policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To our knowledge this is the first 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='04430v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='LG] 11 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1: Illustration of how compression level affects round duration, number of rounds and wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' policy for compression that adapts to the stochastic variations of the underlying network congestion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Under appropri- ate assumptions on the FL algorithm and underlying network congestion and delays, we provide a proof of the asymptotic optimality of NAC-FL in terms of minimizing the mean time until the convergence criterion is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To our knowledge this is the first theoretical result of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Finally we demonstrate via simulation the performance gains and robustness of NAC-FL vs alternative fixed compres- sion and/or fixed error per round policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We explore a variety of models for network congestion, finding that in particular NAC-FL excels in the practically relevant setting where the network sees positive correlations in the network congestion accross time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Related Work Perhaps the most related papers to our work are [13]–[17] which explored adaptive compression schemes for FL settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In [13]–[15] the authors propose adapting compression to network congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In these works, the algorithm to select compression has a per round budget, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', a budget on delay (or compression error) per round, and possibly heterogeneous compression levels are chosen across the clients based on the current network congestion to minimize the compression error (or delay) for the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' These works exploit the diversity of network congestion across the clients, but not across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Meanwhile [16], [17] have observed that using a higher amount of compression at the start and gradually reducing compression through time may improve the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Our proposed policy is novel in that it learns how to best exploit congestion variation across clients and across time to optimize the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Another line of work that aims to reduce the overall commu- nication cost is client sampling [18]–[21], where at each round, only a subset of the clients are chosen to participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The authors of [21] propose a client sampling and power control policy that adapts to time varying channels of clients sharing a single base station and optimizes a proxy for wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Overall we veiw lossy compression and client sampling as alternative approaches geared at addressing communication bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A study of how to jointly adapt lossy compression and client sampling to changing network congestion is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Paper Organization In Section II, we introduce our system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In Section III, we propose our NAC-FL algorithm for lossy compression and under appropriate assumptions prove it is asymptotically optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Section IV is devoted to exploring the method for several problem instances and in particular for various models for the underlying network congestion in terms of correlation across clients and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In Section V, we comment on the practical aspects of estimating the file transfer delay of clients when deploying NAC-FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Finally, in Section VI, we close the paper with some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Throughout this document, unless otherwise men- tioned, quantities denoted with lowercase letters correspond to constants, and uppercase letters correspond to random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bold symbols correspond to vectors, and regular symbols indicate scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For example, x is a constant vector, X is a random vector, x is a constant scalar, and X is a random scalar/variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lowercase and uppercase forms of the same letter correspond to constant and random variable notions of the same quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A sequence indexed by n will be denoted as (xn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' MODEL SETUP In this paper, we focus on a federated architecture, where a server aims to find a model that performs well with respect to the data of a group of m clients, and in which nodes exchange updates based on their local information with only the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' More precisely, suppose the loss function associated with client j is denoted by fj(w), where w represents the weights of the model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', the weights of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The goal is to find the model that minimizes the average loss across clients f(w) = 1 m m � j=1 fj(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The FL algorithm proceeds in rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Each round consists of two stages: (i) a local stage in which each client updates the most recent model received from the server via gradient-based updates based on its local data and (ii), an aggregation stage in which the server updates the global model by aggregating the local updates received from clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We shall let wn denote the global model at the server at round n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, we let τ n denote the total number of local steps (such as gradient steps) that each client performs at round n, and let wτ n,n j denote the resulting local model at node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In this paper, we are interested in the setting where each client sends a compressed version ˜gn Qj of its local model wτ n,n j to the server using a lossy compression algorithm (or, compressor) Q(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The compressor accepts a vector x and a parameter q ∈ [0, qmax] indicating the amount of compression with the maximum value being qmax, and outputs ˆX = Q(x, q) which is an approximation of x, but has a decreased file size as compared to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' ˆX is capitalized to highlight that the compressor Q(·, ·) may use randomness in its compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We shall denote by qn j the compression 2 Wall Clock Time Round Duration No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' of Rounds Compression Amount Compression Amountparameter used by client j for round n, and denote by qn ≜ (qn j )m j=1 the vector of parameters used by the clients in round n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' After receiving updates from all the clients, the server aggregates the compressed local models and produces the next global model wn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Given a target tolerance ε > 0, the goal of FL is to generate a sequence of global models until on some round rε a prespecified stopping criterion is first met, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', the norm of the global loss function gradient is at most ϵ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', ∥∇f(wrε)∥ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Our goal is to find an adaptive compression policy that dynamically adapts to the possibly time varying network states such that the target accuracy is achieved with a minimum overall wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We formalize the overall wall clock time, denoted tε, required to achieve the target accuracy as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The duration d(τ n, qn, cn) of a round n depends on: τ n, the number of local computations performed by clients which we will assume to be the same across clients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' qn, an m dimensional vector of clients’ compression parameters ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' cn, the network state which models network congestion and is assumed to be an element of a finite set C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This allows some flexibility, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', the round’s duration may depend on the max delay to deliver the model update from clients to server, or the sum of the delays if clients share a single resource in TDMA (Time Division Multiple Access) fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The total wall clock time is then given by tε = rε � n=1 d (τ n, qn, cn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (1) In our system model, the sequence of network states, (cn)n, is assumed to be exogenous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', not be controlled by the server or the clients nor their choices of τ n and qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The delays associated with the server multicasting global models to clients are assumed to be exogeneous i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', can not be controlled by the FL server/clients and are not compressed, whence are not part of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Still, in this work, based on observing the network state we will devise an approach to select the clients compression parameters so as to minimize the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As discussed in Section V, in practice observation of the network state may involve light weight in band estimation by probing delays of message bits as they are delivered in a given round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A policy for choosing compression parameters is called a state dependent stationary policy if it can be expressed as a function π of the current network state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', qn = π(cn) for all rounds n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Such a policy will be referred to simply as policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Given a random sequence of network states, (Cn)n, let Rπ ε be the random variable denoting the minimum number of rounds needed to converge to error tolerance ε under policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, the corresponding wall clock time, denoted by T π ε , is expressed as, T π ε = Rπ ε � n=1 d (τ n, π (Cn) , Cn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' NETWORK ADAPTIVE COMPRESSION FOR FEDERATED LEARNING (NAC-FL) Our approach to designing a policy to adapt clients’ com- pression parameters centers on recognizing that the expected wall clock time can be broken up into a product of the expected number of rounds rε needed to converge to an error tolerance ε and the average duration of each round ˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We start by characterizing the relationship between rε, ˆd, and the sequence of selected quantization parameters (qn)n and network states (cn)n for a given FL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Below we state an assumption relating rε to (qn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To that end we introduce a strictly increasing, continuous and bounded scalar function hε : [0, qmax] → R+ of compression parameter q and an associated vector function hε : [0, qmax]×m → Rm + of a compression vector q where hε,j(q) = hε(qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We let h−1 ε denote the inverse of this vector function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For a given FL algorithm there exists a strictly increasing, continuous and bounded function hε(q) and norm ∥·∥ such that given a sequence of compression parameters (qn)n, the FL algorithm has reached the desired error tolerance ε by round r if and only if, r > 1 r r � n=1 ∥hε (qn)∥ for some norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The above assumption implies that the expected number of rounds can be written as the average of an increasing function of the sequence of selected quantization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Roughly speaking, given a lossy compression policy that generates a stationary parameter sequence (Qn)n whose marginal distri- bution is the same as the random vector Q, the above criterion means that the expected number of rounds to converge to the desired error tolerance is approximately E[∥hε (Q)∥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This is a general condition that is motivated by convergence bounds of several FL algorithms with compression, including, [5], [8], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In particular in Appendix A, we illustrate this motivation for an extension of the FedCOM algorithm [11], when q indicates the normalized-variance introduced by the compressor, the scalar function is hε(q) = O(√q + 1/ε) and the norm is the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For any sequence of compression parameters (qn)n the minimum number of rounds rε needed to converge to an error tolerance ε is such that rε = Θ(1/poly(ε)), where poly(ε) denotes a polynomial of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 2 is a natural assumption for gradient based optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' It requires the convergence guaran- tees for the FL algorithm to be such that when we require a more accurate solution, the number of required communication rounds grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This argument indeed holds even for the settings that we do not exchange compressed signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We also make the following additional assumption about the round duration function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2: Illustration of a round duration as a function of compression parameter q for a fixed local computation τ and network state c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Given a network state c, number of local computations τ, and compression parameters q = h−1 ε (r), the round duration d (τ, q, c) = d � τ, h−1 ε (r), c � is bounded, convex in r and decreasing in every coordinate of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In Assumption 3, the round duration being decreasing in r is reasonable, since we expect more rounds as well as smaller file sizes with higher compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The convexity is motivated by the notion that we use a “good compressor” as illustrated next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Consulting Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2, for any two parameters q1, q2 and 0 < α < 1, a new time-sharing compressor Q′ may be derived which outputs Q(x, q1) with probability α and outputs Q(x, q2) with probability (1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This compressor has expected round duration αd(τ, q1, c) + (1 − α)d(τ, q2, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' And, in certain cases, its compression parameter is qα = αq1+ (1−α)q2 (such as when the stochastic quantizer parameterized by its normalized variance [5] is used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' If Q is a “good compressor”, then its round duration, d(τ, qα, c), should be lower compared to that of the simple time-shared compressor, αd(τ, q1, c)+(1−α)d(τ, q2, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, the convexity of the round duration function is a reasonable assumption for “good compressors” (considering hε(q) ∝ q for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The sequence of network states (Cn)n forms an irreducible aperiodic stationary Markov Chain on a finite state space C with invariant distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 4 is a natural assumption made to facilitate the analysis of algorithms (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Expected Wall Clock Time Formulation Given the above mentioned assumptions, we are now ready to introduce the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We begin by showing that we need only consider state dependent stationary policies for choosing compression parameters when optimizing the overall wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Under Assumptions 1-4 there exists a state depen- dent stationary policy to select compression parameters which is asymptotically optimal in terms of minimizing the wall clock time to reach a desired error tolerance of ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The proof of Lemma 1 depends on two critical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' First, since by Assumption 2 the number of rounds needed to converge grows large as ε → 0, one can expect the empirical distribution of the network states modelled by the finite state Markov Chain to concentrate around the invariant prior to the stopping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Second, due to the convexity of the round duration function in Assumption 3, given a sequence of network states there exists a state dependent stationary policy that is near optimal and depends solely on the empirical distribution of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The proof is in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Here, we will focus on the setting where ε is small, hence by Lemma 1, we only need to consider state dependent stationary policies, qn = π(cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Under Assumptions 1-4 and a fixed number of local computations per round τ, for every δ > 0, there exists an εth > 0 such that, for all ε < εth and any state-dependent stationary policy π, the expected wall clock time is bounded as, 1 − δ ≤ E [T π ε ] E[∥hε (π(C))∥] E[d (τ, π(C), C)] ≤ 1 + δ, (2) where, C denotes a random variable whose distributions is µ (see Assumption 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lemma 2 is proved in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Define, ˆtπ ε ≜ E[∥hε (π(C))∥] E[d (τ, π(C), C)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (3) Due to Lemma 2, for small enough ε, ˆtπ ε provides an accurate approximation for E[T π ε ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, from here onwards, we shall assume implicitly that that a small ε is considered and focus on finding a policy to optimize ˆtπ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Suppose the distribution of C is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, one could compute expected wall clock time as given in (3) for any state dependent stationary policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In this case, we could determine an optimal policy π∗ by solving the optimization problem, min π∈Qm|C| ˆtπ ε = E[∥hε (π(C))∥] E [d (τ, π(C), C)] , (4) where Qm|C| is the set of all state-dependent stationary poli- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Alas, in practice, we often cannot directly solve the above problem, as the distribution of C is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Hence, below, we propose a stochastic approximation like algorithm that achieves the optimal wall clock time of π∗ asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' NAC-FL: Informal Description The idea underlying NAC-FL is to keep running estimates for E [∥hε (Q)∥] and E [d(τ, Q, C)] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', ˆrn ε = 1 n n � k=1 ���hε � q(k)���� , ˆdn = 1 n n � k=1 d � τ, q(k), c(k)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4 d(T, q, C) q1 q2 qa bGiven a network state of cn+1 at round n + 1, and, a possible choice for compression parameters q, the running averages would be updated as follows, ˆrn+1 ε = n n + 1 ˆrn ε + 1 n + 1 ∥hε (q)∥ , ˆdn+1 = n n + 1 ˆdn + 1 n + 1d � τ, q, cn+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (5) As seen in (3), to minimize the wall clock time one should minimize ˆrn+1 ε ˆdn+1, which can be expanded as, ˆrn+1 ε ˆdn+1 = n (n + 1)2 � rn ε d � τ, q, cn+1� + ˆdn ∥hε(q)∥ � + n2 (n + 1)2 rn ε ˆdn + O � 1 (n + 1)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Given the fact that ˆrn ε and ˆdn are constants, and neglecting the term O � 1/(n + 1)2� , an optimal choice for qn+1 is qn+1 = argmin q ˆrn ε d � τ, q, cn+1� + ˆdn ∥hε (q)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (6) The NAC-FL policy is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To retrieve policy informally described above the tunable param- eters (βn)n and α should be set to βn = 1 n and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Consider two possible network states c and c′ at a round n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' If the delay under state c is higher compared to c′ for any compression parameters, then NAC-FL would choose a higher compression amount q for state c compared to compression amount q′ for state c′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', q > q′ elementwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This may be concluded from the selection policy of (6), and noting that rε(q) is increasing in q (Assumption 1), and d(τ, q, c) is decreasing in q (Assumption 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Observe that since the estimates ˆrn ε and ˆdn will initially change across rounds, NAC-FL may choose different compres- sion parameters in two rounds for which the network was in the same state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', NAC-FL is not a state-dependent stationary policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Still, we will show NAC-FL is asymptotically near optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To develop this result we shall next present NAC-FL in a more formal manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Algorithm 1: NAC-FL Input : Initialization: ˆr(0) ε , ˆd(0) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' step size schedule {βn}∞ n=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1 for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' , until termination do 2 Server observes network state cn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3 qn = argmin q αˆr(n−1) ε d (τ, q, cn) + ˆd(n−1) ∥hε (q)∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4 ˆrn ε = (1 − βn)ˆr(n−1) ε + βn ∥hε (qn)∥ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 5 ˆdn = (1 − βn) ˆd(n−1) + βnd(τ, qn, cn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 6 end C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' NAC-FL: Formal Description Our NAC-FL approach is also inspired by the Frank-Wolfe Algorithm [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We start by reformulating the optimization program in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Denote by set Vε all possible pairs of expec- tations (ˆrε, ˆd), Vε = � (ˆrε, ˆd) : ∃ π ∈ Qm|C| s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' ˆrε = E [∥hε (π(C))∥] , ˆd = E [d (τ, π(C), C)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (7) Using the set Vε, and denoting H(r, d) ≜ rd, we may write the optimization (4) characterizing the optimal policy π∗ as min ˆrε, ˆd {H(ˆrε, ˆd) : (ˆrε, ˆd) ∈ Vε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (8) In this case, from a point (ˆrn ε , ˆdn), the Frank-Wolfe update would be given as, (ˆrε, ˆd) = argmin (r,d)∈Vε ∇H � ˆrn ε , ˆdn�⊤ �r d � , (9) ˆrn+1 ε = (1 − β)ˆrn ε + βˆrε, ˆdn+1 = (1 − β) ˆdn + β ˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The gradient ∇H(ˆrε, ˆd) is, ∇H(ˆrε, ˆd) = � ˆd ˆrε �⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Vε is a set of feasible averages of ˆrε and ˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, at round (n+1), not all the pairs (r, d) ∈ Vε may be achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Hence, NAC- FL approximates equation (9) as, qn+1 = argmin q ˆrn ε d � τ, q, cn+1� + ˆdn ∥hε (q)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We have thus retrieved our proposed NAC-FL algorithm based on the Frank-Wolfe update, with one difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The above derivation suggests the use of a fixed step-size β at all rounds while the previously derived algorithm used a decaying the step-size βn = 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In our simulations, we will embrace the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The following assumption is required to show the asymp- totic optimality of NAC-FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A state dependent stationary policy π maps from a domain of finite size |C|, to a range positive-real vectors of dimension m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, the policy may be represented by a positive-real vector, π, of dimension m |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, a vector rπ may be obtained by applying hε(·) elementwise to the policy vector π, rπ ≜ hε(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This representation is used in the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The objective function ˆtπ ε of the optimization problem in (4) is a strictly quasiconvex function in π in the following sense, rπ⊤ � ∇rπˆtπ ε � = 0 =⇒ rπ⊤ � ∇2 rπˆtπ ε � rπ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (10) Assumption 5 ensures that there is a unique state dependent stationary policy π∗ which optimizes (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We have observed that the considered network model, compression model and the ∥hε (q)∥ function associated with the FedCOM algorithm indeed satisfy this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Next we shall establish an optimality property for NAC- FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To that end we shall consider executing NAC-FL without termination with βn = β for all n and let � Qn β � n, ˆRn ε,β and 5 ˆDn β be the corresponding sequence of compression parameters and the associated estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let π∗ be the solution and ˆtπ∗ ε the minimum of the optimization problem in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' If Assumptions 1-5 hold, then there exists a positive sequence (βi)∞ i=1 with βi → 0 as i → ∞, such that for every ρ > 0, there exists a thereshold nth(ρ) such that, lim i→∞ sup n≥nth(ρ)/βi P ������ � ˆRn ε,βi − E[∥hε (π∗(C))∥] ˆDn βi − E[d (τ, π∗(C), C)] ������ > ρ � = 0, The proof of Theorem 1 is included in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Theorem 1 should be interpreted with some subtlety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Say the desired error-tolerance ε is very small such that the number of rounds needed to converge under any compression policy is such that rε ≫ nth(ρ)/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, based on Theorem 1, one can show that NAC-FL compression choices will be near optimal after nth(ρ)/β rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Thereafter, since rε is large, NAC-FL will make near optimal choices for long enough leading to a near optimal expected wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We further remark on the meaning of the asymptotic result in the context of minimizing the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In appli- cations that require a very low error-tolerance ε, one needs to have a large number (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', in the asymptotic region) of communication rounds rε for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, even though the wall clock time obtained by using NAC-FL may be large in this setting, it is near-optimal compared to other methods of choosing compression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' SIMULATION In this section, we present our simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We begin by describing additional model details used in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Additional Model Details 1) Compression Model: We shall use the stochastic quan- tizer in [5] which we will denote as Qq(·, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The quantizer has a parameter b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' , 32} corresponding to the number of bits used to represent each co-ordinate, in addition to the bit used to denote signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' When input a vector x, it outputs, Qq(x, b) = ∥x∥∞ sign(x)ζ(x, b) (11) where sign(x) is the element-wise sign operator and where the function ζ(x, b) uniformly quantizes each co-ordinate amongst 2b − 1 levels between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' That is, if xi/∥x∥∞ ∈ � l 2b−1, l+1 2b−1 � , then it is quantized as, ζi(x, b) = � l+1 2b−1, with prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' |xi| ∥x∥∞ (2b − 1) − l, l 2b−1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' When x is quantized to b-bits per co-ordinate, its file size is given by the function, s(b) = ∥x∥0 (b + 1) + 32 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Here, the zero-norm, ∥x∥0, gives the length of the vector, the 1 indicates the bit used to denote the sign, and the 32 bits are for a floating point number denoting the norm, ∥x∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Finally, if client j uses the parameter bj, then the vector of parameters used by the clients is denoted as, b = (bj)m j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2) Network Congestion Model: For purposes of evaluating the performance of various algorithms over different types of network congestion we propose the following general, albeit idealized, model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We let Cn be a m dimensional random vector denoting the Bit Transmission Delay (BTD) for clients during round n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We further let Cn = exp (Zn) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', coordinate-wise exponentiation of an m dimensional first order autoregressive process given by (Zi)∞ i=0 where Z0 = 0, where Zn = A Z(n−1) +En, ∀n ≥ 1, (12) where A is an m×m deterministic matrix, and En ∼ N(µ, Σ) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', m dimensional normal random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Different correlations across time and clients may be modelled by varying A, µ and Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The marginal distributions of Cn are thus log-normal but can be correlated in different ways based on the underlying autoregressive process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In particular: Homogeneous Independent: the parameters are set to A = 0, µ = 1, and Σ = σ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This results in a process which is independent and identically distributed across clients and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Heterogeneous Independent: the parameters are set to A = 0, µi = 0 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' , 5} and µi = 2 for i ∈ {6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' , 10}, and Σ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This results in a process which is independent across clients and time, with the BTD being lower for the first 5 clients compared to the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Perfectly correlated: the parameters are set to A such that Ai,j = a m where a ∈ (0, 1), µ = 0, and Σ such that Σi,j = σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This results in a process where all clients see the same positively correlated time-varying delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Partially correlated: the parameters are set to A such that Ai,j = a m, µ = 0, and Σ such that Σi,i = 1 and Σi,j = 1/2 for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This results in a process where delays are positively correlated accross clients and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3) Model for Round Durations: We will model the duration of a round as the maximum across clients’ delays, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', d(τ, b, c) = max j [θτ + cjs(bj)], where θ represents the compute time per local computation, and cjs(bj) the BTD of client j times the size of the client j’s file capturing the time taken to communicate its update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For simplicity we will set θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4) Compression Level Choice Policies: We compare NAC- FL to the following policies, a) Fixed Bit: Here, a number b is fixed, and all the clients use the stochastic quantizer Qq(x, b) from (11) with the parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We present results for b ∈ {1, 2, 3}, as we didn’t notice a performance improvement for larger parameters in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' b) Fixed Error: This method was suggested in [13] and is parameterized by a number q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' At each round n, the parameters bn of the stochastic quantizers are such that the average normalized-variance ¯qn (see equation (15)) is smaller than 6 q, and the duration of the round d(τ, qn, cn) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We fix q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='25 in all our experiments after finding it to be performing well across different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 5) Machine Learning Model: We consider m = 10 clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We consider the MNIST dataset [24] which may be distributed homogeneously or heterogeneously amongst the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since data is heterogeneous across clients in most FL applications, we consider the heterogenous data case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' That is, each client has data corresponding to 1 unique label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The MNIST dataset has 60,000 training samples, 10,000 test samples and 10 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The clients and the server aim to train a fully connected neural network with the architecture (784, 250, 10) with the sigmoid activation for the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The learning rate is initialized to η0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='07, and is decayed by a factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 every 10 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The aggregation rate and local computations per round are fixed throughout the training to γ = 1 and τ = 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As for the parameters of the NAC-FL policy, we set βn = 1 n, and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We measure the performance of the global model using the following, a) Training Loss: The training loss of the global model is the empirical cross entropy loss across the entire set of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' b) Test Accuracy: The test accuracy is measured over all the test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Here, in some experiments, we run 20 simu- lations with different random seeds, and report the mean, 90th percentile and 10th percentile times to reach a test accuracy of 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The 90th and 10th percentile scores are reported to capture the variation in performance across the 20 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We also report a gain metric, which is sample mean of the time gained to reach 90% accuracy by NAC-Fl compared to a another policy reported in percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For instance, let xi, yi be the times under NAC-FL and another policy for a random seed i, then the gain is 100 ∗ ��20 i=1 yi/xi − 1 � /20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Simulation Results 1) Homogeneous Independent BTD: We simulated over σ2 ∈ {1, 2, 3} in order to study the change in performance over increasing variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We observe that in all the cases, NAC-FL and the Fixed Error policy have very similar perfor- mance across all the considered statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This is because the Fixed Error policy was designed to operate well in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', network delay case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' However, both NAC-FL and Fixed Error policy perform better than all the Fixed Bit policies according to all the statistics across all the considered parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' More- over, we observed that the gap in the performance to Fixed Bit policies increased with increasing variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For instance, the gain of the best Fixed Bit policy increased from 145% to 250% when the variance was increased from 1 to 3, while the gain of the worst fixed bit policy increased from 314% to 881%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This is as expected because both NAC-FL and Fixed Error policy adapt to the heterogenous delay of clients at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Surprisingly, NAC-FL lagged behind Fixed Error policy in some metrics, but it performed better in terms of the gain metric in all the 3 cases, with the gain over Fixed Error policy ranging from 1% to 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' σ2 1 bit 2 bits 3 bits Fixed Error NAC-FL 1 Mean 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='60 90th 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='72 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='05 10th 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='14 Gain 314% 145% 168% 3% 2 Mean 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 90th 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 10th 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='82 Gain 522% 216% 240% 8% 3 Mean 799 430 458 165 168 90th 1430 752 665 318 320 10th 418 157 148 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 Gain 881% 270% 250% 1% TABLE I: Performance comparison of policies with homoge- neous independent BTD in terms of the mean, 90th percentile and 10th percentile times to reach 90% test accuracy under the different policies, and their average sample-path gain compared to NAC-FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' All the numbers represented are in 107 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2) Heterogeneous Independent BTD: We considered this case since the first 5 clients would have consistently worse delay, NAC-FL and the Fixed Error policy would consis- tently compress the updates of those clients heavily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since the data distribution is heterogeneous, it may be possible heavy compression of updates from specific clients throughout the training may hurt the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' On the other hand, the Fixed Bit policies use the same amount of compression across all clients equally irrespective of their delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Still, we observed that NAC-FL and the Fixed Error policy perform better than the Fixed Bit policies as can be seen in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In fact, performance in terms of the gain metric is very comparable to the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', network delay case with σ2 = 1 in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1 bit 2 bits 3 bits Fixed Error NAC-FL Mean 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='85 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='48 90th 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='54 10th 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='54 Gain 319% 146% 173% 4% TABLE II: Performance comparison of policies with heteroge- nous independent BTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The numbers shown are the mean, 90th percentile and 10th percentile times to reach 90% test accuracy under the different policies, and their average sample- path gain compared to NAC-FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' All the numbers represented are in 108 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3) Perfectly Correlated BTD: We will demonstrate that NAC-FL performs better than Fixed Error and Fixed Bit policies under increasing correlated delay across time since they are not designed to optimize the wall clock time under this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To study the variation of network delay across rounds, consider the marginal auto-regressive process of 1 client which may be represented by the following scalar autoregressive process, Zn = a′Z(n−1) + En, (13) where En ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We define metric called asymptotic 7 variance, denoted σ2 ∞, which is designed to capture the variance, and long and short term correlations of a random process, σ2 ∞ ≜ lim n→∞ E �� Z(1) + · · · + Zn�2� n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (14) For the autoregressive process in (13), it may be computed to be, σ2 ∞ = 1/(1 − a′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Table III shows the performance of the different policies un- der varying asymptotic variance of the marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We observe that in addition to beating the baseline fixed bit policies on all the metrics, the NAC-FL performs better than the Fixed Error policy in most metrics as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Considering the gain metric, we observe gain of 13% over the Fixed Error policy for low asymptotic variance of σ2 ∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='56, and is as large as 27% for higher asymptotic variance of σ2 ∞ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Notably, in terms of the 10th percentile time to reach 90% accuracy, the Fixed Error policy required 40%, 23% and 32% more time compared to NAC-FL in the σ2 ∞=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='56, 4 and 16 cases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' σ2 ∞ 1 bit 2 bits 3 bits Fixed Error NAC-FL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='56 Mean 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='11 90th 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='32 10th 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='02 Gain 191% 58% 75% 13% 4 Mean 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='23 90th 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='77 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='00 10th 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='981 Gain 252% 82% 107% 27% 16 Mean 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='36 90th 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='94 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 10th 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='87 Gain 316% 72% 98% 21% TABLE III: Performance comparison of policies with perfectly correlated BTD in terms of the mean, 90th percentile and 10th percentile times to reach 90% test accuracy under the different policies, and their average sample-path gain compared to NAC-FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' All the numbers represented are in 107 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4) Partially Correlated BTD: In Table IV, we show results for the partially correlated BTD case with asymptotic variance σ2 ∞ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We consider this case to demonstrate that NAC-FL is effective with positive (but, not 100%) correlation across clients as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Indeed, we observe NAC-FL performing better compared to all the other policies across all the considered metrics, with a gain of 10% over the Fixed Error policy, and 129% over the best fixed bit policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Notably, in terms of the 10th percentile and 90th percentile metrics, NAC-FL outper- formed Fixed Error policy by 30% and 15% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Figure 3 contains sample path plots of Training Loss and Accuracy vs Wall Clock Time for the independent homo- geneous (σ2 = 2), heterogeneous and perfectly correlated (σ2 ∞ = 4) BTD cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Both accuracy and loss plots for NAC-FL and Fixed Error are overlapping in the independent homogeneous and heterogeneous BTD cases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' However, in the perfectly correlated BTD case, NAC-FL dominates the performance of Fixed Error policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In summary, we observe that NAC-FL’s performance is robust under a range of network models considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' NAC-FL 1 bit 2 bits 3 bits Fixed Error NAC-FL Mean 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='33 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='51 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='83 90th 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='46 10th 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='51 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='47 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='64 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='02 Gain 307% 129% 159% 10% TABLE IV: Performance comparison of policies with partially correlated BTD in terms of the mean, 90th percentile and 10th percentile times to reach 90% test accuracy under the different policies, and their average sample-path gain compared to NAC-FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' All the numbers represented are in 107 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' vastly outperformed the baseline Fixed Bit policies in all the network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The performance of NAC-FL was observed to be similar to that of Fixed Error policy in the independent BTD setting, albeit, it outperformed Fixed Error policy in terms of the gain metric under all the network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Notably, the gap between NAC-FL and Fixed Error policy was observed to be noticeably high in the perfectly and paritally correlated BTD settings, where NAC-FL was able to adapt to positive correlations of BTD across time, whereas Fixed Error could not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' NAC-FL IN PRACTICE In this section we briefly comment on some practical aspects underlying estimating model update delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This involves estimating the network’s current average BTD to each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A simple approach to doing so is to observe that for the stochastic quantizer described in Section IV-A1, clients always send the vector of signs of their updates, no matter what are the bits per coordinate that will be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' So, as the clients send their signs, the server may probe the delay characteristics to estimate the BTD of clients without having to request vacuous (non update related) bits to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' It may then use these estimates to perform the optimization in (6) for the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' CONCLUSION Due to their distributed character FL algorithms are exposed to congestion across a potentially large number of network resources, whence one might say they are exposed to network congestion and variability at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Building adaptive algo- rithms that minimize the impact of time varying congestion across clients presents a significant challenge, particularly when the aim is to directly optimize the expected wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' NAC-FL exemplifies a new class of robust algorithms to optimally adapt clients’ lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This paper further provides the technical roadmap to formalizing and showing asymptotic optimality for such algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' McMahan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Ramage, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Hampson, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in Artificial intelligence and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' PMLR, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1273– 1282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Sahu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Talwalkar, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Smith, “Federated learning: Challenges, methods, and future directions,” IEEE Signal Processing Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 50–60, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 8 (a) (b) (c) (d) (e) (f) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3: Plots of Training Loss and Test Accuracy vs Wall Clock time on different network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Figures (a) and (d) correspond to homogeneous independent BTD case (σ2 = 2), Figures (b) and (e) correspond to the heterogeneous independent BTD case, and Figures (c) and (f) correspond to the perfectly correlated BTD case (σ2 ∞ = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Kairouz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' McMahan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Avent, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bellet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bennis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bhagoji, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bonawitz, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Charles, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Cormode, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', “Advances and open problems in federated learning,” Foundations and Trends® in Machine Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1–2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1–210, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Koneˇcn`y, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' McMahan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Yu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Richt´arik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Suresh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bacon, “Federated learning: Strategies for improving communication efficiency,” arXiv preprint arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='05492, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Alistarh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Grubic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Tomioka, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Vojnovic, “QSGD: Communication-efficient SGD via gradient quantization and encoding,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Basu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Data, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Karakus, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Diggavi, “Qsparse-local-SGD: Dis- tributed SGD with quantization, sparsification and local computations,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Stich, “Local SGD converges fast and communicates little,” arXiv preprint arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='09767, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Stich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Cordonnier, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Jaggi, “Sparsified SGD with memory,” Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Ghadimi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lan, “Stochastic first-and zeroth-order methods for nonconvex stochastic programming,” SIAM Journal on Optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2341–2368, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Reisizadeh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Mokhtari, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Hassani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Jadbabaie, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Pedarsani, “Fedpaq: A communication-efficient federated learning method with periodic averaging and quantization,” in International Conference on Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' PMLR, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2021–2031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Haddadpour, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Kamani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Mokhtari, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Mahdavi, “Feder- ated learning with compression: Unified analysis and sharp guarantees,” arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='01154, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Sattler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Wiedemann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' M¨uller, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Samek, “Robust and communication-efficient federated learning from non-iid data,” IEEE transactions on neural networks and learning systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3400–3413, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bouzinis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Diamantoulakis, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Karagiannidis, “Wireless quantized federated learning: A joint computation and communication design,” arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='05878, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [14] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Yan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Chen, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bao, “Federated learning with adaptive communication compression under dynamic bandwidth and unreliable networks,” Information Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 540, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 242–262, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Ma, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Hu, “Adaptive federated learning with gra- dient compression in uplink NOMA,” IEEE Transactions on Vehicular Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 16 325–16 329, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Jhunjhunwala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Gadhikar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Joshi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Eldar, “Adaptive quantization of model updates for communication-efficient federated learning,” in ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3110–3114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' H¨onig, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Zhao, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Mullins, “DAdaQuant: Doubly-adaptive quan- tization for communication-efficient federated learning,” in International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' PMLR, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 8852–8866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [18] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Horvath, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Richtarik, “Optimal client sampling for federated learning,” arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='13723, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Saad, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Yin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Poor, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Cui, “A joint learning and communications framework for federated learning over wireless networks,” IEEE Transactions on Wireless Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 269–283, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Ribero and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Vikalo, “Communication-efficient federated learning via optimal client sampling,” arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='15197, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Perazzone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Ji, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Chan, “Communication-efficient device scheduling for federated learning using stochastic optimization,” in IEEE INFOCOM 2022-IEEE Conference on Computer Communica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' IEEE, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1449–1458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Stolyar, “On the asymptotic optimality of the gradient scheduling algorithm for multiuser throughput allocation,” Operations research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 12–25, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Frank and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Wolfe, “An algorithm for quadratic programming,” Naval research logistics quarterly, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 95–110, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' LeCun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bottou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Bengio, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Haffner, “Gradient-based learning applied to document recognition,” Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 86, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2278–2324, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 9 101 NAC-FL b=1 b=2 b=3 Training Loss Fixed Error 100 10-1 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='0 Wall Clock Time 1e9101 NAC-FL b=1 b=2 b=3 Training Loss Fixed Error 100 10-1 10-2 0 1 2 3 4 5 Wall Clock Time 1e8101 Training Loss 100 10-1 NAC-FL b=1 b=2 b=3 Fixed Error 10-2 0 1 2 3 4 5 6 Wall Clock Time 1e70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 Test Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 NAC-FL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4 b=1 b=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3 b=3 Fixed Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='0 Wall Clock Time 1e90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 Test Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 NAC-FL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4 b=1 b=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3 b=3 Fixed Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 0 1 2 3 4 5 Wall Clock Time 1e80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='8 Test Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4 NAC-FL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3 b=1 b=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 b=3 Fixed Error 0 L 2 3 4 5 6 Wall Clock Time 1e7[25] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Sparling, “Honors calculus notes.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Available: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='pitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='edu/ sparling/23014/23014notes6/node24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='html [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Billingsley, Convergence of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' John Wiley & Sons, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Ethier and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Kurtz, Markov processes: characterization and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' John Wiley & Sons, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' El Gamal and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Kim, Network information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Cambridge university press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Chung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lam, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Liu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Mitzenmacher, “Chernoff- Hoeffding bounds for markov chains: Generalized and simplified,” arXiv preprint arXiv:1201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='0559, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' [30] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Levin and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Peres, Markov chains and mixing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', 2017, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 10 APPENDIX A FEDERATED LEARNING WITH ADAPTIVE COMPRESSION (FLAC) In this section, we consider a variant of the FedCOM algorithm [11], which we will call FedCOM-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' FedCOM is based on fixing a quantization parameter throughout run of the FL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' On the other hand, FedCOM-V allows for an arbitrary sequence of quantization parameters (qn)n, in order to account for adaptive compression policies such as NAC-FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' FedCOM-V is presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Algorithm 2: FedCOM-V Input : number of local computations schedule (τn)∞ n=1, local learning rate schedule (ηn)∞ n=1, adaptively chosen global learning rate schedule (γn)∞ n=1, adaptively chosen number of rounds r, initial global model w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Result: wr+1: Final model 1 for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' , r do 2 for each client j ∈ [m] do 3 Set w1,n j = wn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4 for a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' , τn do 5 Sample a minibatch Za,n j and compute ˜ga,n j ≜ ∇f(wa,n j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Za,n j ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 6 wa+1,n j = wa,n j −ηn˜ga,n j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 7 end 8 Device sends ˜gn Qj = Q((wn − wτn+1,n j )/ηn, qn j ) back to the server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 9 end 10 Server computes, ˜gn Q = 1 m �m j=1 ˜gn Qj ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 11 Server computes wn+1 = wn −ηnγn˜gn Q and broadcasts to all devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 12 end In order to study the convergence properties of FedCOM-V, we make the following standard assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 6 (Smoothness and Lower Boundedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The objective function f(·) is differentiable and L-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' That is, ∥∇f(x) − ∇f(y)∥ ≤ L ∥x − y∥, for every x, y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moreover, the optimal value of f is lower bounded, f ∗ = minw f(w) > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 7 (Bounded Variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For all clients j ∈ [m] and rounds n and local step a, we can sample an independent mini- batch Za,n j of size |Za,n j |= b and compute an unbiased stochastic gradient ˜ga,n j = ∇f(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Za,n j ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', EZa,n j [˜gj] = ∇f(wa,n j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moreover, the variance is bounded by a constant σ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', EZa,n j ���˜ga,n j − ∇f � wa,n j ���2� ≤ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Assumption 8 (Compression Model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The output of the compressor Q(x, q) is an unbiased estimator of x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', E[Q(x, q)|x] = x, and, its variance is bounded as, E[∥Q(x, q) − x∥2 |x] ≤ q ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We denote the maximum normalized-variance by qmax and the average normalized-variance used at round n by ¯qn = 1 m m � j=1 qn j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (15) The following Theorem states the relationship between (qn)n, ε and rε and is proved in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let Algorithm 2 be run with a sequence of compressors such that the average normalized-variance at round n is ¯Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, assume that the sequence � ¯Qn� n forms a stationary process with the stationary distribution represented by a random variable Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To obtain E[∥∇f(w)∥2] ≤ ε, we can choose, rε = O � log(1/ε)E �√Q + 1 � ε � , τ n = O (n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The upper bound on rε in Theorem 2 provides a justification for Assumption 1 with hε(q) = O(√q + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Here, τ n is a function of n, but for the purposes of NAC-FL we may use the average of τ (1) to τ (rε) in the expression of the duration function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' One may obtain a similar expression for other popular FL algorithms [5], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 11 APPENDIX B PROOF OF THEOREM 1 In this section we show that NAC-FL converges to the optimal solution asymptotically as β ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In order to consider the effect of β ↓ 0 on NAC-FL estimates ˆRn ε and ˆDn in (9), we shall denote these as ˆRn ε,β and ˆDn β respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let conv(Vε) be the convex hull of the set Vε defined in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Recall the positive sequence (βi)i with βi → 0 from the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Letting Xn β ≜ ( ˆRn ε,β ˆDn β)⊤, and H(x) ≜ x1x2 over the domain R2 +, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let the initialization X0 β be equal to x0 ∈ R2 + almost surely for any 0 < β < 1, then, for any s > 0, limi→∞ X⌊s/βi⌋ βi exists, is almost surely deterministic and denoted as x(s) ≜ limi→∞ X(⌊s/βi⌋) βi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, for any x0 ∈ R2 +, x(s) obeys the following differential equation, x(0) = x0, ˙x(s) = v(s) − x(s), s > 0, v(s) = argmin v∈conv (Vε) ∇H (x(s))⊤ v, s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (16) The proof of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 is very similar to that of the main result of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For the sake of completeness, we briefly show the proof at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' From hereon, (16) will be referred to as the Fluid-Frank-Wolfe (FFW) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Under Assumption 5, the FFW process in (16) has a unique fixed point x∗ ∈ conv(Vε) such that, x∗ = argmin x ∈conv (Vε) ∇H (x∗)⊤ x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moreover, x∗ ∈ Vε, and x∗ is the minimizer of H over the set Vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 is proved in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Denote, G(x) = minv∈conv(Vε) ∇H(x)⊤(v − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since, ∇H is a continuous function of x, G(x) is a continuous function of x as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As a consequence of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2, there exists a unique point x∗ ∈ conv(V )ε such that G(x∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For all other x ∈ conv(Vε), G(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We will, in fact, prove a stronger result that G(x) is bounded away from 0 for points that are a distance away from x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Claim 1: for any ω > 0, there exists a ξ > 0 such that if x ∈ conv(Vε) and ∥x − x∗∥ ≥ ω, then G(x) < −ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We prove this claim by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Suppose there exists an ω > 0 such that for all ξ > 0, the set, X ξ ≜ � xξ : xξ ∈ conv(Vε), ��xξ − x∗�� ≥ ω and G(xξ) ≥ −ξ � , = conv(Vε) � � xξ : ��xξ − x∗�� ≥ ω � � � xξ : G(xξ) ≥ −ξ � , is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' conv(Vε) is a compact set because it is the convex hull of a compact set, Vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, the sets � xξ : ��xξ − x∗�� ≥ ω � and � xξ : G(xξ) ≥ −ξ � are also closed because they are the pre-image of continuous functions over closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, X ξ is a closed set since it is the intersection of three closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, it is also bounded because conv(Vε) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, X ξ is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Consider ξ1 > ξ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since, G(x) ≥ −ξ2 implies that G(x) ≥ −ξ1, we have that X ξ1 ⊃ X ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Consider a decreasing sequence (ξi)i∈N with limi→∞ ξi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, (X ξi)i∈N is a decreasing sequence of compact and non-empty sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We know that a decreasing sequence of non empty compact sets has a limit, and the limit is non-empty [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, X 0 ≜ ∞ � i=1 X ξi, exists and is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since ξi ↓ 0, this means that any x ∈ X 0 satisfies G(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since ∥x − x∗∥ ≥ ω and x ∈ conv(Vε) for any x ∈ X 0, this is a contradiction to the fact that x∗ is a unique point in conv(Vε) with G(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, there must exist some ξ > 0 for which X ξ is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Next we proceed to study the asymptotic convergence of the process x(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Note that since Vε is apriori unknown, the initialization x0 may not be in the set Vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Nevertheless, the FFW process x(·) eventually reaches the set conv(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In order to formalize this, let convζ(Vε) denote the ζ-thickening of the set conv(Vε), convζ(Vε) = {y : ∃ x ∈ conv(Vε) such that ∥y − x∥2 ≤ ζ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 12 Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Consider the FFW process defined in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For every ζ > 0, there exists an sζ > 0 such that, x(s) ∈ convζ(Vε) for all s > sζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The proof is the same as that of Corollary 2 in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Claim 2: x(s) → x∗ as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' First we prove that lim inf s→∞ ∥x(s) − x∗∥ = 0 by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As a contradiction assume that there exists an ω > 0 and sω > 0 such that ∥x(s) − x∗∥ > ω for all s > sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let ξ > 0 be the constant according to Claim 1 which ensures that G(x) < −ξ for all x in conv(Vε) satisfying ∥x − x∗∥ > ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moreover, due to continuity of G(·), there exists a ξ′ > 0 and a small enough ζ > 0 such that G(x) < −ξ′ for all x in convζ(Vε) that satisfy ∥x − x∗∥ ≥ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3, there exists a constant sζ > 0 such that x(s) ∈ convζ(Vε) for all s > sζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Define, sω ∗ = sω + sζ + (H(x(sζ + sω)) + 1)/ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, H (x (sω ∗ )) = H(x(sζ + sω)) + � sω ∗ sζ+sω dH(x(s)), = H(x(sζ + sω)) + � sω ∗ sζ+sω ∇H(x(s))⊤ ˙x(s)ds, = H(x(sζ + sω)) + � sω ∗ sζ+sω ∇H(x(s))⊤(v(s) − x(s))ds, = H(x(sζ + sω)) + � sω ∗ sζ+sω G(x(s))ds, < H(x(sζ + sω)) + � sω ∗ sζ+sω −ξ′ds, = H(x(sζ + sω)) − H(x(sζ + sω)) − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since, H is a positive function, this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, there exists a time s > sω + sζ such that ∥x(s) − x∗∥ < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since this is true for every ω > 0 and sω > 0, we have proved that lim inf s→∞ ∥x(s) − x∗∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Next we prove that lims→∞ x(s) = x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Define, Hω = max x∈convζ(Vε) ∥x − x∗∥≤ω H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since lim inf s→∞ ∥x(s) − x∗∥ = 0, there exists a constant sω th > sζ such that ∥x(sω th) − x∗∥ ≤ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3, for all s > sω th, we have x(s) ∈ convζ(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, if for any s > sω th, H(x(s)) > Hω is true, then x(s) satisfies x(s) ∈ convζ(Vε) and ∥x(s) − x∗∥ > ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, due to Claim 1 at all such points, the gradient satisfies, dH(x(s))/ds = G(x(s)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This implies that H(x(s)) ≤ Hω for all s > sω th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moreover, by the continuity of H(·), Hω → H(x∗) as ω ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' And, by definition of the minimum x∗, H(x(s)) ≥ H(x∗) for any s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, by the Sandwich Theorem, lims→∞ H(x(s)) = H(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, by the continuity of H(·) and the uniqueness of the minimum x∗, lims→∞ H(x(s)) = H(x∗) implies that lims→∞ x(s) = x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Claim 2 proves that the Fluid-Frank-Wolfe process converges to the optimal solution x∗ asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In particular, for any ρ > 0, there exists an nth(ρ) > 0 such that, sup s>nth(ρ) ∥x(s) − x∗∥ ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Denote, xβ(s) = X⌊s/β⌋ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, since the functions converge as follows, (xβi) → x as i → ∞, from the Continous Mapping Theorem [26, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='7], we have, lim i→∞ sup s>nth(ρ) P (∥xβi(s) − x∗∥ > ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The above implies the Theorem statement, lim i→∞ sup n>nth(ρ)/βi P ���Xn βi − x∗�� > ρ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 Define the “scaled process” as, xβ(s) ≜ X⌊s/β⌋ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Denote DR2[0, ∞) as the set of functions with domain [0, ∞), range R2, and which are right continuous with left limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Observe that xβ has sample paths in DR2[0, ∞) for any 0 < β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Denote, V n β ≜ �∥hε(qn)∥ d(τ, qn, c) � , which is the action taken by the NAC-FL algorithm (Algorithm 1) at round n, and vβ(s) ≜ V ⌊s/β⌋ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Defining, K = max � x0, max q∈[0,qmax],C∈C ���� � ∥hε(q)∥ d(τ, q, C) ����� � , by the update rule of NAC-FL, xβ(s) = (1 − β) xβ(s − β) + βvβ(s − β), we have xβ(s) ≤ K for any 0 < β < 1 and s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Further, rearranging the NAC-FL update rule as, xβ(s) − xβ(s − β) = β (vβ(s − β) − xβ(s − β)) we obtain, ∥xβ(s) − xβ(s − β)∥ ≤ 2βK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' More generally, for any s1, s2 > 0, we have, ∥xβ(s1) − xβ(s2)∥ ≤ 2K max(β, |s1 − s2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This implies the “asymptotic Lipschitz” property, lim β→0 ∥xβ(s1) − xβ(s2)∥ ≤ 2K|s1 − s2|, ∀s1, s2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, by Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4 in Chapter 3 of [27], the set of stochastic processes {xβ(·)}0<β<1 is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, there exists a sequence (βi)i with βi → 0 as i → ∞ such that xβi(·) → x(·) as i → ∞ for some stochastic process x(·) with sample paths in DR2[0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Next, we need to prove that x(·) behaves according to (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To do so, observe that due to the “continuity property” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', ���Xn β − Xn−1 β ��� ≤ 2Kβ), for any δ > 0, there exists a small enough β > 0 and ∆ > 0 such that, for any integer n in the range [s/β, (s + ∆)/β], we have, | � ∇H(Xn β) �⊤ V n β − Y ∗ Cn| ≤ δ, where, Y ∗ C ≜ min q∈[0,qmax] (∇H(xβ(s)))⊤ � ∥hε(q)∥ d(τ, q, C) � , C ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The above equations say that the optimal action at any round in the considered range is very close to the optimal action at the start of the range, for an appropriate selection of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Summing across n in the range [s/β, (s + ∆)/β] we obtain, ������ � s/β≤n≤(s+∆)/β � ∇H(Xn β) �⊤ V n β − � s/β≤n≤(s+∆)/β Y ∗ Cn ������ ≤ δ∆/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Multiplying by β on both sides, from the definition of the scaled process, we have, ������ � s+∆ s (∇H(xβ(ξ)))⊤ vβ(ξ)dξ − � s/β≤n≤(s+∆)/β βY ∗ Cn ������ ≤ δ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' From the Law of Large Numbers for Markov Chains, we have limβ→0 � s/β≤n≤(s+∆)/β βY ∗ Cn = ∆ � C∈C µ(C)Y ∗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Similar to the convergence of xβ shown above, one can prove convergence of vβ to a process v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, taking limit i → ∞ along the sequence (βi)i, we get, ����� � s+∆ s (∇H(x(ξ)))⊤ v(ξ)dξ − ∆ � C∈C µ(C)Y ∗ C ����� ≤ δ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Observe that � C∈C µ(C)Y ∗ C = minv∈conv(Vε) (∇H(x(s)))⊤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, by choosing a ∆ small enough, we get, ����(∇H(x(s)))⊤ v(s) − min v∈conv(Vε) (∇H(x(s)))⊤ v ���� ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since δ can also be chosen arbitrarily small, we have, (∇H(x(s)))⊤ v(s) = min v∈conv(Vε) (∇H(x(s)))⊤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 14 APPENDIX C PROOF OF LEMMA 1 In this section we show that a state-dependent stationary policy asymptotically optimizes the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' To do so, we first define the notion of a type for sequences of network states and compression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, we show that for a given network state sequence, a policy for choosing compression parameters which depends on the sequence type optimizes the wall clock type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Finally, because the type asymptotically concentrates for markov processes, we show that a state-dependent stationary policy asymptotically optimizes the wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We start by defining the notion of an empirical distribution, called type, and its associated expectation and conditional expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Definition 1 (Type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The type of a finite sequence, x[r] ≜ (xn)r n=1 with elements in domain X, is a function, ˆp � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r]� : X → [0, 1], defined as, ˆp � x ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r]� = �r n=1 1 (xn = x) r , ∀x ∈ X, where 1(x = y) = 1 if x = y, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Similarly, the conditional type and the joint type are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Definition 2 (Joint Type and Conditional Type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The joint type of two finite sequences, x[r] and y[r] with domains X and Y respectively, is a function, ˆp � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]� : X × Y → [0, 1], defined as, ˆp � x, y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]� = �r n=1 1 (xn = x , yn = y) r , ∀x ∈ X, y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The conditional type ˆp � |· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]� : X × Y → [0, 1] is defined as, ˆp � x|y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]� = �r n=1 1 (xn = x , yn = y) �r n=1 1 (yn = y) , ∀x ∈ X, y ∈ Y such that ˆp(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' y[r]) > 0, = ˆp � x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]� ˆp � y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' y[r]� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, the expectation and conditional expectation with respect to the type may be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Definition 3 (Expectation and Conditional Expectation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The expectation of a non-negative function g : X → R+ with respect to type ˆp(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r]) is defined as1, ˆE � g(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r]� ≜ � x∈X g(x)ˆp(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r]), where X denotes a random variable with distribution ˆp(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Similarly, the conditional expectation of a non-negative function l : X → R+ with respect to the type ˆp(·|·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]) is defined as, ˆE � l(X)|Y = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]� ≜ � x∈X l(x)ˆp(x|y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]), ∀y ∈ Y such that ˆp(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' y[r]) > 0, where the random variable pair (X, Y ) has joint distribution ˆp(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x[r], y[r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Suppose Assumptions 1 and 3 hold, and let (cn)n denote an observed sequence of network states and (qn) denote a sequence of compression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, for any positive integer r and positive ε with the associated function hε(·) defined in Assumption 1, there exists a sequence dependent, state dependent stationary policy π such that, r � n=1 ∥hε (qn)∥ ≥ r � n=1 ∥hε (π (cn))∥ , (17) and, r � n=1 d (τ, qn, cn) ≥ r � n=1 d (τ, π (cn) , cn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (18) 1If X is uncountably infinite, then, � x∈X g(x) ≜ sup �� x∈F g(x) : F ⊂ X, F is finite � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 15 Proof of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Given the sequence (qn, cn)r n=1, we obtain the joint type p(·, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Thus, one may interpret the sequence as given an observed network state c, the policy plays the compression parameters q with probability ˆp(q|c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Define the state-dependent stationary policy π as playing the conditional mean (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', the function hε) given any network state c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' That is, π(c) = h−1 ε � ˆE � hε(Q)| C = c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]�� , ∀ c ∈ C such that ˆp(c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' c[r]) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (19) Such a choice for π(c) always exists because hε(·) is continuous, bounded and strictly increasing coordinate-wise applied function which implies that the inverse operator of hε(·) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Note that hε(π(c)) = ˆE � hε(Q)| C = c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' So, due to the convexity of ∥·∥, ∥hε (π(c))∥ ≤ ˆE � ∥hε (Q)∥ |C = c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]� , ∀ c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (20) Then, r � n=1 ∥hε (π(cn))∥ = rˆE � ∥hε(π(C))∥ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' c[r]� , (a) ≤ rˆE � ˆE � ∥hε(Q)∥ |C = c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' c[r]� , (b) = rˆE � ∥hε(Q)∥ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r]� , = r � n=1 ∥hε (qn))∥ , (21) where (a) follows from (20), and (b) follows from the Tower-rule of expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Next we bound d(τ, π(c), c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' By the definition of π in (19), for all c ∈ C, d (τ, π(c), c) = d � τ, h−1 ε (hε(π(c))) , c � , (a) = d � τ, h−1 ε � ˆE � hε(Q)| C = c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]�� , c � , (b) ≤ ˆE � d � τ, h−1 ε (hε(Q)) , c � |C = c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]� , = ˆE � d (τ, Q, c) |C = c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]� , (22) where (a) follows from the definition of policy π, and (b) follows from the convexity of d(τ, h−1 ε (·), c) (Assumption 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (22) is analogous to (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, we may repeat the same calculation in (21) for the delay, r � n=1 d(τ, π(cn), cn) = rˆE � d(τ, π(C), C) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' c[r]� , ≤ rˆE � ˆE � d(τ, Q, C)|C = c ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' c[r]� , = rˆE � d(τ, Q, C) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' q[r], c[r]� , = r � n=1 d(τ, qn, cn), Equation (17) in Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 states that if the FL algorithm with a sequence of compression parameters (qn)n has reached an error tolerance of ε by round r, then, under Assumption 1, it has also reached error tolerance ε under sequence of compression parameters (π(cn))n by around r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moreover, (18) states that (π(cn))n takes lesser amount of time up to round r compared to (qn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' However, this construction of π is sequence dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' More specifically, it is dependent on the type ˆp(· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' c[r]) of the network state sequence observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In order to prove Lemma 1, we need to construct a state-dependent but sequence-independent stationary policy that is near-optimal in minimizing the expected wall clock time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, in the following, we first define the notion of a typical set and show in Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 that the type of an observed network state sequence concentrates around its mean with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Definition 4 (Typical Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For a distribution p on network sets, a typical set with parameters (r, ν), is defined as, T r ν (p) ≜ {cr : |ˆp(c|cr) − p(c)| ≤ νp(c), for all c ∈ C} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 16 We will use the following result called the Typical Averaging Lemma for typical sets [28, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let cr ∈ T r ν (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, for any non-negative function g : C → R+, (1 − ν) E [g(C)] ≤ 1 r r � n=1 g (cn) ≤ (1 + ν) E [g(C)] , where C is a random variable with distribution p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Next, due to ergodicity of stationary Markov chains, we have the following proposition which is proved at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let Assumption 4 be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, there exist positive constants κ1 and κ2 such that, for every ν > 0, and r ∈ N, P � ∃r′ ≥ r such that Cr′ ̸∈ T r′ ν (µ) � ≤ κ1 exp � −κ2ν2r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 will be used to argue that if two network-state sequences have similar types (they belong to T r ν (µ)), then a state dependent stationary policy π will have a similar expected wall clock to converge under both sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 will be used to argue that one observes a typical network state sequence with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We proceed to prove this formally in the following proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Denote hmin ε and hmax ε as the minimum and maximum of the bounded function hε(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, Assumption 1 implies that the number of rounds needed to converge to an error tolerance ε under any sequence of compression parameters is bounded between hmin ε and hmax ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For a positive ν, let ε be small enough such that hmin ε > 2(1 + ν)/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' First, we will consider network state sequences which are typical with repsect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Specifically, we consider a sequence (cn)n such that cr belongs to T r ν (p) for every r > hmin ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1, there exists a state-dependent stationary policy that optimizes the wall clock time to reach error tolerance ε with respect to the sequence (cn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let π′ represent this policy, and rπ′ ε be the minimum number of rounds taken to converge by π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, the wall clock time for π′ can be lower bounded as, rπ′ ε � n=1 d (τ, π′ (cn) , cn) = � � 1 rπ′ ε rπ′ ε � n=1 d (τ, π′ (cn) , cn) � � rπ′ ε , (a) ≥ (1 − ν) E [d(τ, π′(C), C)] rπ′ ε , (b) ≥ (1 − ν) E [d(τ, π′(C), C)] � � 1 rπ′ ε rπ′ ε � n=1 ∥hε (π (cn))∥ � � , (c) ≥ (1 − ν)2 E [d(τ, π′(C), C)] E [∥hε (π′(C))∥] , (d) ≥ (1 − ν)2 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (23) (a) and (c) follow from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1, (b) follows from Assumption 1, and (d) follows by the following definition, π∗ = arg min π ∈Π E [d(τ, π(C)), C)] E [∥hε (π(C))∥] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Performing a similar calculation for π∗, rπ∗ ε� n=1 d (τ, π∗ (cn) , cn) = � � 1 rπ∗ ε rπ∗ ε� n=1 d (τ, π∗ (cn) , cn) � � rπ∗ ε , (a) ≤ (1 + ν) E [d(τ, π∗(C), C)] rπ∗ ε , (b) ≤ (1 + ν)2 E [d(τ, π∗(C), C)] � � 1 rπ∗ ε rπ∗ ε� n=1 ∥hε (π∗ (cn))∥ � � , (c) ≤ (1 + ν)3 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (24) 17 (a) and (c) follow from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 and (b) follows from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3 proved at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The expected wall clock time to reach error tolerance ε under the state-dependent stationary policy π∗ can be upper bounded as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' E � T π∗ ε � (a) ≤ P � CRπ∗ ε ∈ T Rπ∗ ε ν (µ) � (1 + ν)3 E [d(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' π∗(C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' C)] E [∥hε (π∗(C))∥] + P � CRπ∗ ε ̸∈ T Rπ∗ ε ν (µ) � hmax ε dmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (b) ≤ (1 + ν)3 E [d(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' π∗(C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' C)] E [∥hε (π∗(C))∥] + P � ∃r′ ≥ hmin ε ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Cr′ ̸∈ T r′ ν (µ) � hmax ε dmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (c) ≤ (1 + ν)3 E [d(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' π∗(C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' C)] E [∥hε (π∗(C))∥] + κ1 exp � −κ2ν2hmin ε � hmax ε dmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (25) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (a) follows by using (25) for typical sequences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' and upper bounding Rπ∗ ε by hmax ε and the round duration by dmax for non-typical sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (b) follows by upper bounding P � CRπ∗ ε ∈ T Rπ∗ ε ν (µ) � by 1, and upper bounding P � CRπ∗ ε ̸∈ T Rπ∗ ε ν (µ) � by P � ∃r′ ≥ hmin ε , Cr′ ̸∈ T r′ ν (µ) � because, almost surely, Rπ∗ ε ≥ hmin ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (c) follows from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let T ∗ ε be the random variable representing the wall-clock time to reach error tolerance ε when one uses the optimal sample- path sequence dependent policy on random network-state sequence (Cn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, denoting R∗ ε as the corresponding random variable denoting the number of rounds needed to reach error tolerance ε, we have, E [T ∗ ε ] (a) ≥ P � CR∗ ε ∈ T R∗ ε ν (µ) � (1 − ν)2 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] , (b) ≥ P � ∀r′ ≥ hmin ε , Cr′ ∈ T r′ ν (µ) � (1 − ν)2 E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] , (c) ≥ (1 − ν)2 � 1 − κ1 exp � −κ2ν2hmin ε �� E [d(τ, π∗(C), C)] E [∥hε (π∗(C))∥] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (26) (a) follows due to (23), and (b) follows since, almost surely, R∗ ε ≥ hmin ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (c) follows from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to Assumption 2, both hmin ε and hmax ε are Θ(1/poly(ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' And, ν can be made as small as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, from (25) and (26), E � T π∗ ε � → E [T ∗ ε ] as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Under Assumptions 1 and 2, for ν > 0, if ε is small enough such that hmin ε > 2(1 + ν)/ν, then, for any state-dependent stationary policy π, the minimum number of rounds rπ ε to reach an error tolerance ε is such that, rπ ε ≤ (1 + ν) 1 rπ ε rπ ε � n=1 ∥hε (qn)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to Assumption 1, rπ ε satisfies, rπ ε > 1 rπ ε rπ ε � n=1 ∥hε (qn)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Moreover, since rπ ε is the earliest round at which error tolerance ε is reached, due to Assumption 1, round rπ ε − 1 satisfies, rπ ε − 1 ≤ 1 rπ ε − 1 rπ ε −1 � n=1 ∥hε (qn)∥ , ≤ 1 rπ ε − 1 rπ ε � n=1 ∥hε (qn)∥ , = � rπ ε rπ ε − 1 � 1 rπ ε rπ ε � n=1 ∥hε (qn)∥ , =⇒ rπ ε ≤ � rπ ε rπ ε − 1 �2 1 rπ ε rπ ε � n=1 ∥hε (qn)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (27) 18 Now, we upper bound rπ ε /(rπ ε − 1), rπ ε rπ ε − 1 = 1 + 1 rπ ε − 1, (a) < 1 + ν 2 + ν , = 1 + ν 1 + ν/2, (b) ≤ √ 1 + ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (28) (a) follows by rearranging the assumption rπ ε > 2(1+ν)/ν to obtain rπ ε −1 > (2+ν)/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (b) follows since 1+ν/2 ≥ √1 + ν for ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Substituting (28) in (27) completes the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In order to prove Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2, we use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 from [29] which we re-state below for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let Assumption 4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Define rmix as the 1/8 mixing time2 of the Markov chain (Cn)n and f : C → [0, 1] be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let, µf ≜ E � f(C(1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, there exists a constant κc such that for every 0 ≤ ν ≤ 1 and r ∈ N, P ������ 1 r r � n=1 f(Cn) − µf ����� ≥ νµf � ≤ κc exp � −ν2µfr 72rmix � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For some c ∈ C, define f(c′) = 1(c′ = c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, due to Theorem 3, there exist a constant κc such that, P � � ������ 1 r′ r′ � n=1 1(Cn = c) − µ(c) ������ ≥ νµ(c) � � ≤ κc exp � −ν2µ(c)r′ 72rmix � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (29) Denote, κ ≜ � c∈C κc and µmin = minc∈C µ(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since the Markov chain is irreducible, µmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, using (29) and taking a union bound over all c ∈ C, we obtain, P � Cr′ ̸∈ T r′ ν (µ) � ≤ κ exp � −ν2µminr′ 72rmix � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Define κ2 ≜ µmin/(72rmix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Taking a further union bound over all r′ ≥ r, P � ∃r′ ≥ r such that Cr′ ̸∈ T r′ ν (µ) � ≤ κ 1 − exp (−κ2ν2) exp � −κ2ν2r � Defining κ1 ≜ κ/(1 − exp(−κ2ν2)) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2Denoting M as the transition-matrix of the Markov chain, and µ as its stationary distribution, rmix ≜ maxψ � r : ∥Mrψ − µ∥T V ≤ 1/8 � , where ∥·∥T V denotes the TV-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='9 of [30], rmix is finite for an aperiodic and irreducible Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 19 APPENDIX D PROOF OF LEMMA 2 Here we prove Lemma 2 which states that the expected wall clock is approximately equal to the product of the expected number of rounds and the expected round duration asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The proof is very similar to the proof of Lemma 1 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As such, we will use the notation introduced in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Denote hmin ε and hmax ε as the minimum and maximum of the bounded function hε(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' And, let dmin and dmax denote the minimum and maximum of the positive, bounded function d(·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Similar to the proof of Lemma 1, we consider network state sequences which are typical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In order to define the parameters for the typical set, for any given δ > 0, we choose a small enough εth > 0 and δ′ > 0 such that, 1) εth is small enough such that hmin εth > 2(1 + δ′)/δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 2) δ′ > 0 is such that for all 0 < ε ≤ εth, (1 − δ) < (1 − δ′)2 � 1 − κ1 exp � −κ2(δ′2hmin ε ) �� , and, max � � �(1 + δ′)3, 1 + κ1 exp � −κ2δ′2hmin ε � hmax ε dmax hmin ε dmin � � � < (1 + δ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Such a choice of δ′ is possible because hmin ε = Θ(1/poly(ε)), and exp(−κ2δ′2hmin ε )hmax ε /hmin ε = exp(−Ω(δ′2/poly(ε))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We consider a sequence (cn)n such that cr ∈ T r δ′(µ) for every r ≥ hmin εth .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For a policy π, let rπ ε denote the minimum number of rounds needed to converge to error tolerance ε < εth given network state sequence (cn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, the wall clock time for π can be lower bounded as, rπ ε � n=1 d (τ, π (cn) , cn) = � � 1 rπ ε rπ ε � n=1 d (τ, π (cn) , cn) � � rπ ε , (a) ≥ (1 − δ′) E [d(τ, π(C), C)] rπ ε , (b) ≥ (1 − δ′) E [d(τ, π(C), C)] � � 1 rπ ε rπ ε � n=1 ∥hε (π (cn))∥ � � , (c) ≥ (1 − δ′)2 E [d(τ, π(C), C)] E [∥hε (π(C))∥] , (30) (a) and (c) follow from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1 since rπ ε > hmin εth , and (b) follows from Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Performing a similar calculation for the upper bound, rπ ε � n=1 d (τ, π (cn) , cn) = � � 1 rπ ε rπ ε � n=1 d (τ, π (cn) , cn) � � rπ ε , (a) ≤ (1 + δ′) E [d(τ, π(C), C)] rπ ε , (b) ≤ (1 + δ′)2 E [d(τ, π(C), C)] � � 1 rπ ε rπ ε � n=1 ∥hε (π (cn))∥ � � , (c) ≤ (1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (31) (a) and (c) follow from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='1, and (b) follows from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 20 Let Rπ ε be the random variable denoting the minimum number number of rounds needed to converge to an error tolerance ε < εth when policy π is used on the sequence (Cn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The expected wall clock times of π can be lower bounded as, E [T π ε ] ≥ P � CRπ ε ∈ T Rπ ε δ′ (µ) � (1 − δ′)2 E [d(τ, π(C)), C)] E [∥hε (π(C))∥] , (a) ≥ P � ∀r′ ≥ hmin ε , Cr′ ∈ T r′ ν (µ) � (1 − δ′)2 E [d(τ, π(C), C)] E [∥hε (π(C))∥] (b) ≥ (1 − δ′)2 � 1 − κ1 exp � −κ2δ′2hmin ε �� E [d(τ, π(C), C)] E [∥hε (π(C))∥] (c) ≥ (1 − δ) E [d(τ, π(C), C)] E [∥hε (π(C))∥] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (32) (a) follows since Rπ ε ≥ hmin ε almost surely, (b) follows from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 and (c) follows from the choice of δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The expected wall clock times of π can be upper bounded as, E [T π ε ] (a) ≤ P � CRπ ε ∈ T Rπ ε δ′ (µ) � (1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] + P � CRπ ε ̸∈ T Rπ ε δ′ (µ) � hmax ε dmax, (b) ≤ (1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] + P � ∃r′ ≥ hmin ε Cr′ ̸∈ T r′ δ′ (µ) � hmax ε dmax, (c) ≤ (1 + δ′)3 E [d(τ, π(C), C)] E [∥hε (π(C))∥] + κ1 exp � −κ2δ′2hmin ε � hmax ε dmax, (d) ≤ (1 + δ) E [d(τ, π(C), C)] E [∥hε (π(C))∥] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (33) (a) follows by using (31) to upper bound the wall clock time for typical sequences, and using the worst-case upper bound hmax ε dmax for non-typical sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (b) follows because, almost surely, Rπ ε ≥ hmin ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (c) follows from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (d) follows from the choice of δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' (32) and (33), jointly, conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 21 APPENDIX E PROOF OF PROPOSITION B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 In this section, we prove Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 which states that the Fluid-Frank-Wolfe (FFW) process has a unique stationary point in the set conv(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' It further states that the stationary point lies in the set Vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In order to demonstrate the main arguments of the proof, we will first consider the case with a single client (m = 1) and a single network state, C = {c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Later, in Subsection B, we will generalize these arguments to complete the proof of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Warmup: 1 Client and 1 Network State (1C1NS) Case Recall that the set Vε is the set of pairs of achievable expected number of rounds ˆrε and expected round duration ˆd of state-dependent stationary policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Here, we observe that conv(Vε) may be interpreted as the corresponding feasibility set for (possibly random) stationary policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We start by making this observation precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In the 1C1NS case, a stationary policy may be represented by a one-dimensional random variable Π that denotes the possibly randomly selected compression parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The space of possible stationary policies is denoted by the set of distributions Q1, Q1 = {fΠ : fΠ is a distribution over [0, qmax]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Under a policy corresponding to Π and a small enough error tolerance ε, due to Lemma 2, the expression for the expected wall clock time for stationary policies is given by, E[T Π ε ] ≈ ˆtΠ ε = E [hε(Π)] E [d(τ, Π, c)] , where T Π ε is the wall clock time to reach error tolerance ε under compression parameter Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, the feasible set conv(Vε) for stationary policies may be written for this case as, conv(Vε) = � (ˆrε, ˆd) : ∃fΠ ∈ Q1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', Π ∼ fΠ satisfies ˆrε = E [hε(Π)] , ˆd = E [d(τ, Π, c)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Here, stationary policies may be separated into two categories, deterministic policies: here, Π is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' stochastic policies: here, Π is non-deterministic random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Our aim is to prove that there exists a unique fixed-point of the FFW update in conv(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As we will see, this will prove Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='2 for the special case of 1 client and 1 network state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Before delving into the proof of this statement, we state a couple of properties of conv(Vε) which are useful in proving the existence of a unique fixed-point of the FFW update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For any hmin ε ≤ h ≤ hmax ε , there exists a deterministic policy that minimizes ˆtΠ ε given a constraint E[hε(Π)] = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' As a contradiction, assume that there is no deterministic policy that minimizes the expected wall clock time given a constraint E [hε(Π)] = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Let Π∗ be a stochastic policy that minimizes the expected wall clock time with the constraint E[hε(Π)] = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, consider an alternate policy with deterministic compression parameter π chosen as, hε(π) = E [hε(Π∗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Such a π exists due to the Intermediate Value Theorem since hε(·) is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In this case, by the strict convexity of the duration function assumed in Assumption 3, we have, E [d(τ, π, c)] < E [d (τ, Π∗, c)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, the relation between their expected wall clock times is, ˆtπ ε = E [hε(π)] E [d(τ, π, c)] < E [hε(Π∗)] E [d (τ, Π∗, c)] = ˆtΠ∗ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' This is a contradiction to the assumption that a stochastic policy minimizes the expected wall clock time given the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For notational brevity, denote, ¯d(ˆrε) = d(τ, h−1 ε (ˆrε), c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Recall that we may denote points (ˆrε, ˆd) ∈ conv(Vε) by a two-dimensional vector x = (ˆrε ˆd)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Also, recall the function H(x) = x1x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The following proposition states several equivalent ways of describing a point x in the set Vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The following statements are equivalent I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x ∈ conv(Vε) is of the form (ˆrε, ¯d(ˆrε)) for some hmin ε ≤ ˆrε ≤ hmax ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x ∈ conv(Vε) is such that α x ̸∈ conv(Vε) for any 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x = (ˆrε, ˆd) ∈ conv(Vε) is such that, ˆd = min{d′ : (ˆrε, d′) ∈ conv(Vε)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x = (ˆrε, ˆd) ∈ conv(Vε) is such that, ˆrε ˆd = min{ˆrεd′ : (ˆrε, d′) ∈ conv(Vε)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4: Feasibility set for stationary policies conv(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' In the 1C1NS case, a point (ˆrε, ˆd) ∈ conv(Vε) corresponds to a policy Π such that, ˆrε = E[hε(Π)] and ˆd = E[d(τ, Π, c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The function ¯d(ˆrε) is represented by the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' x is an extreme point3 of conv(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' The equivalences may be inferred from the structure of the feasibility set conv(Vε) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' We briefly describe the arguments required to prove the equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' I ⇐⇒ III because (ˆrε, ¯dε) corresponds to a deterministic policy by definition, and, due to part b, a deterministic policy minimizes the wall clock time ˆrε ˆd amongst the set of policies {fΠ ∈ Q1 : s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=', Π ∼ fΠ satisfies E[hε(Π)] = ˆrε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' I, III ⇐⇒ II because ¯d(ˆrε) is a strictly decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' I ⇐⇒ V because ¯d is a strictly convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' III and IV are trivially equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' From here on, we will call points of the form described in Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5 as extreme points, and all other points in conv(Vε) as non-extreme points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Note that in the 1C1NS case, the set of extreme points is equal to the set Vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' However, we refrain from using this fact here because in the general case of multiple clients and multiple network states, this is no longer true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Next, we prove the existence of a unique fixed point of the FFW update in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' First, we show that a non-extreme point cannot be a fixed point of the FFW update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' If a point x ∈ conv(Vε) is not an extreme point of conv(Vε), then it is not a fixed-point of the FFW update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='5, x being a non-extreme point implies that there exists a constant 0 < α < 1 such that α x ∈ conv(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Observe that ∇H(x) = (x2 x1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since, hε() and d() are non-negative functions, we have that ∇H(x) has non-negative entries for any x ∈ Vε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Recall that x is a fixed-point of the FFW update if and only if x = arg min x′ ∈conv (Vε) ∇H(x)⊤ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to the elementwise non-negativity of ∇H, ∇H(x)⊤(α x) < ∇H(x)⊤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Since α x ∈ conv(Vε), x is not a fixed point of the FFW-update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='6, we focus on only extreme points in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Before proving the existence of a unique fixed point of the FFW update amongst the set of extreme points, we state a result about an equivalent description of a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' For this purpose, define ˆt(ˆrε) = ˆrε ¯d(ˆrε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Under Assumption 5, an extreme-point x = � ˆrε, ¯d(ˆrε) �⊤ with hmin ε < ˆrε < ˆhmax ε is a fixed point for the FFW-update if and only if ˆt′(ˆrε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' First, as a contradiction, assume that there exists an ˆrε with ˆt′(ˆrε) = 0, but x = � ˆrε ¯d(ˆrε) �⊤ is not a fixed point of the FFW update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Recall that x is a fixed point of the FFW update if x = arg min x′ ∈conv (Vε) ∇H(x)⊤ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Therefore, this implies that there exists a different point z = (ˆr′ ¯d(ˆr′))⊤ in conv(Vε) such that, ∇H(x)⊤(z − x) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 5 for an illustration of the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Due to strict convexity of the curve ¯d(·), there exists a point w = (ˆr ¯d(ˆr′))⊤ with ˆr being in between ˆrε and ˆr′ such that ∇H(x)⊤(w − x) = −ξ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' Then, for any 0 < θ < 1, denote 3A point x ∈ conv(Vε) is called an extreme point if it cannot be written as the convex combination of two other points in conv(Vε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE3T4oBgHgl3EQfSQko/content/2301.04430v1.pdf'} +page_content=' 23 푟0), log(sin(푖)) ∈ [−3, 0], log(푞) ∈ [−1, 1], log(휎퐿) ∈ +[2, 4], +log(푒) ∈ [−5, 0], log(휙0) ∈ [−5, +log(2 × 휋)]. The +MNRAS 000, 1–6 (2022) + +4 +Zhang +Table 1. parameters of the emission line components +model parameters of elliptical accretion disk model for broad H훼 +푟0 = 2035 ± 240, 푟1 = 3766 ± 500, sin(푖) = 0.71 ± 0.04 +푞 = 3.35 ± 0.19, 푒 = 0.81 ± 0.08, 휎퐿 = 796 ± 70km/s, 휙0 = 190 ± 6◦ +model parameters of Gaussian emission components +line +휆0 +휎 +flux +broad H훼 +6505.6±1.1 +41.4±1.2 +897±25 +6643.9±1.1 +34.9±1.2 +699±24 +Narrow H훼 +6564.2±0.5 +5.6±0.6 +311±54 +Narrow H훽 +4862.4±0.3 +4.2±0.4 +45±8 +[O iii]휆5007Å +5008.8±0.3 +3.9±0.3 +172±10 +[O i]휆6300Å +6301.9±1.1 +7.6±1.2 +113±14 +[N ii]휆6583Å +6585.5±0.2 +6.5±0.3 +642±55 +[S ii]휆6716Å +6719.2±0.9 +6.6±0.9 +260±33 +[S ii]휆6731Å +6734.5±0.8 +4.6±0.7 +157±29 +Notice: For the Gaussian emission components, the first column shows which +line is measured, the Second, third, fourth columns show the measured line +parameters: the center wavelength 휆0 in unit of Å, the line width (second +moment) 휎 in unit of Å and the line flux in unit of 10−17 erg/s/cm2. +determined best fitting results and corresponding residuals to the +emission line around H훼 are shown in right panels of Fig. 2 with +휒2/푑표 푓 ∼ 0.48. The MCMC technique determined posterior dis- +tributions of the model parameters in the elliptical accretion disk +model are shown in Fig. 3. And the half width at half maximum of +each parameter distribution is accepted as uncertainty of the param- +eter. The determined parameters and corresponding uncertainties of +each model parameter are listed in Table 1. Moreover, as discussed +in Zhang (2022a), clean double-peaked broad line emission features +can lead to solely determined model parameters in the elliptical ac- +cretion disk model. Therefore, there are no further discussions on +whether is there solely determined model parameter of sin(푖). +4 MAIN DISCUSSIONS +In the section, two points are mainly considered. First, it is neces- +sary to determine that the accretion disk origination is favoured to +explain the double-peaked broad H훼 in SDSS J1607+3319, rather +than a BBH system. Second, it is necessary to determine that the +large broad Balmer decrement (flux ratio of broad H훼 to broad +H훽) is due to serious obscurations, rather than due to local phys- +ical conditions, because that BLRs modeled with relatively low opti- +cal depths and low ionization parameters can reproduce large broad +Balmer decrements, as well discussed in Kwan & Krolik (1981); +Canfield & Puetter (1981); Goodrich (1990) without considering se- +rious obscurations and see the unobscured central regions in a Type- +1.9 AGN in Barcons et al. (2003). +For the first point on BBH system, the following discussions are +given. The double-peaked broad H훼 can also be well described by +two broad Gaussian functions shown as dashed purple lines in top +right panel of Fig. 2 with model parameters listed in Table 1. Under +the assumption of BBH system in SDSS J1607+3319, considering +the strong linear correlation between broad H훼 luminosity and con- +tinuum luminosity as discussed in Greene & Ho (2005), there are to- +tally equal (ratio about 897:699 from emission fluxes of the two broad +Figure 4. CSS V-band light curve of SDSS J1607+3319. Horizontal solid +and dashed red lines show the mean value and corresponding 2RMS scatters +of the light curve. +Gaussian components) continuum luminosities related to central two +BH accreting systems, indicating there should be strong variabilities +with QPOs due to orbital rotating effects. However, there are none +variabilities in the collected 8.4years-long CSS (Catalina Sky Survey, +Drake et al. (2009)) V-band light curve shown in Fig. 4 with almost +all data points lying within 2RMS scatter ranges. Therefor, rather +than the BBH system, the elliptical accretion disk model is preferred +to explain the double-peaked broad H훼 in SDSS J1607+3319. +For the second point, properties of virial BH mass are mainly +discussed. Based on accepted virialization assumptions to prop- +erties of observed broad H훼 as discussed in Vestergaard (2002); +Peterson et al. (2004); Greene & Ho (2005); Shen et al. (2011); +Mejia-Restrepo et al. (2022), virial BH mass can be estimated by +푀퐵퐻 = 15.6 × 106( +퐿퐻 훼 +1042erg/s)0.55( +휎퐻 훼 +1000km/s )2.06M⊙ += (5.5 ± 0.6) × 107M⊙ +(1) +with 퐿퐻 훼 = (1.39 ± 0.05) × 1041erg/s as line luminosity of ob- +served broad H훼 and 휎퐻 훼 = (3100 ± 110)km/s as second mo- +ment of observed broad H훼, after considering more recent em- +pirical R-L relation to estimate BLRs sizes in Bentz et al. (2013). +Uncertainty of virial BH mass is determined by uncertainties of +the 퐿퐻 훼 and 휎퐻 훼. If large broad Balmer decrement was due to +local physical conditions, the estimated virial BH mass should be +simply consistent with the 푀BH − 휎 relation (Ferrarese & Merritt +2000; Gebhardt et al. 2000; Kormendy & Ho 2013; Batiste et al. +2017; Bennert et al. 2021) expected value, otherwise, there should +be smaller virial BH mass. Then, Fig. 5 shows virial BH mass prop- +erties of SDSS J1607+3319 in the 푀BH − 휎 space. In order to show +clearer results, the 89 quiescent galaxies from Savorgnan & Graham +(2015) and the 29 reverberation mapped (RM) AGN from Woo et al. +(2015) and the 12 tidal disruption events (TDEs) from Zhou et al. +(2021) are considered to draw the linear correlation between stellar +velocity dispersion and BH mass +log( 푀퐵퐻 +M⊙ +) = (−2.89 ± 0.49) + (4.83 ± 0.22) × log( 휎★ +km/s) +(2) +through +the +Least +Trimmed +Squares +robust +technique +(Cappellari et al. 2013). And then the 3휎, 4휎 and 5휎 confi- +dence bands to the linear correlation are determined and shown +in Fig. 5. Therefore, the estimated viral BH mass of SDSS +J1607+3319 is lower than 푀BH − 휎 expected value with confidence +level higher than 4휎. Therefore, locate physical conditions are +MNRAS 000, 1–6 (2022) + +Opening angle of Dust Torus +5 +Figure 5. On the correlation between stellar velocity dispersion measured +through absorption features and virial BH mass of SDSS J1607+3319. +Solid five-point-star in dark green shows the virial BH mass of SDSS +J1607+3319 determined by properties of observed broad H훼. Dot-dashed +lines in magenta and in black represent the 푀BH − 휎 relations through the +quiescent galaxies in Kormendy & Ho (2013) and through the RM AGNs in +Woo et al. (2015), respectively. Solid circles in red, in blue and in pink show +the values for the 89 quiescent galaxies in Savorgnan & Graham (2015), the +29 RM AGNs in Woo et al. (2015) and the 12 TDEs in Zhou et al. (2021), re- +spectively. Thick solid red line shows the best fitting results to all the objects, +and thick dashed, dotted and dot-dashed red lines show corresponding 3휎, +4휎 and 5휎 confidence bands to the best fitting results. +disfavored to explain the large broad Balmer decrement in SDSS +J1607+3319. +Based on the double-peaked broad H훼 in the Type-1.9 DPAGN +SDSS J1607+3319, half opening angle of central dust torus is well +estimated as (46±4)◦ (sin(푖) ∼ 0.71 ± 0.04), roughly consistent with +statistical mean value in Zhuang et al. (2018). Therefore, it is inter- +esting to study properties of opening angles of dust torus through +Type-1.9 DPAGN in the near future, after many efforts to disfavour +BBH systems to explain their double-peaked broad H훼 and to dis- +favour local physical conditions to explain disappearance of broad +H훽. +Before ending of the manuscript, an additional point is noted. +Before giving clear physical information of materials in the central +dust torus, it is hard to confirm that the accretion disk origination +determined inclination angle is completely consistent with the half +opening angle of the central dust torus in Type-1.9 DPAGN. If ma- +terial densities in regions around upper boundary of the central dust +torus were too low to lead the broad H훽 being totally obscured, the +determined inclination angle should be lower than the intrinsic half +opening angle of the central dust torus. Moreover, it is not clear +whether are there different radial dependent material densities in the +direction perpendicular to the equatorial plane related to central AGN +activities, which should also have effects on the consistency between +the accretion disk origination determined inclination angle and the +half opening angle of the central dust torus in AGN with different +central AGN activities. In the near future, through studying a sample +of Type-1.9 DPAGN as one of our ongoing projects, clearer clues +and detailed discussions will be given on the consistency between +the inclination angle and the half opening angle of the central dust +torus. +5 CONCLUSIONS +An independent method is proposed to estimate the opening an- +gle of the central dust torus in Type-1.9 DPAGN through unique +double-peaked features of broad H훼, accepted the assumptions of +obscurations of the central dust torus on BLRs leading to disappear- +ance of broad H훽 and of the double-peaked broad H훼 with accre- +tion disk originations. Then, among the reported DPAGN, the SDSS +J1607+3319 is collected due to its apparent broad double-peaked +broad H훼 but no broad H훽. Moreover, long-term optical variabilities +can be applied to disfavour the BBH system in SDSS J1607+3319 to +explain the double-peaked broad H훼. And properties of virial BH +mass can be applied to determine that local physical conditions are +not favoured to explain the large broad Balmer decrement in SDSS +J1607+3319. Then, based on the well applied elliptical accretion +disk model applied to describe the double-peaked broad H훼 in SDSS +J1607+3319, the half opening angle of the central dust torus can be +well estimated as (46±4)◦ in SDSS J1607+3319. The results in the +manuscript strongly indicate that the proposed independent method +is practicable, and can be applied to study detailed properties of the +opening angles of the central dust torus through a sample of Type-1.9 +DPAGN, which will be studied in the near future. +ACKNOWLEDGEMENTS +Zhang +gratefully +acknowledges +the +anonymous +referee +for +giving us constructive comments and suggestions to greatly +improve our paper. Zhang gratefully acknowledges the kind +funding +support +NSFC-12173020. +This +research +has +made +use of the data from the SDSS (https://www.sdss.org/) +funded by the Alfred P. Sloan Foundation, the Participating +Institutions, the National Science Foundation and the U.S. De- +partment of Energy Office of Science, and use of the data from +CSS +http://nesssi.cacr.caltech.edu/DataRelease/. +The +research +has +made +use +of +the +MPFIT +package +https://pages.physics.wisc.edu/~craigm/idl/cmpfit.html, +and +of +the +LTS_LINEFIT +package +https://www-astro.physics.ox.ac.uk/~cappellari/software/, +and of the emcee package https://pypi.org/project/emcee/. +DATA AVAILABILITY +The data underlying this article will be shared on request to the +corresponding author (aexueguang@qq.com). +REFERENCES +Antonucci, R., 1993, ARA&A, 31, 473 +Almeida, C. R., Ricci, C., 2017, Nat Astron, 1, 679 +Alonso-Herrero, A.; Ramos Almeida, C.; Mason, R., et al., 2011, ApJ, 736, +82 +Arshakian, T. G., 2005, A&A, 436, 817 +Batiste, M.; Bentz, M. C.; Raimundo, S. I.; Vestergaard, M.; Onken, C. A., +2017, ApJL, 838, 10 +Bennert, V. N.; Treu, T.; Ding, X.; et al., 2021, ApJ, 921, 36 +Barcons, X.; Carrera, F. J.; Ceballos, M. T., 2003, MNRAS, 339, 757 +Bentz, M. C.; Denney, K. D.; Grier, C. J, et al., 2013, ApJ, 767, 149 +Bruzual, G.; Charlot, S. 2003, MNRAS, 344, 1000 +Burtscher, L.; Meisenheimer, K.; Tristram, K. R. W., et al., 2013, A&A, 558, +149 +Canfield, R. C.; Puetter, R. C., 1981, ApJ, 243, 390 +Cappellari, M.; Scott, N.; Alatalo, K., et al., 2013, MNRAS, 432, 1709 +Cappellari, M., 2017, MNRAS, 466, 798 +Chen, K. Y., & Halpern, J. P., 1989, ApJ, 344, 115 +MNRAS 000, 1–6 (2022) + +6 +Zhang +Cid Fernandes, R.; Mateus, A.; Sodre, L.; Stasinska, G.; Gomes, J. M., 2005, +MNRAS, 358, 363 +Drake, A. J.; Djorgovski, S. G.; Mahabal, A.; et al., 2009, ApJ, 696, 870 +Eracleous, M., Livio, M., Halpern, J. P., Storchi-Bergmann, T.,1995, ApJ, +438, 610 +Ezhikode, S. H.; Gandhi, P.; Done, C.; Ward, M.; Dewangan, G. C.; Misra, +R.; Philip, N. S, 2017, MNRAS, 472, 3492 +Ferrarese, F.; Merritt, D., 2000, ApJL, 539, 9 +Foreman-Mackey, D.; Hogg, D. W.; Lang, D.; Goodman, J., 2013, PASP, 125, +306 +Gebhardt, K.; Bender, R.; Bower, G, et al., 2000, ApJL, 539, 13 +Graham, M. J.; Djorgovski, S. G.; Stern, D., et al., 2015a, Natur, 518, 74 +Graham, M. J., Djorgovski, S. G., Stern, D., et al., 2015b, MNRAS, 453, 1562 +Gratadour, D.; Rouan, D.; Grosset, L.; Boccaletti, A.; Clenet, Y., 2015, A&A, +581, 8 +Greene, J. E.; Ho, L. C., 2005, ApJ, 630, 122 +Goodrich, R. W., 1990, ApJ, 355, 88 +Kauffmann, G.; Heckman, T. M.; Tremonti, C., et al. 2003, MNRAS, 346, +1055 +Kwan, J.; Krolik, J. H., 1981, ApJ, 250, 478 +Kormendy, J.; Ho, L. C., 2013, ARA&A, 51, 511 +Kuraszkiewicz, J.; Wilkes, B. J.; Atanas, A.; et al., 2021, ApJ, 913, 134 +Liu, H.; Liu, W.; Dong, X.; Zhou, H.; Wang, T.; Lu, H.; Yuan, W., 2019, +ApJS, 243, 21 +Mandal, A. K.; Rakshit, S.; Kurian, K. S., et al., 2018, MNRAS, 475, 5330 +Marco, O.; Alloin, D., 2000, A&A, 353, 465 +Marin, F.; Goosmann, R. W.; Petrucci, P. O., 2016, A&A, 519, 23 +Mateos, S.; Carrera, F. J.; Barcons, X., et al., 2017, ApJL, 841, 18 +Matt, G.; Iwasawa, K., 2019, MNRAS, 482, 151 +Mejia-Restrepo, J. E.; Trakhtenbrot, B.; Koss, M. J., et al., 2022, ApJS, 261, +5 +Moran, E. C.; Barth, A. J.; Kay, L. E.; Filippenko, A. V., 2020, ApJL, 540, 73 +Netzer, H., 2015, ARA&A, 53, 365 +Netzer, H.; Lani, C.; Nordon, R.; Trakhtenbrot, B.; Lira, P.; Shemmer, O., +2016, ApJ, 819, 123 +Ogawa, S.; Ueda, Y.; Tanimoto, A.; Yamada, S., 2021, ApJ, 906, 84 +Osterbrock, D. E., 1981, ApJ, 249, 462 +Peterson, B. M.; Ferrarese, L.; Gilbert, K. M., et al., 2004, ApJ, 613, 682 +Ricci, C.; Ananna, T. T.; Temple, M. J.; et al., 2022, ApJ, 938, 67 +Rouan, D.; Rigaut, F.; Alloin, D.; Doyon, R.; Lai, O.; Crampton, D.; Gendron, +E.; Arsenault, R., 1998, A&A, 339, 687 +Savic, D.; Goosmann, R.; Popovic, L. C.; Marin, F.; Afanasiev, V. L., 2018, +A&A, 614, 120 +Savorgnan, G. A. D.; Graham, A. W., 2015, MNRAS, 446, 2330 +Shen, Y.; Loeb, A., 2010, ApJ, 725, 249 +Shen, Y.; Richards, G. T.; Strauss, M. A.; et al., 2011, ApJS, 194, 45 +Simpson, C., 2005, MNRAS, 360, 565 +Stalevski, M., Ricci, C., Ueda, Y., Lira, P., Fritz, J., Baes, M., 2016, MNRAS, +458, 2288 +Storchi-Bergmann, T., et al., 2003, ApJ, 489, 8 +Storchi-Bergmann, T., et al., 2017, ApJ, 835, 236 +Tristram, K. R. W.; Meisenheimer, K.; Jaffe, W., et al., 2007, A&A, 474, 837 +Tran, H. D., 2003, ApJ, 583, 632 +Urry, C. M.; Padovani, P., 1995, PASP, 107, 803 +Vestergaard, M., 2002, ApJ, 571, 733 +Woo, J.; Yoon, Y.; Park, S.; Park, D.; Kim, S. C., 2015, ApJ, 801, 38 +Zhang, X. G., 2021d, MNRAS, 502, 2508 +Zhang, X. G., 2021a, ApJ, 909, 16, ArXiv:2101.02465 +Zhang, X. G., 2021b, ApJ, 919, 13, ArXiv:2107.09214 +Zhang, X. G., 2021c, MNRAS Letter, 500, 57 +Zhang, X. G., 2022a, ApJS, 260, 31 +Zhang, X. G., 2022b, ApJS, 261, 23 +Zhang, X. G., 2022c, ApJ, 937, 105, ArXiv:2209.02164 +Zhang, X. G., 2022, MNRAS accepted, Arxiv:2202.11995 +Zhou, Z. Q.; Liu, F. K.; Komossa, S., et al., 2021, ApJ, 907, 77 +Zhuang, M.; Ho, L. C.; Shangguan, J., 2018, ApJ, 862, 118 +MNRAS 000, 1–6 (2022) + diff --git a/BNAzT4oBgHgl3EQf__-y/content/tmp_files/load_file.txt b/BNAzT4oBgHgl3EQf__-y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b930cc9167f655e7b6df3c2cb57ccdb80b8fe96 --- /dev/null +++ b/BNAzT4oBgHgl3EQf__-y/content/tmp_files/load_file.txt @@ -0,0 +1,731 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf,len=730 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='01957v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='GA] 5 Jan 2023 MNRAS 000, 1–6 (2022) Preprint 6 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='0 A practicable estimation of opening angle of dust torus in Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN with double-peaked broad H훼 Xue-Guang Zhang1★ 1 School of Physical Science and Technology, GuangXi University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 100, Daxue Road, 530004, Nanning, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' China 6 January 2023 ABSTRACT In this manuscript, an independent method is proposed to estimate opening angle of dust torus in AGN, through unique properties of Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN with double-peaked broad H훼 (Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN) coming from central accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN without broad H훽 can be expected by the commonly accepted unified model of AGN, considering central BLRs seriously obscured by dust torus with its upper boundary in the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' For the unique Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN, accretion disk originations of double-peaked broad H훼 can be applied to determine the inclination angle of the central accretion disk, which is well accepted as substitute of the half opening angle of the central dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Then, among low redshift Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN in SDSS, SDSS J1607+3319 at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='063 is collected, and the half opening angle of the central dust torus is determined to be around 46±4◦, after considering disfavoured BBH system to explain the double-peaked broad H훼 through long-term none variabilities and disfavoured local physical conditions to explain disappearance of broad H훽 through virial BH mass properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The results indicate that more detailed studying on dust torus of AGN can be appropriately done through Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Key words: galaxies:active - galaxies:nuclei - quasars:emission lines - quasars: individual (SDSS J1607+3319) 1 INTRODUCTION An unified model of Active Galactic Nuclei (AGN) is well known and widely accepted to explain different spectroscopic phenomena between Type-1 AGN with optical both broad and narrow emission lines and Type-2 AGN with only optical narrow emission lines, af- ter mainly considering obscurations on central Broad Line Regions (BLRs) by central dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The unified model has been firstly dis- cussed in Antonucci (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Urry & Padovani (1995), and more re- cently reviewed and discussed in Netzer (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Kuraszkiewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Zhang (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The Unified model has been strongly sup- ported by clearly detected polarized broad emission lines and/or clearly detected broad infrared emission lines in some Type-2 AGN (Tran 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Savic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Moran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Moreover, there are observational/theoretical evidence to support central dust torus as one fundamental structure in the unified model, such as the re- sults in NGC1068 in Rouan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Marco & Alloin (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Gratadour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2015) through direct Near-IR images and polari- metric images, the resolved dust torus in the Circinus galaxy in Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2007), the reported diversity of dusty torus in AGN in Burtscher (2013), the estimated covering factors of central dust torus in local AGN in Ezhikode et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2017), the determined size of central dust torus in H0507+164 in Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2018), the well dis- cussed X-ray clumpy torus model in Ogawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2021) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='. More recent review on dust torus can be found in Almeida & Ricci (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Under the framework of the unified model, considering different orientations of central dust torus in the line of sight, there is a spe- cial kind of AGN, Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN (firstly discussed in Osterbrock ★ Corresponding author Email: aexueguang@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='com (1981)), with broad H훼 emission lines but no broad H훽 indicat- ing central BLRs seriously obscured by dust torus with its upper boundary in the line of sight, besides the Type-1 and Type-2 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Commonly, as a transition type, Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN are considered as the best candidates on studying properties, especially properties of spatial structures, of the unified model expected central dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Actually, there are some reports on the opening angles (covering factor) of the central dust torus in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Arshakian (2005) have proposed a receding torus model, based on statistically signif- icant correlation between the half opening angle of the torus and [O iii] emission-line luminosity, and then followed and discussed in Simpson (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Alonso-Herrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2018) have reported that the half opening angle of the torus declines with increasing accretion rate until the Eddington ratio reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='5, above which the trend reverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Netzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Stalevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2016) have found no evidence for a luminosity dependence of the torus covering factor in AGN not to support the receding torus model, similar conclusions can also be found in Mateos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' More recent interesting discussions on central obscurations by dust torus can be found in Ricci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2022) to support a radiation-regulated unification model in AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Until now, there are rare reports on the opening angles of the cen- tral dust torus in AGN through direct spatial resolved images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' How to measure/determine the opening angle of the central dust torus in an individual AGN is still an interesting challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Here, based on unique properties of Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN with BLRs being seriously ob- scured by the central dust torus, an independent method is proposed to estimate the opening angle of the central dust torus in a special kind of Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN, the Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN with double-peaked broad H훼 (Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' © 2022 The Authors 2 Zhang Section 2 presents our main hypothesis to estimate the half opening angle of the central dust torus in special Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Sec- tion 3 shows the spectroscopic results of the Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN SDSS J160714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='40+331909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='12 (=SDSS J1607+3319) at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Sec- tion 4 gives the main discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Section 5 gives our final con- clusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' And the cosmological parameters have been adopted as 퐻0 = 70km · s−1Mpc−1, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='7 and Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2 MAIN HYPOTHESIS Accretion disk originations have been well accepted to double- peaked broad emission lines, as well discussed in Chen & Halpern (1989);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Eracleous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Storchi-Bergmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2003, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The inclination angle of the central accretion disk can be well estimated through double-peaked broad line emission features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Meanwhile, considering the serious obscurations from central dust torus in Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN, the accretion disk origination determined inclination angle should be well accepted to trace the half opening angle of the central dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Certainly, beside the accretion disk origination, commonly known binary black hole (BBH) system can also be applied to explain double-peaked broad emission lines, such as the results shown in Shen & Loeb (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' However, assumed BBH system should lead to optical quasi-periodic oscillations (QPOs) with periodicities about hundreds to thousands of days, such as the results shown in Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2015a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Zhang (2022), and will be dis- cussed to disfavor the BBH system in the target in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Moreover, it should be confirmed that the seriously obscured broad H훽 are not due to local intrinsic physical conditions (such as the case in H1320+551 discussed in Barcons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2003)), but due to serious obscurations by the central dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' It is exciting to check whether the method can be applied to esti- mate the opening angle of the central dust torus in Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN, which is the main objective of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' And the following three criteria are accepted to collect targets of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' First, the targets are Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN, with apparent double-peaked broad H훼 but no apparent broad H훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Second, there are no signs for optical QPOs in the targets, indicating BBH systems not preferred to explain the double-peaked broad H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Third, after considering the BH mass properties which will be well discussed in the Section 4, serious ob- scurations by the central dust torus are well accepted to explain the seriously obscured broad H훽 in the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Among the low redshift (푧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='35) broad line AGN listed in Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2011) with SPECIAL_INTEREST_FLAG=1 and in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2019) with flag MULTI_PEAK=2, there are 561 low red- shift DPAGN with reliable broad H훼 emission lines (both reported line width and line luminosity at least five times larger than their re- ported uncertainties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Based on the main hypothesis and correspond- ing criteria above, Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN SDSS J160714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='40+331909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='12 (=SDSS J1607+3319) at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='063 is collected as the unique target of the manuscript, based on two main unique features through properties of its spectroscopic and long-term variabilities, well dis- cussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' On the one hand, among the Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN, the SDSS J1607+3319 has the most apparent double- peaked features in broad H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' On the other hand, there are no appar- ent variabilities in SDSS J1607+3319, which can be well applied to disfavour the BBH system in the SDSS J1607+3319, combining its double-peaked features in broad H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Top panel shows the SSP method determined descriptions (solid red line) to the SDSS spectrum (solid dark green line) with emission lines being masked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In top panel, solid blue line and dashed blue line show the determined host galaxy contributions and power law AGN continuum emissions, respectively, solid cyan line shows the line spectrum calculated by the SDSS spectrum minus the sum of host galaxy contributions and AGN continuum emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Bottom panels show the best fitting results (solid red line) to absorption features (solid dark green line) of Ca ii H+K (left panel), Mg i (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In each panel, the determined 휒2/푑표 푓 and stellar velocity dispersion are marked in red characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 3 SPECTROSCOPIC RESULTS OF THE TYPE-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN SDSS J1607+3319 SDSS J1607+3319 has its SDSS spectrum (plate-mjd-fiberid=1419- 53144-0453) with signal-to-noise about 34 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In or- der to measure emission lines, the commonly accepted SSP (Sim- ple Stellar Population) method is applied to determine host galaxy contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' More detailed descriptions on the SSP method can be found in Bruzual & Charlot (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Cid Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Cappellari (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' And the SSP method has been applied in our previous papers Zhang (2021a,b,d, 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Here, we show simple descriptions on SSP method as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The 39 simple stellar population templates from Bruzual & Charlot (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2003) have been exploited, combining with a power law component applied to describe intrinsic AGN continuum emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' When the SSP method is applied, optical narrow emission lines are masked out by full width at zero intensity about 450 km/s, and the spectrum with wavelength range from 6250 to 6750Å are also masked out due to the strongly broad H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Then, through the Levenberg-Marquardt least-squares minimization technique, SDSS spectra with emission lines being masked out can be well described by combinations of broadened stellar population templates and the power law component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The best descriptions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 1 with 휒2/푑표 푓 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='91 (the summed squared residuals divided by de- gree of freedom) and with determined stellar velocity dispersion (the broadening velocity) about 224±5 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Moreover, in order to determine reliable stellar velocity dispersion, absorption features of around Ca ii H+K from 3750 to 4200Å and around Mg i from 5050 to 5250Å are applied to re-measure stel- lar velocity dispersions, through the same SSP method above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The best fitting results are shown in bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 1 with deter- mined stellar velocity dispersions in units of km/s about 222±11 and 208±26 through the Ca ii H+K and Mg i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Therefore, in MNRAS 000, 1–6 (2022) Opening angle of Dust Torus 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Top panels show the best fitting results (solid red line) to the emission lines (solid dark green line), and bottom panels show the corresponding residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In top left panel, solid blue line shows the determined narrow H훽, solid green lines show the determined [O iii] doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In top right panel, solid blue line shows the determined narrow H훼, solid cyan line shows the determined double-peaked broad H훼 described by the elliptical accretion disk model, solid green lines show the determined [O i], [N ii] and [S ii] doublets, dashed purple lines show the determined broad H훼 described by two broad Gaussian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In each bottom panel, horizontal dashed lines show residuals=0, ± 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' MCMC technique determined two-dimensional posterior distributions in contour of the model parameters in the elliptical accretion disk model applied to describe the double-peaked broad H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In each panel, sold circle plus error bars in red mark the positions of the accepted values and corresponding uncertainties of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The number densities related to different colors are shown in color bar in top region of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' the manuscript, the inverse variance weighted mean stellar velocity dispersion 휎★ =222±26 km/s in SDSS J1607+3319 is accepted, which is consistent with the SDSS pipeline reported 230 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' After subtractions of host galaxy contributions and AGN contin- uum emissions, emission lines in the line spectrum can be well mea- sured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Similar as what we have previously done in Zhang (2021a,b, 2022a,b,c), for the emission lines within rest wavelength range from 4600 to 5150Å, there are one broad and one narrow Gaussian func- tions applied to describe probable broad and apparent narrow H훽, two Gaussian functions applied to describe [O iii]휆4959, 5007Å dou- blet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' When the functions above are applied, each component has line intensity not smaller than zero, and the [O iii] components have the same redshift and the same line width and have flux ratio to be fixed to the theoretical value 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Then, through the Levenberg-Marquardt least-squares minimization technique, the best fitting results to the emission lines and the corresponding residuals (line spectrum minus thebest fittingresultsandthendividedbyuncertaintiesofSDSS spec- trum) are shown in left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2 with 휒2/푑표 푓 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Based on the fitting results, it is not necessary to consider broad Gaussian component in H훽, because the determined line width and line flux (around to zero) of the broad Gaussian component are smaller than their corresponding uncertainties, indicating there are no apparent broad H훽 in SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Meanwhile, Gaussian functions can be applied to describe the narrow emission lines within rest wavelength range from 6200 to 6850Å, the [O i], [N ii], [S ii] and narrow H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' But the commonly ac- cepted elliptical accretion disk model with seven model parameters well discussed in Eracleous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (1995) is applied to describe the double-peaked broad H훼, because the model can be applied to ex- plain almost all observational double-peaked broad H훼 of the SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The seven model parameters are inner and out bound- aries [푟0, 푟1] in the units of 푅퐺 (Schwarzschild radius), inclination angle 푖 of disk-like BLRs, eccentricity 푒, orientation angle 휙0 of elliptical rings, local broadening velocity 휎퐿 in units of km/s, line emissivity slope 푞 ( 푓푟 ∝ 푟−푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Meanwhile, we have also applied the very familiar elliptical accretion disk model in our more recent studies on double-peaked lines in Zhang (2021c, 2022a), and there are no further discussions on the elliptical accretion disk model in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Then, in order to obtain more reliable uncertainties of model parameters in the complicated model functions, rather than the Levenberg-Marquardt least-squares Minimization technique, the Maximum Likelihood method combining with the MCMC (Markov Chain Monte Carlo) technique (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2013) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The evenly prior distributions of the seven model pa- rameters in the elliptical accretion disk model are accepted with the following limitations, log(푟0) ∈ [2, 4], log(푟1) ∈ [2, 6] (푟1 > 푟0), log(sin(푖)) ∈ [−3, 0], log(푞) ∈ [−1, 1], log(휎퐿) ∈ [2, 4], log(푒) ∈ [−5, 0], log(휙0) ∈ [−5, log(2 × 휋)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The MNRAS 000, 1–6 (2022) 4 Zhang Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' parameters of the emission line components model parameters of elliptical accretion disk model for broad H훼 푟0 = 2035 ± 240, 푟1 = 3766 ± 500, sin(푖) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='04 푞 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='19, 푒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='08, 휎퐿 = 796 ± 70km/s, 휙0 = 190 ± 6◦ model parameters of Gaussian emission components line 휆0 휎 flux broad H훼 6505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='2 897±25 6643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='2 699±24 Narrow H훼 6564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6 311±54 Narrow H훽 4862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='4 45±8 [O iii]휆5007Å 5008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='3 172±10 [O i]휆6300Å 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='2 113±14 [N ii]휆6583Å 6585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='3 642±55 [S ii]휆6716Å 6719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 260±33 [S ii]휆6731Å 6734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='7 157±29 Notice: For the Gaussian emission components, the first column shows which line is measured, the Second, third, fourth columns show the measured line parameters: the center wavelength 휆0 in unit of Å, the line width (second moment) 휎 in unit of Å and the line flux in unit of 10−17 erg/s/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' determined best fitting results and corresponding residuals to the emission line around H훼 are shown in right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2 with 휒2/푑표 푓 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The MCMC technique determined posterior dis- tributions of the model parameters in the elliptical accretion disk model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' And the half width at half maximum of each parameter distribution is accepted as uncertainty of the param- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The determined parameters and corresponding uncertainties of each model parameter are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Moreover, as discussed in Zhang (2022a), clean double-peaked broad line emission features can lead to solely determined model parameters in the elliptical ac- cretion disk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Therefore, there are no further discussions on whether is there solely determined model parameter of sin(푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 4 MAIN DISCUSSIONS In the section, two points are mainly considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' First, it is neces- sary to determine that the accretion disk origination is favoured to explain the double-peaked broad H훼 in SDSS J1607+3319, rather than a BBH system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Second, it is necessary to determine that the large broad Balmer decrement (flux ratio of broad H훼 to broad H훽) is due to serious obscurations, rather than due to local phys- ical conditions, because that BLRs modeled with relatively low opti- cal depths and low ionization parameters can reproduce large broad Balmer decrements, as well discussed in Kwan & Krolik (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Canfield & Puetter (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Goodrich (1990) without considering se- rious obscurations and see the unobscured central regions in a Type- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 AGN in Barcons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' For the first point on BBH system, the following discussions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The double-peaked broad H훼 can also be well described by two broad Gaussian functions shown as dashed purple lines in top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2 with model parameters listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Under the assumption of BBH system in SDSS J1607+3319, considering the strong linear correlation between broad H훼 luminosity and con- tinuum luminosity as discussed in Greene & Ho (2005), there are to- tally equal (ratio about 897:699 from emission fluxes of the two broad Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' CSS V-band light curve of SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Horizontal solid and dashed red lines show the mean value and corresponding 2RMS scatters of the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Gaussian components) continuum luminosities related to central two BH accreting systems, indicating there should be strong variabilities with QPOs due to orbital rotating effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' However, there are none variabilities in the collected 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='4years-long CSS (Catalina Sky Survey, Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2009)) V-band light curve shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 4 with almost all data points lying within 2RMS scatter ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Therefor, rather than the BBH system, the elliptical accretion disk model is preferred to explain the double-peaked broad H훼 in SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' For the second point, properties of virial BH mass are mainly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Based on accepted virialization assumptions to prop- erties of observed broad H훼 as discussed in Vestergaard (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Greene & Ho (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Mejia-Restrepo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2022), virial BH mass can be estimated by 푀퐵퐻 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6 × 106( 퐿퐻 훼 1042erg/s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='55( 휎퐻 훼 1000km/s )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='06M⊙ = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='6) × 107M⊙ (1) with 퐿퐻 훼 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='05) × 1041erg/s as line luminosity of ob- served broad H훼 and 휎퐻 훼 = (3100 ± 110)km/s as second mo- ment of observed broad H훼, after considering more recent em- pirical R-L relation to estimate BLRs sizes in Bentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Uncertainty of virial BH mass is determined by uncertainties of the 퐿퐻 훼 and 휎퐻 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' If large broad Balmer decrement was due to local physical conditions, the estimated virial BH mass should be simply consistent with the 푀BH − 휎 relation (Ferrarese & Merritt 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Kormendy & Ho 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Batiste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Bennert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2021) expected value, otherwise, there should be smaller virial BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Then, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 5 shows virial BH mass prop- erties of SDSS J1607+3319 in the 푀BH − 휎 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In order to show clearer results, the 89 quiescent galaxies from Savorgnan & Graham (2015) and the 29 reverberation mapped (RM) AGN from Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2015) and the 12 tidal disruption events (TDEs) from Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2021) are considered to draw the linear correlation between stellar velocity dispersion and BH mass log( 푀퐵퐻 M⊙ ) = (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='49) + (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='22) × log( 휎★ km/s) (2) through the Least Trimmed Squares robust technique (Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' And then the 3휎, 4휎 and 5휎 confi- dence bands to the linear correlation are determined and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Therefore, the estimated viral BH mass of SDSS J1607+3319 is lower than 푀BH − 휎 expected value with confidence level higher than 4휎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Therefore, locate physical conditions are MNRAS 000, 1–6 (2022) Opening angle of Dust Torus 5 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' On the correlation between stellar velocity dispersion measured through absorption features and virial BH mass of SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Solid five-point-star in dark green shows the virial BH mass of SDSS J1607+3319 determined by properties of observed broad H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Dot-dashed lines in magenta and in black represent the 푀BH − 휎 relations through the quiescent galaxies in Kormendy & Ho (2013) and through the RM AGNs in Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2015), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Solid circles in red, in blue and in pink show the values for the 89 quiescent galaxies in Savorgnan & Graham (2015), the 29 RM AGNs in Woo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2015) and the 12 TDEs in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2021), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Thick solid red line shows the best fitting results to all the objects, and thick dashed, dotted and dot-dashed red lines show corresponding 3휎, 4휎 and 5휎 confidence bands to the best fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' disfavored to explain the large broad Balmer decrement in SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Based on the double-peaked broad H훼 in the Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN SDSS J1607+3319, half opening angle of central dust torus is well estimated as (46±4)◦ (sin(푖) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='04), roughly consistent with statistical mean value in Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Therefore, it is inter- esting to study properties of opening angles of dust torus through Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN in the near future, after many efforts to disfavour BBH systems to explain their double-peaked broad H훼 and to dis- favour local physical conditions to explain disappearance of broad H훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Before ending of the manuscript, an additional point is noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Before giving clear physical information of materials in the central dust torus, it is hard to confirm that the accretion disk origination determined inclination angle is completely consistent with the half opening angle of the central dust torus in Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' If ma- terial densities in regions around upper boundary of the central dust torus were too low to lead the broad H훽 being totally obscured, the determined inclination angle should be lower than the intrinsic half opening angle of the central dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Moreover, it is not clear whether are there different radial dependent material densities in the direction perpendicular to the equatorial plane related to central AGN activities, which should also have effects on the consistency between the accretion disk origination determined inclination angle and the half opening angle of the central dust torus in AGN with different central AGN activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' In the near future, through studying a sample of Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN as one of our ongoing projects, clearer clues and detailed discussions will be given on the consistency between the inclination angle and the half opening angle of the central dust torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 5 CONCLUSIONS An independent method is proposed to estimate the opening an- gle of the central dust torus in Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN through unique double-peaked features of broad H훼, accepted the assumptions of obscurations of the central dust torus on BLRs leading to disappear- ance of broad H훽 and of the double-peaked broad H훼 with accre- tion disk originations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Then, among the reported DPAGN, the SDSS J1607+3319 is collected due to its apparent broad double-peaked broad H훼 but no broad H훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Moreover, long-term optical variabilities can be applied to disfavour the BBH system in SDSS J1607+3319 to explain the double-peaked broad H훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' And properties of virial BH mass can be applied to determine that local physical conditions are not favoured to explain the large broad Balmer decrement in SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Then, based on the well applied elliptical accretion disk model applied to describe the double-peaked broad H훼 in SDSS J1607+3319, the half opening angle of the central dust torus can be well estimated as (46±4)◦ in SDSS J1607+3319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The results in the manuscript strongly indicate that the proposed independent method is practicable, and can be applied to study detailed properties of the opening angles of the central dust torus through a sample of Type-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='9 DPAGN, which will be studied in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Zhang gratefully acknowledges the anonymous referee for giving us constructive comments and suggestions to greatly improve our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Zhang gratefully acknowledges the kind funding support NSFC-12173020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' This research has made use of the data from the SDSS (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='org/) funded by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Sloan Foundation, the Participating Institutions, the National Science Foundation and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' De- partment of Energy Office of Science, and use of the data from CSS http://nesssi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='cacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='edu/DataRelease/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' The research has made use of the MPFIT package https://pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='edu/~craigm/idl/cmpfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='html, and of the LTS_LINEFIT package https://www-astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='uk/~cappellari/software/, and of the emcee package https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='org/project/emcee/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on request to the corresponding author (aexueguang@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' REFERENCES Antonucci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1993, ARA&A, 31, 473 Almeida, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Ricci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2017, Nat Astron, 1, 679 Alonso-Herrero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ramos Almeida, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Mason, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2011, ApJ, 736, 82 Arshakian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2005, A&A, 436, 817 Batiste, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Bentz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Raimundo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Vestergaard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Onken, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2017, ApJL, 838, 10 Bennert, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ding, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021, ApJ, 921, 36 Barcons, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Carrera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ceballos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2003, MNRAS, 339, 757 Bentz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Denney, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Grier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2013, ApJ, 767, 149 Bruzual, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Charlot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2003, MNRAS, 344, 1000 Burtscher, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Meisenheimer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Tristram, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2013, A&A, 558, 149 Canfield, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Puetter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1981, ApJ, 243, 390 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Scott, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Alatalo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2013, MNRAS, 432, 1709 Cappellari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2017, MNRAS, 466, 798 Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', & Halpern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1989, ApJ, 344, 115 MNRAS 000, 1–6 (2022) 6 Zhang Cid Fernandes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Mateus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Sodre, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Stasinska, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Gomes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2005, MNRAS, 358, 363 Drake, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Djorgovski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Mahabal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2009, ApJ, 696, 870 Eracleous, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Livio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Halpern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Storchi-Bergmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=',1995, ApJ, 438, 610 Ezhikode, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Gandhi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Done, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ward, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Dewangan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Misra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Philip, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' S, 2017, MNRAS, 472, 3492 Ferrarese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Merritt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2000, ApJL, 539, 9 Foreman-Mackey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Hogg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Lang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Goodman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2013, PASP, 125, 306 Gebhardt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Bender, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Bower, G, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2000, ApJL, 539, 13 Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Djorgovski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Stern, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2015a, Natur, 518, 74 Graham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Djorgovski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Stern, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2015b, MNRAS, 453, 1562 Gratadour, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Rouan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Grosset, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Boccaletti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Clenet, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2015, A&A, 581, 8 Greene, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2005, ApJ, 630, 122 Goodrich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1990, ApJ, 355, 88 Kauffmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Tremonti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' 2003, MNRAS, 346, 1055 Kwan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Krolik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1981, ApJ, 250, 478 Kormendy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2013, ARA&A, 51, 511 Kuraszkiewicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Wilkes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Atanas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021, ApJ, 913, 134 Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Dong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Yuan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2019, ApJS, 243, 21 Mandal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Rakshit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Kurian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2018, MNRAS, 475, 5330 Marco, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Alloin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2000, A&A, 353, 465 Marin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Goosmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Petrucci, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2016, A&A, 519, 23 Mateos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Carrera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Barcons, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2017, ApJL, 841, 18 Matt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Iwasawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2019, MNRAS, 482, 151 Mejia-Restrepo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Trakhtenbrot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Koss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2022, ApJS, 261, 5 Moran, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Barth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Kay, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Filippenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2020, ApJL, 540, 73 Netzer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2015, ARA&A, 53, 365 Netzer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Lani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Nordon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Trakhtenbrot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Lira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Shemmer, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2016, ApJ, 819, 123 Ogawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ueda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Tanimoto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Yamada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021, ApJ, 906, 84 Osterbrock, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1981, ApJ, 249, 462 Peterson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ferrarese, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Gilbert, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2004, ApJ, 613, 682 Ricci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ananna, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Temple, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2022, ApJ, 938, 67 Rouan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Rigaut, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Alloin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Doyon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Lai, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Crampton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Gendron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Arsenault, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1998, A&A, 339, 687 Savic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Goosmann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Popovic, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Marin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Afanasiev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2018, A&A, 614, 120 Savorgnan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Graham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2015, MNRAS, 446, 2330 Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Loeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2010, ApJ, 725, 249 Shen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Strauss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2011, ApJS, 194, 45 Simpson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2005, MNRAS, 360, 565 Stalevski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Ricci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Ueda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Lira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Fritz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', Baes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2016, MNRAS, 458, 2288 Storchi-Bergmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2003, ApJ, 489, 8 Storchi-Bergmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2017, ApJ, 835, 236 Tristram, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Meisenheimer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Jaffe, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2007, A&A, 474, 837 Tran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2003, ApJ, 583, 632 Urry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Padovani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 1995, PASP, 107, 803 Vestergaard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2002, ApJ, 571, 733 Woo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Yoon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Park, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2015, ApJ, 801, 38 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021d, MNRAS, 502, 2508 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021a, ApJ, 909, 16, ArXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='02465 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021b, ApJ, 919, 13, ArXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='09214 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021c, MNRAS Letter, 500, 57 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2022a, ApJS, 260, 31 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2022b, ApJS, 261, 23 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2022c, ApJ, 937, 105, ArXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='02164 Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2022, MNRAS accepted, Arxiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content='11995 Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Komossa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2021, ApJ, 907, 77 Zhuang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Ho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=' Shangguan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} +page_content=', 2018, ApJ, 862, 118 MNRAS 000, 1–6 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNAzT4oBgHgl3EQf__-y/content/2301.01957v1.pdf'} diff --git a/ENE0T4oBgHgl3EQfQgDK/content/tmp_files/2301.02195v1.pdf.txt b/ENE0T4oBgHgl3EQfQgDK/content/tmp_files/2301.02195v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..56dca90230e3436fe5d133e502844f1ae8f89194 --- /dev/null +++ b/ENE0T4oBgHgl3EQfQgDK/content/tmp_files/2301.02195v1.pdf.txt @@ -0,0 +1,954 @@ +Towards Autoformalization of Mathematics and Code Correctness: +Experiments with Elementary Proofs +Garett Cunningham +School of EECS +Ohio University +Athens, OH 45701 +gc974517@ohio.edu +Razvan C. Bunescu +Department of Computer Science +UNC Charlotte +Charlotte, NC 28223 +razvan.bunescu@uncc.edu +David Juedes +School of EECS +Ohio University +Athens, OH 45701 +juedes@ohio.edu +Abstract +The ever-growing complexity of mathemati- +cal proofs makes their manual verification by +mathematicians very cognitively demanding. +Autoformalization seeks to address this by +translating proofs written in natural language +into a formal representation that is computer- +verifiable via interactive theorem provers. In +this paper, we introduce a semantic parsing ap- +proach, based on the Universal Transformer +architecture, that translates elementary math- +ematical proofs into an equivalent formaliza- +tion in the language of the Coq interactive the- +orem prover. +The same architecture is also +trained to translate simple imperative code dec- +orated with Hoare triples into formally veri- +fiable proofs of correctness in Coq. +Exper- +iments on a limited domain of artificial and +human-written proofs show that the models +generalize well to intermediate lengths not +seen during training and variations in natural +language. +1 +Introduction +To the uninitiated, the notion of mathematical proof +represents simply an argument written by people +to convince others of mathematical truth. How- +ever, in a real sense, mathematical proof must have +formal underpinnings that go beyond the written +argument. Arguments that lack such underpinnings +might have fatal errors or even logical inconsisten- +cies (see, for example, Russell’s Paradox (Irvine +and Deutsch, 2021)). Nevertheless, mathematical +arguments written in natural language are the norm +and they have great value. +In Tymoczko (1979)’s well-known paper that dis- +cusses a somewhat controversial (at the time) proof +of the Four Color Theorem (Appel and Haken, +1977; Appel et al., 1977), he explores “what is a +mathematical proof?” He posits that all mathemati- +cal proofs must be (i) convincing, (ii) surveyable, +and (iii) formalizable. The first two points are for +the reader—proofs must be convincing to and com- +prehensible by mathematicians. For the third point, +he notes that, “Most mathematicians and philoso- +phers believe that any acceptable proof can be for- +malized. We can always find an appropriate formal +language and theory in which the informal proof +can be embedded and ‘filled out’ into a rigorous +formal proof.” For most mathematicians, this third +part is crucial for ensuring that subtle, but fatal, +errors in logic do not exist in mathematical proof. +Great progress has been made since the 1970’s +in fully formalizing significant mathematical re- +sults. For instance, the Feit-Thompson Theorem +(Gonthier et al., 2013; Gonthier, 2013) and the Four +Color Theorem (Gonthier, 2008) have been for- +mally verified using the proof assistant Coq (Bertot +and Castéran, 2013), and the Kepler Conjecture +(Hales, 2005; Hales et al., 2017) has been formally +verified using the proof assistants Isabelle and HOL +Light (Nipkow et al., 2002). Moreover, proof assis- +tants have demonstrated immense utility for soft- +ware verification, such as the full certification of a +C compiler (Leroy, 2009). Proofs demonstrating +the correct behavior of code share a similar struc- +ture to proofs in pure mathematics, where systems +like Hoare logic replace standard first-order logic. +Thus, Tymoczko’s criteria for mathematical proof +can be extended to the verification of programs. +For many experts, LaTeX provides an excellent +tool for satisfying the first two criteria. In addition, +carefully written LaTeX (Higham, 2020) provides +a rich structure for establishing the third criterion. +The vast majority of modern mathematics is ex- +pressed using natural language (NL), with the over- +whelming majority typeset in LaTeX. Fully for- +malizing mathematics using proof assistants is still +a difficult and time consuming task. This paper +takes some preliminary steps toward bridging this +gap by exploring how modern machine learning +techniques can be used to convert carefully writ- +ten LaTeX into equivalent, and formally verified +arXiv:2301.02195v1 [cs.CL] 5 Jan 2023 + +mathematics in Coq, a process referred to as auto- +formalization in the literature (Szegedy, 2020). +Wang et al. (2018, 2020) explored the similar +task of translating mathematical statements from +LaTeX into Mizar, using LSTM-based models with +attention. To generate aligned LaTeX-Mizar pairs, +they use a tool (Bancerek, 2006) that translates +top-level Mizar statements into artificial LaTeX +sentences, a task that is facilitated by the fact that +Mizar is human readable and similar in length with +the corresponding LaTeX version. Carman (2021) +evaluated the competency of LSTMs toward for- +malizing a restricted set of artificially generated the- +orems about simple arithmetic expressions, report- +ing reasonable success over expression lengths seen +during training. More recently, Wu et al. (2022) +evaluated Codex and PaLM on a significantly more +limited, but human-written set of theorems in alge- +bra and number theory. +In contrast to prior work, we address the auto- +formalization of both theorems and their proofs, +and extend the scope to proofs of code correctness. +We use a number of manually written mathemati- +cal statements to abstract a complex grammar that +is then used to generate a dataset of substantially +longer and more diverse mathematical theorems +and proofs. +We develop an architecture based +on the Universal Transformer (Dehghani et al., +2018) and adapt a copying mechanism (Gu et al., +2016) to handle arbitrary numbers and variable +names at test time. The models are evaluated exten- +sively on their ability to systematically generalize +to statement lengths not seen during training, for +which we report sequence-level accuracy as well +as a semantic-level accuracy calculated by combin- +ing sequence-level accuracy for the theorem and +running Coq to determine if the generated proof +is correct. Code and data are made available at +https://github.com/gc974517/autoformalization. +2 +Dataset of Theorems and Proofs +We create two independent datasets of mathemat- +ical statements that overall correspond to four +classes of theorems and proofs: the first dataset con- +tains three classes of arithmetic statements (EVEN- +ODD, COMPOSITES, and POWERS), described in +detail in Section 2.1, and the second dataset contain- +ing statements about code correctness via Hoare +logic (POLY), described in detail in Section 2.2. +In each example, the input theorem-proof pair is +given in LaTeX, whereas the formalized output is +represented in Coq. This work focuses on the proof +assistant Coq (Bertot and Castéran, 2013) because +(a) there is a rich set of mathematical libraries that +have been developed for it, (b) it has been used +successfully to reason about significant computa- +tion artifacts, such as the ComperCert C compiler +(Leroy, 2009)), and (c) it benefits from a rich set of +training material for the proof assistant related to +software verification (Pierce et al., 2010). +Each class of examples demonstrates features +necessary for the successful autoformalization of +mathematical theorems and proofs. For example, +POWERS and COMPOSITES examples may define +useful terminology to make the theorems shorter, +e.g. proving that 4 is a square, or conversely they +may state theorems directly without any prelim- +inary definitions, e.g. proving ∃n. n2 = 4. As +shown in Figures 3 and 4, this corresponds in Coq +to aliasing propositions using the Definition key- +word. Additionally, the examples in the dataset +provide a stress test of the copying mechanism de- +scribed in Section 3.1, testing its ability to learn +the correct order and number of terms to include +in mathematical expressions, as well as their place- +ment in theorems and proofs, in a way that general- +izes to arbitrary tokens in mathematical language. +For each of the four classes of theorems and +proofs, we manually created a few examples our- +selves in order to guide the construction of a com- +plex grammar that is then used to generate a dataset +of substantially longer and more diverse mathemat- +ical theorems and proofs. Each dataset is generated +using its corresponding grammar in an identical +way. First, a random seed is sampled that controls +the overall structure of the theorem, proof, and +definition, if any. Then, the skeleton structure of +the proof is completed with phrases that are sam- +pled from a separate context-free grammar. The +coarse control of the skeleton structure allows the +construction of examples with interesting features +like sublemmas, forward or backward proof direc- +tion, coreference, or additional conditions for the +theorem, among others. +Many of the difficulties in formalizing mathe- +matical statements from NL into Coq stem from +the wide variability in the level of detail of mathe- +matical proofs, and the frequent mismatch between +what is considered an acceptable inference step in +NL proofs vs. an inference step in Coq. Further- +more, there may be multiple Coq proofs for any +given theorem, at different levels of granularity. We + +LaTeX Input Sequence +Theorem. 28M + 308 is even. +Proof. We know the summation between even numbers +in N will be an even number. Observe that 308 is known +to be even. Additionally, note that the pair M × 28 is +trivially even. This is true because the coefficient 28 is +even. +Coq Output Sequence +Require Import Arith. +Theorem M28_308: +forall M : nat, Nat.even(28 * M + 308) = true. +Proof. +intros. +repeat rewrite Nat.even_add. +assert (H1: Nat.even 308 = true). +{ auto. } +assert (H2: Nat.even 28 = true). +{ auto. } +assert (H3: Nat.even (28 * M) = true). +{ rewrite Nat.even_mul. +rewrite H2. +auto. } +rewrite H1. +rewrite H3. +auto. +Qed. +Figure 1: Generated example from the EVEN-ODD set. +address this ambiguity by requiring the structure of +the Coq proof to match the overall structure of the +NL proof. This is achieved by quasi-synchronously +generating the LaTeX and Coq versions of mathe- +matical statements, while still allowing for some +simple re-orderings in order to improve general- +ization performance, e.g. swapping arguments of +commutative operations. +In total, the grammar-based method for gen- +erating examples can theoretically produce over +283 million unique arithmetic examples and over +491,000 unique code examples, before consider- +ing variations in phrasing by sampling from the +context-free grammar. +2.1 +Arithmetic Statements +We generated three classes of mathematical state- +ments, i.e. theorem-proof pairs: +• EVEN-ODD: an expression is even or odd. +• COMPOSITES: a number is composite. +• POWERS: a number is an integer power of n. +EVEN-ODD examples contain arithmetic expres- +sions of n variables with even coefficients that are +summed with a constant term, meaning that the +parity of this constant determines the parity of the +LaTeX Input Sequence +Theorem. 450 + a · 192 + j · 462 is guaranteed to be +even for any natural terms j, and a. +Proof. It can be justified that 192 · a + j · 462 is trivially +even. Note that 192a is an even number in N because +multiplying between an even integer with an arbitrary +number in N is guaranteed to be even. Likewise, 462j +is trivially an even number in N. The claim is proven as +a consequence of the fact that the sum of even numbers +with an even number will be in itself an even number. +Therefore, our theorem holds. +Coq Output Sequence +Require Import Arith. +Theorem a450_192j_450_even: +forall j a : nat, +Nat.even (192 * a + 462 * j + 450) = true. +Proof. +intros. +rewrite Nat.even_add. +assert (H1: Nat.even (192 * a) = true). +{ rewrite Nat.even_mul. +auto. } +assert (H2: Nat.even (462 * j) = true). +{ rewrite Nat.even_mul. +auto. } +assert (H3: Nat.even +(192 * a + 462 * j) = true). +{ repeat rewrite Nat.even_add. +rewrite H1. +rewrite H2. +auto. } +rewrite H3. +auto. +Qed. +Figure 2: Instance of sublemma use in the EVEN-ODD +dataset. The proof that the sum of non-constant terms is +even (assertion H3) is given before proving the theorem. +whole expression. Proofs make use of this fact +with varying rigor based on our manually designed +grammar, an example of which is shown by Figure +1. The Coq program is generated concurrently with +the paired LaTeX example. The example shown in +Figure 2 illustrates the generation and use of prior +facts to prove an implicit sublemma, in both the +natural language and matching Coq version. +Examples of theorems and proofs for POWERS +and COMPOSITES share a similar structure in both +their LaTeX and Coq forms, as shown in Figures 3 +and 4, respectively. The theorems assert the ex- +istence of a natural number such that a defining +property holds and their proofs are constructive, +with the distinction that examples for composites +prove factorization into n factors. +For both training and testing, we generate 5,000 +even-odd, 5,000 composites, and 2,000 powers ex- + +LaTeX Input Sequence +Definition. We define that w ∈ N is a composite natural +number if taking some R, Q ∈ N we have Q, R ≥ 2 and +Q × R = w. +Theorem. 35 is a composite whole number. +Proof. Remember that a composite natural number is the +multiplication between Q and R such that Q and R ≥ 2. +Allow R = 7, Q = 5. We justify the result is valid as +35 = Q · R. +Coq Output Sequence +Require Import Lia. +Definition composite +(w : nat) := +exists R Q : nat, +(Q >= 2) /\ (R >= 2) +/\ (Q * R = w). +Theorem w_composite: +composite 35. +Proof. +unfold composite. +exists 7. +exists 5. +lia. +Qed. +Figure 3: Generated COMPOSITES example. +LaTeX Input Sequence +Definition. We say o is a square whenever there exists +some whole number Z such that Z ≥ 2 and o = Z2. +Theorem. o = 64 is a square. +Proof. Let Z = 8. Observe that 64 = 82. Also notice +Z = 8 is more than or equal to 2. This yields 64 is a +square whole number. +Coq Output Sequence +Require Import Lia. +Definition square +(o : nat) := +exists Z : nat, +(Z >= 2) /\ (o = Z^2). +Theorem square_64: +square 64. +Proof. +unfold square. +exists 8. +assert (H1: 8 >= 2). +{ lia. } +repeat split. +apply H1. +Qed. +Figure 4: Generated example from the POWERS set. +amples. We train on values of n ∈ {2, 3, 5, 7, 9} +and test on values n ∈ {2, 3, . . . , 12}, where n rep- +resents the number of variables in the arithmetic +expression, the number of factors, or the power, +respectively. This is done in order to evaluate the +model’s ability to generalize to unseen arithmetic +expression lengths and numbers of factors. +2.1.1 +Handwritten Examples +We also created a small collection of 45 human- +written LaTeX theorem-proof pairs to evaluate per- +formance on examples outside of our manually +generated grammar. These are distinct from the +original manually written examples that were used +to guide the development of the generative gram- +mar. There are 15 examples for each type of proof +from the arithmetic set, using the same vocabulary +with a number of unseen grammatical structures. +2.2 +Code Correctness Statements +We create a dataset of correctness proofs about +short programs written in the imperative program- +ming language Imp (Pierce et al., 2018), which we +call POLY. The programs represent various algo- +rithms for evaluating a polynomial, and their proofs +of correctness verify that the programs correctly +model the polynomial as a mathematical function. +Proofs are conducted as either fully decorated pro- +grams or as sequences of Hoare triples with natural +language justifying steps in between. An example +is shown in Figure 5. +For both training and testing data, we generate +5,000 examples. We train on programs containing +2, 3, 5, 7, 9, and 11 lines, then test on programs con- +taining from 2 up to 14 lines to evaluate the model’s +ability to generalize to novel program lengths. +3 +Semantic Parsing Architecture +To formalize LaTeX statements into Coq, we de- +veloped an encoder-decoder architecture based on +the Universal Transformer (Dehghani et al., 2018). +Similar to Csordás et al. (2021), we do so by adding +recursive passes into the encoder and decoder of +a base Transformer (Vaswani et al., 2017), thus +making the model analogous to a Universal Trans- +former without adaptive computation time (ACT). +Further, we introduce a copying mechanism and +support for out-of-vocabulary mathematical terms. +3.1 +Copying Mechanism +Mathematical language contains features uncom- +mon or non-existent in natural language, such as +numbers, variables, and carefully defined terminol- +ogy. In order to address the use of general math- +ematical jargon, these tokens are replaced in the + +LaTeX Input Sequence +Coq Output Sequence +Theorem. Consider the following series of +commands such that +S := 3; +S := 3 + S * Z; +S := 1 + S * Z +Allow Z = y, for any natural number y, ahead +of running this code then S = 3×y2+3×y+1 +after the set of instructions has executed. +Proof. By application of usual Hoare logic: +{Z = y} +S := 3; +{Z = y ∧ S = 3} +S := 3 + S * Z; +{Z = y ∧ S = 3 × y + 3} +S := 1 + S * Z +{Z = y ∧ S = 3 × y2 + 3 × y + 1} +Hence, this program is shown to be correct. +Require Import String. +From PLF Require Import Imp. +From PLF Require Import Hoare. +Theorem poly_code_correct: +forall y : nat, +{{ Z = y }} +S := 3; +S := 3 + S * Z; +S := 1 + S * Z +{{ S = 3 * y ^ 2 + 3 * y + 1 }}. +Proof. +intros. +apply hoare_seq with +(Q := ( +(Z = y /\ S = 3) +)%assertion). +apply hoare_seq with +(Q := ( +(Z = y /\ S = 3 * y + 3) +)%assertion). +apply hoare_seq with +(Q := ( +(Z = y /\ S = 3 * y^2 + 3 * y + 1) +)%assertion). +all: eapply hoare_consequence_pre; +try (apply hoare_asgn || assn_auto''). +Qed. +Figure 5: Generated POLY example: [Left] the Hoare logic proof; [Right] the code correctness proof in Coq. +LaTeX input with generic forms denoting their us- +age, such as up to for variables, +which effectively ensures generalization to vari- +able renaming (Ferreira et al., 2022), up to + for numbers, or for definitions, cou- +pled with the use of a copying mechanism adapted +from Gu et al. (2016). Note that a different generic +token is introduced for each unique numerical con- +stant or variable literal in the theorem and its proof, +and the corresponding generic token is used in +the Coq version. For example, considering the +⟨LaTeX, Coq⟩ pair in Figure 3, , , +, and would be used to replace the +constants 2, 35, 7, and 5 respectively, everywhere in +the LaTeX and Coq statements. Similarly, , +, and were used to replace variable +literals w, R, and Q. This is in contrast to using +just two generic tokens and every- +where, which would make all numbers coreferent +and all variables coreferent. Preliminary experi- +ments validated the utility of encoding these dis- +tinctions while maintaining the correct coreference +in both LaTeX and Coq statements. +Overall, by using generic tokens for numbers, +variables, and definitions, only a limited set of em- +beddings need to be trained and the model is forced +to utilize contextual information in order to appro- +priately copy tokens into the Coq output. In this +way, the model has the ability to generalize to un- +seen numbers or variable and definition names. +The original CopyNet (Gu et al., 2016) used an +encoder-decoder architecture with a copying mech- +anism to calculate the probabilities of generating +in-vocabulary tokens vs. copying tokens from the +input sequence to the output. Our autoformaliza- +tion task guarantees mutual exclusivity between +generating (g) and copying (c) tokens, which al- +lows using a simplified formula for calculating the +probability of producing a token yt at time step t. +Letting Vc denote the Coq vocabulary, X denote +the input sequence of LaTeX tokens, and X denote +the collection of unique tokens in X, we calculate +the probability of producing yt as: +p(yt) = +� +� +� +� +� +� +� +p(yt, g) = 1 +Zt +eψg(yt), +yt ∈ Vc +p(yt, c) = 1 +Zt +� +xj∈X:xj=yt +eψc(xj), yt ∈ X +where Zt = +� +yt∈Vc +eψg(yt) + +� +xj∈X +eψc(xj). The scor- + +ing functions are given by ψg(yt) = v⊤ +ytWost and +ψc(xj) = tanh +� +h⊤ +j Wc +� +st, where vyt is a one- +hot encoding of yt, hj is the hidden encoder state +for the input token xj, st is the decoder state at step +t, and Wo and Wc are learnable parameters. +3.2 +Encoder-Decoder Architecture +We diverge from the standard Transformer archi- +tecture in a few crucial ways: +• Probabilities are calculated via p(yt) above. +• Absolute positional encodings are removed. +• Self-attention uses relative positional repre- +sentations as in Shaw et al. (2018). +• Stacks of N encoder/decoder blocks have T +recurrent passes. +All other aspects of the model remain unchanged +from the original Transformer. We emphasize rel- +ative positional information over absolute in our +model architecture. Preliminary evaluations on the +EVEN-ODD dataset showed that Transformer mod- +els that use absolute positional encodings obtain +0% sequence-level accuracy on expression lengths +that are not seen at training time. Removing re- +liance on absolute position resolves this type of +systematic generalization. The use of relative posi- +tional encodings for the Transformer-based models +was thus essential for achieving stronger systematic +generalization, which also agrees with the findings +of Csordás et al. (2021) on other NLP tasks. +4 +Experimental Evaluations +To evaluate the performance of trained models, we +ran two primary experiments: first on the collection +of arithmetic examples, then on the collection of +code correctness examples. All models are eval- +uated in terms of sequence-level accuracy, where +an example is considered correctly processed only +if the generated Coq sequence for both the theo- +rem and its proof perfectly matches token by to- +ken the ground truth sequence. We also report +semantic-level accuracy, for which the generated +Coq theorem needs to attains a perfect sequence- +level accuracy and the Coq engine verifies that the +generated Coq proof truly proves the generated +Coq theorm, regardless of whether it matches the +ground truth version of the proof. This empha- +sizes that the model was able to capture the general +meaning of the natural language proof by correctly +translating the theorem and successfully proving it +EVEN-ODD +COMPOSITES +POLY +n +Seq +Sem +Seq +Sem +Both +2 +99.6 +99.8 +76.7 +97.6 +100.0 +3 +99.4 +99.6 +64.6 +94.2 +100.0 +4 +99.4 +99.4 +56.1 +93.9 +82.1 +5 +99.2 +99.6 +54.9 +94.4 +99.2 +6 +98.8 +98.8 +57.1 +94.3 +45.1 +7 +99.1 +99.5 +58.5 +93.4 +96.5 +8 +93.8 +94.0 +53.5 +88.3 +15.7 +9 +98.6 +98.6 +53.7 +93.7 +98.2 +10 +7.0 +7.0 +1.2 +1.6 +35.6 +11 +0.0 +0.0 +0.0 +0.0 +93.5 +12+ +0.0 +0.0 +0.0 +0.0 +0.0 +POWERS +Seq = 100.0 +Sem = 100 +Table 1: +Sequence-level (Seq) and semantic-level +(Sem) accuracy (%) on test examples, split by expres- +sion length, with the exception of POWERS. +using the natural language version as a guide. +All experiments were performed on one NVIDIA +RTX-A6000 GPU with 48GB of memory. +4.1 +Arithmetic Statements +We evaluate a Transformer model on the full data +combining EVEN-ODD + COMPOSITES + POWERS +and using both the theorem and its proof in each +sequence. We tune a model with embedding and +state sizes of 32, a feed forward width of 256, 4 +encoder and decoder blocks with 4 recurrent passes, +4 attention heads, and a clipping value of 2 for self- +attention. We trained this model over minibatches +of size 20, optimized with Adam using β1 = 0.9, +β2 = 0.98, ε = 1e − 9, and an initial learning rate +of 0.001, annealed by a factor of 1/ +√ +10 based on +training loss plateaus with a patience of 5 epochs. +The results in Table 1 show that the model gener- +alizes well to the intermediate lengths of {4, 6, 8}, +with a small number of correctly translated exam- +ples longer than the maximum of 9 used in training. +Otherwise, the model fails to generalize to longer +unseen lengths, which is not surprising, given that +Transformer models are known to fail dramatically +at systematic generalization on longer inputs for +various NLP tasks (Csordás et al., 2021), or to in- +cur substantial decrease in accuracy for longer sym- +bolic integration problems (Welleck et al., 2022). +Switching to semantic-level evaluation leads to a +significant increase in accuracy for COMPOSITES, +with a more modest increase for EVEN-ODD. + +4.2 +Code Correctness Statements +We extend our scope to include data representing +proofs of program correctness using the language +of Hoare logic. We train a separate model with +the same embedding and state sizes, feed forward +width, and learning rates as in Section 4.1. Depth +is increased to 8 encoder and decoder blocks with 8 +recurrent passes, 8 attention heads, and a clipping +value of 8. The model is trained over minibatches +of size 1 with Adam, with a patience of 3 epochs. +The POLY results shown in Table 1 demonstrate +that the model is able to generalize to program line +counts of {4, 6, 8, 10} unseen during training with +diminishing returns as the program length grows, +eventually failing to generalize for lengths longer +than the maximum seen in training. We observe +that increasing the depth of the model significantly +improved generalization. +A model with identi- +cal hyperparameters to the arithmetic experiment +yielded less then half the sequence-level accuracy +for intermediate program lengths. Therefore, fur- +ther increasing the depth of the model could push +performance closer to optimal generalization to in- +termediate lengths at the cost of significantly more +computing resources. Additionally, POLY exam- +ples are far less prone to non-fatal token swapping +errors. We observe that semantic-level accuracy is +identical to sequence-level, as all copying errors +compromised the validity of the proof. Therefore, +accuracies are shown as one column (Both). +4.3 +Handwritten Examples +We also evaluate the semantic-level accuracy of +the trained models on the collection of 45 human- +written LaTeX theorem-proof pairs. This is done +by manually verifying that the generated Coq the- +orem corresponds to the LaTeX version and that +the subsequent proof is correct according to the +Coq interpreter. The fully trained model achieved +53.3% for both EVEN-ODD and COMPOSITES, and +73.3% for POWERS. +Mistakes in almost all cases are confined to the +mishandling of out-of-vocabulary tokens, such as +mis-copying a variable within a definition or the +omission of an assertion in the proof tied to a term. +The model otherwise generated syntactically sound +Coq code. Mistakes strongly correlate with exam- +ples that deviate significantly from the grammatical +structure of the artificial data. Thus, pre-trained lan- +guage models as evaluated by Wu et al. (2022) or +pre-training new models on mathematical corpora +like MATH (Hendrycks et al., 2021) may serve to +alleviate the problems caused by the scarcity of +aligned natural and formal mathematics data. +5 +Concluding Remarks +As we have seen, it is feasible to train machine +learning models to perform autoformalization over +very restricted domains of math and code correct- +ness proofs. These models show capability to sys- +tematically generalize to new expression lengths +and program sizes. Moreover, these models were +able to translate previously unseen hand written +natural language examples, albeit with lower ac- +curacy. We are hopeful that this approach can be +applied to autoformalization of a larger segment of +mathematics and code verification. +As mentioned by Szegedy (2020), "Autoformal- +ization is not just a challenge: successful autofor- +malization would represent a breakthrough for gen- +eral AI with significant implications in various do- +mains." We see an especially significant impact in +education, where integration of autoformalization +into proof assistants for introductory mathematics +and software verification courses would enable the +detection of missing steps or misconceptions in +students’ proofs. +References +K. Appel and W. Haken. 1977. Every planar map is +four colorable, part I: discharging. Illinois Journal +of Mathematics, 21(3):429 – 490. +K. Appel, W. Haken, and J. Koch. 1977. Every planar +map is four colorable. II: Reducibility. Ill. J. Math., +21:491–567. +Grzegorz Bancerek. 2006. +Automatic translation in +formalized mathematics. Mechanized Mathematics +and Its Applications, 5(2):19–31. +Yves Bertot and Pierre Castéran. 2013. Interactive the- +orem proving and program development: Coq’Art: +the calculus of inductive constructions. +Springer +Science & Business Media. +Benjamin Andrew Carman. 2021. Translating LaTeX +to Coq: A Recurrent Neural Network Approach to +Formalizing Natural Language Proofs. Ph.D. thesis, +Ohio University. +Róbert Csordás, Kazuki Irie, and Juergen Schmidhu- +ber. 2021. The devil is in the detail: Simple tricks +improve systematic generalization of transformers. +In Proceedings of the 2021 Conference on Empiri- +cal Methods in Natural Language Processing, pages +619–634. + +Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, +Jakob Uszkoreit, and Łukasz Kaiser. 2018. Univer- +sal transformers. arXiv preprint arXiv:1807.03819. +Deborah Ferreira, Mokanarangan Thayaparan, Marco +Valentino, Julia Rozanova, and Andre Freitas. 2022. +To be or not to be an Integer? Encoding Variables for +Mathematical Text. In Findings of the Association +for Computational Linguistics: ACL 2022, pages +938–948, Dublin, Ireland. Association for Compu- +tational Linguistics. +G. Gonthier. 2013. Engineering mathematics: The odd +order theorem proof. In Conference Record of the +Annual ACM Symposium on Principles of Program- +ming Languages, pages 1–2. Cited By 12. +Georges Gonthier. 2008. Formal proof – the four-color +theorem. Notices of the AMS, 55(11):1382–1393. +Georges Gonthier, +Andrea Asperti, +Jeremy Avi- +gad, Yves Bertot, Cyril Cohen, François Garil- +lot, Stéphane Le Roux, Assia Mahboubi, Russell +O’Connor, Sidi Ould Biha, Ioana Pasca, Laurence +Rideau, Alexey Solovyev, Enrico Tassi, and Lau- +rent Théry. 2013. A machine-checked proof of the +odd order theorem. In Interactive Theorem Proving, +pages 163–179, Berlin, Heidelberg. Springer Berlin +Heidelberg. +Jiatao Gu, Zhengdong Lu, Hang Li, and Victor OK +Li. 2016. +Incorporating copying mechanism in +sequence-to-sequence learning. In Proceedings of +the 54th Annual Meeting of the Association for Com- +putational Linguistics (Volume 1: +Long Papers), +pages 1631–1640. +Thomas Hales, Mark Adams, Gertrud Bauer, Tat Dat +Dang, John Harrison, Le Truong Hoang, Cezary +Kaliszyk, +Victor +Magron, +Sean +McLaughlin, +Tat Thang Nguyen, and et al. 2017. A formal proof +of the Kepler conjecture. Forum of Mathematics, Pi, +5:e2. +Thomas C. Hales. 2005. A proof of the Kepler conjec- +ture. Ann. Math. (2), 162(3):1065–1185. +Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul +Arora, Steven Basart, Eric Tang, Dawn Song, and +Jacob Steinhardt. 2021. +Measuring mathematical +problem solving with the math dataset. NeurIPS. +Nicholas J. Higham. 2020. Handbook of Writing for +the Mathematical Sciences, third edition. +Society +for Industrial and Applied Mathematics, Philadel- +phia, PA, USA. +Andrew David Irvine and Harry Deutsch. 2021. Rus- +sell’s Paradox. In Edward N. Zalta, editor, The Stan- +ford Encyclopedia of Philosophy, Spring 2021 edi- +tion. Metaphysics Research Lab, Stanford Univer- +sity. +Xavier Leroy. 2009. Formal verification of a realistic +compiler. Communications of the ACM, 52(7):107– +115. +Tobias Nipkow, Lawrence C Paulson, and Markus +Wenzel. 2002. Isabelle/HOL: a proof assistant for +higher-order logic, volume 2283. Springer Science +& Business Media. +Benjamin +C +Pierce, +Chris +Casinghino, +Marco +Gaboardi, +Michael +Greenberg, +C˘at˘alin +Hri¸tcu, +Vilhelm Sjöberg, and Brent Yorgey. 2010. Software +foundations. +Webpage: +http://www. cis. upenn. +edu/bcpierce/sf/current/index. html. +Benjamin C. Pierce, Arthur Azevedo de Amorim, Chris +Casinghino, Marco Gaboardi, Michael Greenberg, +Cˇatˇalin Hri¸tcu, Vilhelm Sjöberg, Andrew Tolmach, +and Brent Yorgey. 2018. Programming Language +Foundations. Software Foundations series, volume +2. Electronic textbook. +Version 5.5. http://www. +cis.upenn.edu/~bcpierce/sf. +Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. +2018. Self-attention with relative position represen- +tations. In Proceedings of the 2018 Conference of +the North American Chapter of the Association for +Computational Linguistics: Human Language Tech- +nologies, Volume 2 (Short Papers), pages 464–468, +New Orleans, Louisiana. Association for Computa- +tional Linguistics. +Christian Szegedy. 2020. +A promising path towards +autoformalization and general artificial intelligence. +In Intelligent Computer Mathematics. CICM 2020. +Lecture Notes in Computer Science, volume 12236, +pages 3–20. Springer. +Thomas Tymoczko. 1979. The four-color problem and +its philosophical significance. +Journal of Philoso- +phy, 76(2):57–83. +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob +Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz +Kaiser, and Illia Polosukhin. 2017. Attention is all +you need. Advances in Neural Information Process- +ing Systems, 30:5998–6008. +Qingxiang Wang, Chad Brown, Cezary Kaliszyk, and +Josef Urban. 2020. Exploration of neural machine +translation in autoformalization of mathematics in +Mizar. In Proceedings of the 9th ACM SIGPLAN +International Conference on Certified Programs and +Proofs, CPP 2020, page 85–98, New York, NY, USA. +Association for Computing Machinery. +Qingxiang Wang, Cezary Kaliszyk, and Josef Urban. +2018. First experiments with neural translation of +informal to formal mathematics. In Intelligent Com- +puter Mathematics, pages 255–270, Cham. Springer +International Publishing. +Sean Welleck, Peter West, Jize Cao, and Yejin Choi. +2022. +Symbolic Brittleness in Sequence Models: +On Systematic Generalization in Symbolic Mathe- +matics. Proceedings of the AAAI Conference on Ar- +tificial Intelligence, 36(8):8629–8637. Number: 8. + +Yuhuai Wu, Albert Q Jiang, Wenda Li, Markus N +Rabe, Charles Staats, Mateja Jamnik, and Christian +Szegedy. 2022. +Autoformalization with large lan- +guage models. arXiv preprint arXiv:2205.12615. + diff --git a/ENE0T4oBgHgl3EQfQgDK/content/tmp_files/load_file.txt b/ENE0T4oBgHgl3EQfQgDK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2407165655362b1cc11f2564a3e95055db40ac84 --- /dev/null +++ b/ENE0T4oBgHgl3EQfQgDK/content/tmp_files/load_file.txt @@ -0,0 +1,544 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf,len=543 +page_content='Towards Autoformalization of Mathematics and Code Correctness: Experiments with Elementary Proofs Garett Cunningham School of EECS Ohio University Athens, OH 45701 gc974517@ohio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='edu Razvan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Bunescu Department of Computer Science UNC Charlotte Charlotte, NC 28223 razvan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='bunescu@uncc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='edu David Juedes School of EECS Ohio University Athens, OH 45701 juedes@ohio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='edu Abstract The ever-growing complexity of mathemati- cal proofs makes their manual verification by mathematicians very cognitively demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Autoformalization seeks to address this by translating proofs written in natural language into a formal representation that is computer- verifiable via interactive theorem provers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In this paper, we introduce a semantic parsing ap- proach, based on the Universal Transformer architecture, that translates elementary math- ematical proofs into an equivalent formaliza- tion in the language of the Coq interactive the- orem prover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The same architecture is also trained to translate simple imperative code dec- orated with Hoare triples into formally veri- fiable proofs of correctness in Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Exper- iments on a limited domain of artificial and human-written proofs show that the models generalize well to intermediate lengths not seen during training and variations in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 1 Introduction To the uninitiated, the notion of mathematical proof represents simply an argument written by people to convince others of mathematical truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' How- ever, in a real sense, mathematical proof must have formal underpinnings that go beyond the written argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Arguments that lack such underpinnings might have fatal errors or even logical inconsisten- cies (see, for example, Russell’s Paradox (Irvine and Deutsch, 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Nevertheless, mathematical arguments written in natural language are the norm and they have great value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Tymoczko (1979)’s well-known paper that dis- cusses a somewhat controversial (at the time) proof of the Four Color Theorem (Appel and Haken, 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Appel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 1977), he explores “what is a mathematical proof?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' He posits that all mathemati- cal proofs must be (i) convincing, (ii) surveyable, and (iii) formalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The first two points are for the reader—proofs must be convincing to and com- prehensible by mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For the third point, he notes that, “Most mathematicians and philoso- phers believe that any acceptable proof can be for- malized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We can always find an appropriate formal language and theory in which the informal proof can be embedded and ‘filled out’ into a rigorous formal proof.” For most mathematicians, this third part is crucial for ensuring that subtle, but fatal, errors in logic do not exist in mathematical proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Great progress has been made since the 1970’s in fully formalizing significant mathematical re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For instance, the Feit-Thompson Theorem (Gonthier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Gonthier, 2013) and the Four Color Theorem (Gonthier, 2008) have been for- mally verified using the proof assistant Coq (Bertot and Castéran, 2013), and the Kepler Conjecture (Hales, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2017) has been formally verified using the proof assistants Isabelle and HOL Light (Nipkow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Moreover, proof assis- tants have demonstrated immense utility for soft- ware verification, such as the full certification of a C compiler (Leroy, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proofs demonstrating the correct behavior of code share a similar struc- ture to proofs in pure mathematics, where systems like Hoare logic replace standard first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Thus, Tymoczko’s criteria for mathematical proof can be extended to the verification of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For many experts, LaTeX provides an excellent tool for satisfying the first two criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In addition, carefully written LaTeX (Higham, 2020) provides a rich structure for establishing the third criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The vast majority of modern mathematics is ex- pressed using natural language (NL), with the over- whelming majority typeset in LaTeX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Fully for- malizing mathematics using proof assistants is still a difficult and time consuming task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This paper takes some preliminary steps toward bridging this gap by exploring how modern machine learning techniques can be used to convert carefully writ- ten LaTeX into equivalent, and formally verified arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='02195v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='CL] 5 Jan 2023 mathematics in Coq, a process referred to as auto- formalization in the literature (Szegedy, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2018, 2020) explored the similar task of translating mathematical statements from LaTeX into Mizar, using LSTM-based models with attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' To generate aligned LaTeX-Mizar pairs, they use a tool (Bancerek, 2006) that translates top-level Mizar statements into artificial LaTeX sentences, a task that is facilitated by the fact that Mizar is human readable and similar in length with the corresponding LaTeX version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Carman (2021) evaluated the competency of LSTMs toward for- malizing a restricted set of artificially generated the- orems about simple arithmetic expressions, report- ing reasonable success over expression lengths seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' More recently, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2022) evaluated Codex and PaLM on a significantly more limited, but human-written set of theorems in alge- bra and number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In contrast to prior work, we address the auto- formalization of both theorems and their proofs, and extend the scope to proofs of code correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We use a number of manually written mathemati- cal statements to abstract a complex grammar that is then used to generate a dataset of substantially longer and more diverse mathematical theorems and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We develop an architecture based on the Universal Transformer (Dehghani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2018) and adapt a copying mechanism (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2016) to handle arbitrary numbers and variable names at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The models are evaluated exten- sively on their ability to systematically generalize to statement lengths not seen during training, for which we report sequence-level accuracy as well as a semantic-level accuracy calculated by combin- ing sequence-level accuracy for the theorem and running Coq to determine if the generated proof is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Code and data are made available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='com/gc974517/autoformalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2 Dataset of Theorems and Proofs We create two independent datasets of mathemat- ical statements that overall correspond to four classes of theorems and proofs: the first dataset con- tains three classes of arithmetic statements (EVEN- ODD, COMPOSITES, and POWERS), described in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1, and the second dataset contain- ing statements about code correctness via Hoare logic (POLY), described in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In each example, the input theorem-proof pair is given in LaTeX, whereas the formalized output is represented in Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This work focuses on the proof assistant Coq (Bertot and Castéran, 2013) because (a) there is a rich set of mathematical libraries that have been developed for it, (b) it has been used successfully to reason about significant computa- tion artifacts, such as the ComperCert C compiler (Leroy, 2009)), and (c) it benefits from a rich set of training material for the proof assistant related to software verification (Pierce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Each class of examples demonstrates features necessary for the successful autoformalization of mathematical theorems and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For example, POWERS and COMPOSITES examples may define useful terminology to make the theorems shorter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' proving that 4 is a square, or conversely they may state theorems directly without any prelim- inary definitions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' proving ∃n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' n2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' As shown in Figures 3 and 4, this corresponds in Coq to aliasing propositions using the Definition key- word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Additionally, the examples in the dataset provide a stress test of the copying mechanism de- scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1, testing its ability to learn the correct order and number of terms to include in mathematical expressions, as well as their place- ment in theorems and proofs, in a way that general- izes to arbitrary tokens in mathematical language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For each of the four classes of theorems and proofs, we manually created a few examples our- selves in order to guide the construction of a com- plex grammar that is then used to generate a dataset of substantially longer and more diverse mathemat- ical theorems and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Each dataset is generated using its corresponding grammar in an identical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' First, a random seed is sampled that controls the overall structure of the theorem, proof, and definition, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Then, the skeleton structure of the proof is completed with phrases that are sam- pled from a separate context-free grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The coarse control of the skeleton structure allows the construction of examples with interesting features like sublemmas, forward or backward proof direc- tion, coreference, or additional conditions for the theorem, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Many of the difficulties in formalizing mathe- matical statements from NL into Coq stem from the wide variability in the level of detail of mathe- matical proofs, and the frequent mismatch between what is considered an acceptable inference step in NL proofs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' an inference step in Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Further- more, there may be multiple Coq proofs for any given theorem, at different levels of granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We LaTeX Input Sequence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 28M + 308 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We know the summation between even numbers in N will be an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Observe that 308 is known to be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Additionally, note that the pair M × 28 is trivially even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This is true because the coefficient 28 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Coq Output Sequence Require Import Arith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Theorem M28_308: forall M : nat, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even(28 * M + 308) = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' intros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' repeat rewrite Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even_add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' assert (H1: Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even 308 = true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' { auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' } assert (H2: Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even 28 = true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' { auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' } assert (H3: Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even (28 * M) = true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' { rewrite Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even_mul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' rewrite H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' } rewrite H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' rewrite H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Figure 1: Generated example from the EVEN-ODD set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' address this ambiguity by requiring the structure of the Coq proof to match the overall structure of the NL proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This is achieved by quasi-synchronously generating the LaTeX and Coq versions of mathe- matical statements, while still allowing for some simple re-orderings in order to improve general- ization performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' swapping arguments of commutative operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In total, the grammar-based method for gen- erating examples can theoretically produce over 283 million unique arithmetic examples and over 491,000 unique code examples, before consider- ing variations in phrasing by sampling from the context-free grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 Arithmetic Statements We generated three classes of mathematical state- ments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' theorem-proof pairs: EVEN-ODD: an expression is even or odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' COMPOSITES: a number is composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' POWERS: a number is an integer power of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' EVEN-ODD examples contain arithmetic expres- sions of n variables with even coefficients that are summed with a constant term, meaning that the parity of this constant determines the parity of the LaTeX Input Sequence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 450 + a · 192 + j · 462 is guaranteed to be even for any natural terms j, and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' It can be justified that 192 · a + j · 462 is trivially even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Note that 192a is an even number in N because multiplying between an even integer with an arbitrary number in N is guaranteed to be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Likewise, 462j is trivially an even number in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The claim is proven as a consequence of the fact that the sum of even numbers with an even number will be in itself an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Therefore, our theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Coq Output Sequence Require Import Arith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Theorem a450_192j_450_even: forall j a : nat, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even (192 * a + 462 * j + 450) = true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' intros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' rewrite Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even_add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' assert (H1: Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even (192 * a) = true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' { rewrite Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even_mul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' } assert (H2: Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even (462 * j) = true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' { rewrite Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even_mul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' } assert (H3: Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even (192 * a + 462 * j) = true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' { repeat rewrite Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='even_add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' rewrite H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' rewrite H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' } rewrite H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' auto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Figure 2: Instance of sublemma use in the EVEN-ODD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The proof that the sum of non-constant terms is even (assertion H3) is given before proving the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' whole expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proofs make use of this fact with varying rigor based on our manually designed grammar, an example of which is shown by Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The Coq program is generated concurrently with the paired LaTeX example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The example shown in Figure 2 illustrates the generation and use of prior facts to prove an implicit sublemma, in both the natural language and matching Coq version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Examples of theorems and proofs for POWERS and COMPOSITES share a similar structure in both their LaTeX and Coq forms, as shown in Figures 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The theorems assert the ex- istence of a natural number such that a defining property holds and their proofs are constructive, with the distinction that examples for composites prove factorization into n factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For both training and testing, we generate 5,000 even-odd, 5,000 composites, and 2,000 powers ex- LaTeX Input Sequence Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We define that w ∈ N is a composite natural number if taking some R, Q ∈ N we have Q, R ≥ 2 and Q × R = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 35 is a composite whole number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Remember that a composite natural number is the multiplication between Q and R such that Q and R ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Allow R = 7, Q = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We justify the result is valid as 35 = Q · R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Coq Output Sequence Require Import Lia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Definition composite (w : nat) := exists R Q : nat, (Q >= 2) /\\ (R >= 2) /\\ (Q * R = w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Theorem w_composite: composite 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' unfold composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' exists 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' exists 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' lia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Figure 3: Generated COMPOSITES example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' LaTeX Input Sequence Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We say o is a square whenever there exists some whole number Z such that Z ≥ 2 and o = Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' o = 64 is a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Let Z = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Observe that 64 = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Also notice Z = 8 is more than or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This yields 64 is a square whole number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Coq Output Sequence Require Import Lia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Definition square (o : nat) := exists Z : nat, (Z >= 2) /\\ (o = Z^2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Theorem square_64: square 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' unfold square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' exists 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' assert (H1: 8 >= 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' { lia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' } repeat split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' apply H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Figure 4: Generated example from the POWERS set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We train on values of n ∈ {2, 3, 5, 7, 9} and test on values n ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' , 12}, where n rep- resents the number of variables in the arithmetic expression, the number of factors, or the power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This is done in order to evaluate the model’s ability to generalize to unseen arithmetic expression lengths and numbers of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 Handwritten Examples We also created a small collection of 45 human- written LaTeX theorem-proof pairs to evaluate per- formance on examples outside of our manually generated grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' These are distinct from the original manually written examples that were used to guide the development of the generative gram- mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' There are 15 examples for each type of proof from the arithmetic set, using the same vocabulary with a number of unseen grammatical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 Code Correctness Statements We create a dataset of correctness proofs about short programs written in the imperative program- ming language Imp (Pierce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2018), which we call POLY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The programs represent various algo- rithms for evaluating a polynomial, and their proofs of correctness verify that the programs correctly model the polynomial as a mathematical function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proofs are conducted as either fully decorated pro- grams or as sequences of Hoare triples with natural language justifying steps in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' An example is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For both training and testing data, we generate 5,000 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We train on programs containing 2, 3, 5, 7, 9, and 11 lines, then test on programs con- taining from 2 up to 14 lines to evaluate the model’s ability to generalize to novel program lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 3 Semantic Parsing Architecture To formalize LaTeX statements into Coq, we de- veloped an encoder-decoder architecture based on the Universal Transformer (Dehghani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Similar to Csordás et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2021), we do so by adding recursive passes into the encoder and decoder of a base Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2017), thus making the model analogous to a Universal Trans- former without adaptive computation time (ACT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Further, we introduce a copying mechanism and support for out-of-vocabulary mathematical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 Copying Mechanism Mathematical language contains features uncom- mon or non-existent in natural language, such as numbers, variables, and carefully defined terminol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In order to address the use of general math- ematical jargon, these tokens are replaced in the LaTeX Input Sequence Coq Output Sequence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Consider the following series of commands such that S := 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' S := 3 + S * Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' S := 1 + S * Z Allow Z = y, for any natural number y, ahead of running this code then S = 3×y2+3×y+1 after the set of instructions has executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' By application of usual Hoare logic: {Z = y} S := 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' {Z = y ∧ S = 3} S := 3 + S * Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' {Z = y ∧ S = 3 × y + 3} S := 1 + S * Z {Z = y ∧ S = 3 × y2 + 3 × y + 1} Hence, this program is shown to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Require Import String.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' From PLF Require Import Imp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' From PLF Require Import Hoare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Theorem poly_code_correct: forall y : nat, {{ Z = y }} S := 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' S := 3 + S * Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' S := 1 + S * Z {{ S = 3 * y ^ 2 + 3 * y + 1 }}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' intros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' apply hoare_seq with (Q := ( (Z = y /\\ S = 3) )%assertion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' apply hoare_seq with (Q := ( (Z = y /\\ S = 3 * y + 3) )%assertion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' apply hoare_seq with (Q := ( (Z = y /\\ S = 3 * y^2 + 3 * y + 1) )%assertion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' all: eapply hoare_consequence_pre;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=" try (apply hoare_asgn || assn_auto'')." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Figure 5: Generated POLY example: [Left] the Hoare logic proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' [Right] the code correctness proof in Coq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' LaTeX input with generic forms denoting their us- age, such as up to for variables, which effectively ensures generalization to vari- able renaming (Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2022), up to for numbers, or for definitions, cou- pled with the use of a copying mechanism adapted from Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Note that a different generic token is introduced for each unique numerical con- stant or variable literal in the theorem and its proof, and the corresponding generic token is used in the Coq version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' For example, considering the ⟨LaTeX, Coq⟩ pair in Figure 3, , , , and would be used to replace the constants 2, 35, 7, and 5 respectively, everywhere in the LaTeX and Coq statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Similarly, , , and were used to replace variable literals w, R, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This is in contrast to using just two generic tokens and every- where, which would make all numbers coreferent and all variables coreferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Preliminary experi- ments validated the utility of encoding these dis- tinctions while maintaining the correct coreference in both LaTeX and Coq statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Overall, by using generic tokens for numbers, variables, and definitions, only a limited set of em- beddings need to be trained and the model is forced to utilize contextual information in order to appro- priately copy tokens into the Coq output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In this way, the model has the ability to generalize to un- seen numbers or variable and definition names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The original CopyNet (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2016) used an encoder-decoder architecture with a copying mech- anism to calculate the probabilities of generating in-vocabulary tokens vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' copying tokens from the input sequence to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Our autoformaliza- tion task guarantees mutual exclusivity between generating (g) and copying (c) tokens, which al- lows using a simplified formula for calculating the probability of producing a token yt at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Letting Vc denote the Coq vocabulary, X denote the input sequence of LaTeX tokens, and X denote the collection of unique tokens in X, we calculate the probability of producing yt as: p(yt) = � � � � � � � p(yt, g) = 1 Zt eψg(yt), yt ∈ Vc p(yt, c) = 1 Zt � xj∈X:xj=yt eψc(xj), yt ∈ X where Zt = � yt∈Vc eψg(yt) + � xj∈X eψc(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The scor- ing functions are given by ψg(yt) = v⊤ ytWost and ψc(xj) = tanh � h⊤ j Wc � st, where vyt is a one- hot encoding of yt, hj is the hidden encoder state for the input token xj, st is the decoder state at step t, and Wo and Wc are learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 Encoder-Decoder Architecture We diverge from the standard Transformer archi- tecture in a few crucial ways: Probabilities are calculated via p(yt) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Absolute positional encodings are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Self-attention uses relative positional repre- sentations as in Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Stacks of N encoder/decoder blocks have T recurrent passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' All other aspects of the model remain unchanged from the original Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We emphasize rel- ative positional information over absolute in our model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Preliminary evaluations on the EVEN-ODD dataset showed that Transformer mod- els that use absolute positional encodings obtain 0% sequence-level accuracy on expression lengths that are not seen at training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Removing re- liance on absolute position resolves this type of systematic generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The use of relative posi- tional encodings for the Transformer-based models was thus essential for achieving stronger systematic generalization, which also agrees with the findings of Csordás et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2021) on other NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 4 Experimental Evaluations To evaluate the performance of trained models, we ran two primary experiments: first on the collection of arithmetic examples, then on the collection of code correctness examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' All models are eval- uated in terms of sequence-level accuracy, where an example is considered correctly processed only if the generated Coq sequence for both the theo- rem and its proof perfectly matches token by to- ken the ground truth sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We also report semantic-level accuracy, for which the generated Coq theorem needs to attains a perfect sequence- level accuracy and the Coq engine verifies that the generated Coq proof truly proves the generated Coq theorm, regardless of whether it matches the ground truth version of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This empha- sizes that the model was able to capture the general meaning of the natural language proof by correctly translating the theorem and successfully proving it EVEN-ODD COMPOSITES POLY n Seq Sem Seq Sem Both 2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='9 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='8 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='5 8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='7 9 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='7 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='6 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='5 12+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 POWERS Seq = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='0 Sem = 100 Table 1: Sequence-level (Seq) and semantic-level (Sem) accuracy (%) on test examples, split by expres- sion length, with the exception of POWERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' using the natural language version as a guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' All experiments were performed on one NVIDIA RTX-A6000 GPU with 48GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1 Arithmetic Statements We evaluate a Transformer model on the full data combining EVEN-ODD + COMPOSITES + POWERS and using both the theorem and its proof in each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We tune a model with embedding and state sizes of 32, a feed forward width of 256, 4 encoder and decoder blocks with 4 recurrent passes, 4 attention heads, and a clipping value of 2 for self- attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We trained this model over minibatches of size 20, optimized with Adam using β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='98, ε = 1e − 9, and an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='001, annealed by a factor of 1/ √ 10 based on training loss plateaus with a patience of 5 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The results in Table 1 show that the model gener- alizes well to the intermediate lengths of {4, 6, 8}, with a small number of correctly translated exam- ples longer than the maximum of 9 used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Otherwise, the model fails to generalize to longer unseen lengths, which is not surprising, given that Transformer models are known to fail dramatically at systematic generalization on longer inputs for various NLP tasks (Csordás et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2021), or to in- cur substantial decrease in accuracy for longer sym- bolic integration problems (Welleck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Switching to semantic-level evaluation leads to a significant increase in accuracy for COMPOSITES, with a more modest increase for EVEN-ODD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='2 Code Correctness Statements We extend our scope to include data representing proofs of program correctness using the language of Hoare logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We train a separate model with the same embedding and state sizes, feed forward width, and learning rates as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Depth is increased to 8 encoder and decoder blocks with 8 recurrent passes, 8 attention heads, and a clipping value of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The model is trained over minibatches of size 1 with Adam, with a patience of 3 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The POLY results shown in Table 1 demonstrate that the model is able to generalize to program line counts of {4, 6, 8, 10} unseen during training with diminishing returns as the program length grows, eventually failing to generalize for lengths longer than the maximum seen in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We observe that increasing the depth of the model significantly improved generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' A model with identi- cal hyperparameters to the arithmetic experiment yielded less then half the sequence-level accuracy for intermediate program lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Therefore, fur- ther increasing the depth of the model could push performance closer to optimal generalization to in- termediate lengths at the cost of significantly more computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Additionally, POLY exam- ples are far less prone to non-fatal token swapping errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We observe that semantic-level accuracy is identical to sequence-level, as all copying errors compromised the validity of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Therefore, accuracies are shown as one column (Both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='3 Handwritten Examples We also evaluate the semantic-level accuracy of the trained models on the collection of 45 human- written LaTeX theorem-proof pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' This is done by manually verifying that the generated Coq the- orem corresponds to the LaTeX version and that the subsequent proof is correct according to the Coq interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The fully trained model achieved 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='3% for both EVEN-ODD and COMPOSITES, and 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='3% for POWERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Mistakes in almost all cases are confined to the mishandling of out-of-vocabulary tokens, such as mis-copying a variable within a definition or the omission of an assertion in the proof tied to a term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The model otherwise generated syntactically sound Coq code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Mistakes strongly correlate with exam- ples that deviate significantly from the grammatical structure of the artificial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Thus, pre-trained lan- guage models as evaluated by Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2022) or pre-training new models on mathematical corpora like MATH (Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 2021) may serve to alleviate the problems caused by the scarcity of aligned natural and formal mathematics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 5 Concluding Remarks As we have seen, it is feasible to train machine learning models to perform autoformalization over very restricted domains of math and code correct- ness proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' These models show capability to sys- tematically generalize to new expression lengths and program sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Moreover, these models were able to translate previously unseen hand written natural language examples, albeit with lower ac- curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' We are hopeful that this approach can be applied to autoformalization of a larger segment of mathematics and code verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' As mentioned by Szegedy (2020), "Autoformal- ization is not just a challenge: successful autofor- malization would represent a breakthrough for gen- eral AI with significant implications in various do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='" We see an especially significant impact in education, where integration of autoformalization into proof assistants for introductory mathematics and software verification courses would enable the detection of missing steps or misconceptions in students’ proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' References K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Appel and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Haken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Every planar map is four colorable, part I: discharging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Illinois Journal of Mathematics, 21(3):429 – 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Appel, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Haken, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Koch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Every planar map is four colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' II: Reducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=', 21:491–567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Grzegorz Bancerek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Automatic translation in formalized mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Mechanized Mathematics and Its Applications, 5(2):19–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Yves Bertot and Pierre Castéran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Interactive the- orem proving and program development: Coq’Art: the calculus of inductive constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Benjamin Andrew Carman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Translating LaTeX to Coq: A Recurrent Neural Network Approach to Formalizing Natural Language Proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' thesis, Ohio University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Róbert Csordás, Kazuki Irie, and Juergen Schmidhu- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The devil is in the detail: Simple tricks improve systematic generalization of transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empiri- cal Methods in Natural Language Processing, pages 619–634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Łukasz Kaiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Univer- sal transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' arXiv preprint arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='03819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Deborah Ferreira, Mokanarangan Thayaparan, Marco Valentino, Julia Rozanova, and Andre Freitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' To be or not to be an Integer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Encoding Variables for Mathematical Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: ACL 2022, pages 938–948, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Gonthier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Engineering mathematics: The odd order theorem proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Conference Record of the Annual ACM Symposium on Principles of Program- ming Languages, pages 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Cited By 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Georges Gonthier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Formal proof – the four-color theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Notices of the AMS, 55(11):1382–1393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Georges Gonthier, Andrea Asperti, Jeremy Avi- gad, Yves Bertot, Cyril Cohen, François Garil- lot, Stéphane Le Roux, Assia Mahboubi, Russell O’Connor, Sidi Ould Biha, Ioana Pasca, Laurence Rideau, Alexey Solovyev, Enrico Tassi, and Lau- rent Théry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' A machine-checked proof of the odd order theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Interactive Theorem Proving, pages 163–179, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Springer Berlin Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Jiatao Gu, Zhengdong Lu, Hang Li, and Victor OK Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Incorporating copying mechanism in sequence-to-sequence learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Proceedings of the 54th Annual Meeting of the Association for Com- putational Linguistics (Volume 1: Long Papers), pages 1631–1640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Thomas Hales, Mark Adams, Gertrud Bauer, Tat Dat Dang, John Harrison, Le Truong Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat Thang Nguyen, and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' A formal proof of the Kepler conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Forum of Mathematics, Pi, 5:e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Thomas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Hales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' A proof of the Kepler conjec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' (2), 162(3):1065–1185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Measuring mathematical problem solving with the math dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' NeurIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Nicholas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Higham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Handbook of Writing for the Mathematical Sciences, third edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Society for Industrial and Applied Mathematics, Philadel- phia, PA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Andrew David Irvine and Harry Deutsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Rus- sell’s Paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Edward N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Zalta, editor, The Stan- ford Encyclopedia of Philosophy, Spring 2021 edi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Metaphysics Research Lab, Stanford Univer- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Xavier Leroy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Formal verification of a realistic compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Communications of the ACM, 52(7):107– 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Tobias Nipkow, Lawrence C Paulson, and Markus Wenzel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Isabelle/HOL: a proof assistant for higher-order logic, volume 2283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Benjamin C Pierce, Chris Casinghino, Marco Gaboardi, Michael Greenberg, C˘at˘alin Hri¸tcu, Vilhelm Sjöberg, and Brent Yorgey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Software foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Webpage: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' cis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' edu/bcpierce/sf/current/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Benjamin C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Pierce, Arthur Azevedo de Amorim, Chris Casinghino, Marco Gaboardi, Michael Greenberg, Cˇatˇalin Hri¸tcu, Vilhelm Sjöberg, Andrew Tolmach, and Brent Yorgey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Programming Language Foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Software Foundations series, volume 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Electronic textbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Version 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' cis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='edu/~bcpierce/sf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Self-attention with relative position represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Tech- nologies, Volume 2 (Short Papers), pages 464–468, New Orleans, Louisiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Christian Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' A promising path towards autoformalization and general artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Intelligent Computer Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' CICM 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Lecture Notes in Computer Science, volume 12236, pages 3–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Thomas Tymoczko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' The four-color problem and its philosophical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Journal of Philoso- phy, 76(2):57–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Advances in Neural Information Process- ing Systems, 30:5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Qingxiang Wang, Chad Brown, Cezary Kaliszyk, and Josef Urban.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Exploration of neural machine translation in autoformalization of mathematics in Mizar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, page 85–98, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Qingxiang Wang, Cezary Kaliszyk, and Josef Urban.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' First experiments with neural translation of informal to formal mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' In Intelligent Com- puter Mathematics, pages 255–270, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Sean Welleck, Peter West, Jize Cao, and Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Symbolic Brittleness in Sequence Models: On Systematic Generalization in Symbolic Mathe- matics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Ar- tificial Intelligence, 36(8):8629–8637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Number: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Yuhuai Wu, Albert Q Jiang, Wenda Li, Markus N Rabe, Charles Staats, Mateja Jamnik, and Christian Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' Autoformalization with large lan- guage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} +page_content='12615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf'} diff --git a/EdAzT4oBgHgl3EQfwv7y/content/tmp_files/2301.01729v1.pdf.txt b/EdAzT4oBgHgl3EQfwv7y/content/tmp_files/2301.01729v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e61fcf365e53aa9a9117e5c6e7f24d175fb5615a --- /dev/null +++ b/EdAzT4oBgHgl3EQfwv7y/content/tmp_files/2301.01729v1.pdf.txt @@ -0,0 +1,964 @@ +Locally adaptive aggregation of organisms under death risk in rock-paper-scissors models +J. Menezesa,b, E. Rangelb +aInstitute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands +bSchool of Science and Technology, Federal University of Rio Grande do Norte +Caixa Postal 1524, 59072-970, Natal, RN, Brazil +cDepartment of Computer Engineering and Automation, Federal University of Rio Grande do Norte, Av. Senador Salgado Filho 300, Natal, 59078-970, Brazil +Abstract +We run stochastic simulations of the spatial version of the rock-paper-scissors game, considering that individuals use sensory +abilities to scan the environment to detect the presence of enemies. If the local dangerousness level is above a tolerable threshold, +individuals aggregate instead of moving randomly on the lattice. We study the impact of the locally adaptive aggregation on the +organisms’ spatial organisation by measuring the characteristic length scale of the spatial domains occupied by organisms of a +single species. Our results reveal that aggregation is beneficial if triggered when the local density of opponents does not exceed +30%; otherwise, the behavioural strategy may harm individuals by increasing the average death risk. We show that if organisms +can perceive further distances, they can accurately scan and interpret the signals from the neighbourhood, maximising the effects +of the locally adaptive aggregation on the death risk. Finally, we show that the locally adaptive aggregation behaviour promotes +biodiversity independently of the organism’s mobility. The coexistence probability rises if organisms join conspecifics, even in +the presence of a small number of enemies. We verify that our conclusions hold for more complex systems by simulating the +generalised rock-paper-scissors models with five and seven species. Our discoveries may be helpful to ecologists in understanding +systems where organisms’ self-defence behaviour adapts to local environmental cues. +Keywords: population dynamics, cyclic models, stochastic simulations, behavioural strategies +1. Introduction +Behavioural biology has revealed the mechanisms that or- +ganisms use to improve their fitness, being fundamental for the +stability of the rich biodiversity in nature[1–4]. There is plenty +of evidence that self-preservation strategies are properly exe- +cuted because of the organism’s evolutionary ability to scan the +environment cues, perceiving the presence of a nearby enemy +and the energy expended in the action[5–9]. In this scenario, +living in groups facilitates the defence action since individual +protection against enemies is maximised by collective effort in +surveillance and resistance, demanding less individual energy +expenditure on defense against enemies [10–19]. +Cyclic models of biodiversity have been studied using the +rock-paper-scissors game rules, which successfully describe the +nonhierarchical competition interactions found in many biolog- +ical systems [20–29]. However, experiments with bacteria Es- +cherichia coli revealed that the cyclic dominance among three +bacteria strains is insufficient to stabilise the system. It has been +discovered that coexistence is ensured only if individuals inter- +act locally [30]. This shows the central role of space in the sta- +bility of biological systems, as it has been also observed in com- +munities of lizards and systems of competing coral reefs [31– +33]. Furthermore, cyclic dominance has been shown to play a +fundamental role in the spatial interactions in social systems, +public good with punishment, and human bargaining [34, 35]. +There is plenty of evidence that organisms’ mobility plays a +central role in promoting or jeopardising biodiversity in struc- +tured populations [36–46]. +Evidence shows that organisms’ +foraging behaviour may affect biodiversity in the spatial rock- +paper-scissors game [22, 28]. Organisms’ moving to escape +their enemies and find natural resources to the species perpet- +uation may unbalance the cyclic game or decelerate the pop- +ulation dynamics, thus jeopardising or promoting biodiversity +[28, 29, 47–49]. +Recently, it has been shown that aggregation behaviour is an +efficient antipredator strategy in tritrophic predator-prey cyclic +models [49]. Numerical simulations of the Lotka-Volterra ver- +sion of the rock-paper-scissors game revealed that individu- +als’ predation risk decreases if organisms execute a gregarious +movement, instead of exploring the territory to found prey and +reproduce. In contrast with the standard model where organ- +isms move in a random direction, the grouping strategy pro- +duces spiral-type patterns with organisms of the same species +living in spatial domains whose characteristic length depends +on the the distance the individuals can scan their neighbour- +hood, and their cognitive ability to perform the directional self- +preservation movement tactic [49]. +Although the revealing details of the complexity of the spa- +tial interactions, the model in Ref. [49] considers exclusively a +non-adaptive aggregation tactic, i.e., individuals cannot smartly +adapt their movement to trigger the grouping strategy only +when pressured by an imminent enemy’ attack, as happens, for +example, in spider mites communities [9]. In this case, the +unnecessary expenditure is avoided since organisms can con- +Preprint submitted to Journal of LATEX Templates +January 5, 2023 +arXiv:2301.01729v1 [q-bio.PE] 4 Jan 2023 + +1 +3 +2 +1 +2 +3 +Figure 1: The rock-paper-scissors model rules. The black arrows illustrate the +dominance in the spatial game: individuals of species i eliminate organisms of +species i+1, with i = 1, 2, 3 and i±3 = i. Organisms of the same species aggre- +gate when attacked and move randomly when not in danger. Dark blue, pink, +and green represent individuals of species 1, 2, and 3 moving gregariously; +light blue, pink, and green indicate organisms of species 1, 2, and 3 moving +randomly. +tinue freely advancing on the lattice to conquer territory, allow- +ing the population growth [39]. In this work, we sophisticate +the stochastic model to simulate a locally adaptive aggregation +where organisms move gregariously only under death risk [49]. +We also consider that the decision to aggregate is the individual +competence, meaning that each organism acts autonomously +according to its own local reality. Therefore, each individual +can decide if moving gregariously or randomly, with the con- +gregation being triggered only if the local density of enemies +is higher than a tolerable threshold. In addition, we implement +the behavioural survival strategy using the May-Leonard imple- +mentation of the spatial rock-paper-scissors game. This allows +the generalisation of our results to systems where competition +for natural resources is the goal of the cyclic game [50]. +We aim to answer the following questions: i) how does the +locally adaptive aggregation modify the spiral patterns, char- +acteristic of the standard May-Leonard implementation of the +rock-paper-scissors model?; ii) how does the aggregation trig- +ger influence the organisms’ spatial organisation altering the +size of the typical single-species domains?; iii) how does adap- +tive grouping benefit individuals by reducing the average death +risk?; iv) how does the locally adaptive congregation behaviour +impact species coexistence probability? +The outline of this paper is as follows. In Sec. 2, we in- +troduce our stochastic model and present the methods used to +implement the locally adaptive grouping in our simulation algo- +rithm. In Sec. 3, the changes in the spatial patterns are studied +for various values of aggregation trigger; the autocorrelation +function and characteristic length scales are addressed in Sec. +4. The reduction in the organisms’ average death risk is com- +puted in Sec. 5 for a range of aggregation triggers and mobil- +ity probabilities. Finally, the coexistence probability in terms +of the individual’s mobility is investigated in Sec. 6, while our +comments and conclusions appear in Sec. 7. +2. The Model +We study a cyclic model of three species that outcompete +each other according to the rock-paper-scissors game rules, il- +lustrated in Fig. 1. This means that individuals of species i elim- +inate organisms of species i + 1, with i = 1, 2, 3, with the cyclic +identification i = i + 3 β, where β is an integer. Our model con- +siders that organisms of the same species aggregate to minimize +the probability of being killed in the spatial game. The gre- +garious movement is locally adaptive, triggered whenever the +density of enemies in the organisms’ neighbourhood is higher +than a tolerable threshold. This means that each individual of +species i can scan the environment to perceive the presence of +organisms of species i + 1, thus, accurately deciding if the best +strategy is to search for refuge joining their conspecifics. or +continue moving randomly to explore the territory. The dark +colours in Fg.1 stand for individuals executing the gregarious +movement, whereas the light colours represent organisms mov- +ing randomly. +2.1. Numerical simulations +To perform the numerical simulations, we use square lattices +with periodic boundary conditions; the number of grid sites is +N. We use the May-Leonard implementation, where the total +number of individuals is not conserved. Therefore, as each grid +point is occupied by at most one individual (or it is empty), +the maximum number of organisms in the system is the total +number of grid points N. +Initially, the organisms are randomly distributed in the lat- +tice: each individual is allocated at a random grid site. The ini- +tial conditions are prepared so that the number of individuals is +the same for every species is the same. We define the number of +individuals of each species at the initial state as one-third of the +total number of organisms: Ii(t = 0) ≈ N/3, with i = 1, 2, 3; +the rest of grid sites are left empty in the initial conditions. +Once the random initial conditions are ready, the algorithm +stochastically implements the interactions following the von +Neumann neighbourhood, where each organism can interact +with one of its four immediate neighbours. The spatial inter- +actions are: +• Selection: i j → i ⊗ , with j = i + 1, where ⊗ means +an empty space: an individual of species i eliminates a +neighbour of species i + 1 following the rules illustrated in +Fig.1 - the grid site occupied by the eliminated individual +is left empty. +• Reproduction: i ⊗ → i i : an empty space is filled by a new +organism of any species. +• Mobility: i ⊙ → ⊙ i , where ⊙ means either an individual +of any species or an empty site. An organism moves by +switching positions with another individual of any species +or an empty space. +The interactions are implemented following a fixed set of +probabilities which is the same for every species: s (selec- +tion probability), r (reproduction probability), and m (mobility +probability). During the interaction implementation, the code +follows the steps: +1. an active individual of any species is drawn among all or- +ganisms in the lattice; +2. one interaction is randomly chosen following the set of +probabilities rates (s, r, and m); +2 + +(a) +(b) +(c) +(d) +Figure 2: Snapshots captured from simulations of the rock-paper-scissors game with individuals’ locally adaptive aggregation. The realisations ran in lattice with +2002 grid points for a timespan of 2000 generations, with R = 3, r = s = 0.25 and m = 0.5. Figures 2a, 2b, 2c, and 2d show the organisms’ spatial organisation at +the end of Simulation A (ϕ = 1.0), B (ϕ = 0.1), (ϕ = 0.025), and D (ϕ = 0.0), respectively. The colours follow the scheme in Fig. 1, with blue, pink, and green +depicting individuals of species 1, 2, and 3, respectively. Dark and light colours distinguish organisms performing the congregation strategy and moving randomly. +Yellow dots depict empty sites. +3. one of the four immediate neighbours is drawn to suffer +the action (selection, reproduction, and random mobility) +- the only exception is the adaptive gregarious movement, +where the organism move towards the direction with more +conspecifics. +Every time an interaction is implemented, one timestep is +counted. After N timesteps, one generation is completed - our +time unit is one generation. +To understand the population dynamics during the simula- +tions, we calculate the density of organisms of species i, ρi(t), +with i = 1, 2, 3. This is defined as the fraction of the lattice oc- +cupied by individuals of the species i at time t, ρi(t) = Ii(t)/N. +Also, the temporal dependence of the density of empty spaces +is computed as ρ0 = 1 − ρ1 − ρ2 − ρ3. +2.2. Implementing the locally adaptive aggregation strategy +To implement the locally adaptive grouping tactic, we define +the perception radius, R, to represent the maximum distance an +organism of species i can scan the environment to be aware of +the presence of enemies. Thus, the local density of organisms of +each species is computed within a circular area of radius R, cen- +tred in the organism of species i [29, 49]. In addition, we intro- +duce the aggregation trigger, ϕ, to represent the minimum local +density of individuals of species i − 1 (enemies) that stimulates +the organism of species i to move gregariously. This means that +if the local density of organisms of species i − 1 is lower than +ϕ, the individual moves randomly. +The numerical implementation of the gregarious movement +is performed by dividing the observing disc into four circular +sectors, each section in the directions of the one nearest neigh- +bour of the von Neumann neighbourhood [22, 25, 26, 28, 49, +51]. Next, it is computed how many individuals of species i +exist within each circular sector, with organisms on the circu- +lar sector borders assumed to be part of both circular sectors. +Finally, the organism switches positions with the immediate +neighbour in the direction with more conspecifics; a draw in +the event of a tie. +3. Spatial Patterns +Our first goal is to understand the effects of the locally adap- +tive congregation strategy in spatial patterns. Therefore, we ran +a single simulation for four values of the aggregation trigger: +• Simulation A: ϕ = 1.0 - the absence of organisms’ group- +ing behaviour, i.e., individuals do not aggregate even under +death risk; +• Simulation B: ϕ = 0.1 - organisms’ agglomeration occurs +if, at least, 10% neighbours are enemies; +• Simulation C: ϕ = 0.025 - an individual move gregariously +if at least, 2.5% neighbours are enemies; +• Simulation D: ϕ = 0.0 - the gregarious movement is not +locally adaptive, with individuals always grouping inde- +pendently of the presence of enemies surrounding them. +The realisations were performed in lattices with 2002 grid sites, +running for a timespan of 2000 generations. We set the param- +eters to s = r = 0.25, m = 0.5, and R = 3. +Figures 2a, 2b, 2c, and 2d show the individuals’ spatial or- +ganisation at the end of Simulations A, B. C, and D, respec- +tively. To depict each organism, we use the same colours of +the scheme in Fig. 1: blue, pink, and green dots show the in- +dividuals of species 1, 2, and 3, respectively. The organisms +performing the aggregation strategy are highlighted using dark +colours, while the individuals moving randomly appear in light +shades. We also quantified the dynamics of the species densities +for Simulation A, B, C, and D, which are depicted in Figs. 3a, +3b, 3c, and 3d, respectively. As in Fig.1, blue, pink, and green +lines shows the temporal dependence of densities of individuals +of species 1, 2, and 3, respectively; +Let us first focus on Simulation A, where individuals do not +aggregate to protect themselves against enemies (Fig. 2a). Be- +cause of the random initial conditions, selection interactions are +frequent at the beginning of the simulation. After that, spatial +patterns are formed with organisms of the same species occu- +pying departed patches. Since organisms are unaware of the +3 + +0.2 +0.25 +0.3 +0.35 +0.4 +0 +500 +1000 +1500 +2000 +ϕ = 1.0 +ρi +t (generations) +1 +2 +3 +(a) +0.2 +0.25 +0.3 +0.35 +0.4 +0 +500 +1000 +1500 +2000 +ϕ = 0.1 +ρi +t (generations) +1 +2 +3 +(b) +0.2 +0.25 +0.3 +0.35 +0.4 +0 +500 +1000 +1500 +2000 +ϕ = 0.025 +ρi +t (generations) +1 +2 +3 +(c) +0.2 +0.25 +0.3 +0.35 +0.4 +0 +500 +1000 +1500 +2000 +ϕ = 0.0 +ρi +t (generations) +1 +2 +3 +(d) +Figure 3: Dynamics of species densities during the simulations in Fig. 2. The blue, pink, and green lines in Figs. 3a, 3b, 3c, and 3d depict the temporal dependence +of the density of individuals of species 1, 2, and 3, in Simulations A, B, C, and D, respectively. +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0 +500 +1000 +1500 +2000 +ρ0 +t (generations) +ϕ = 0.000 +ϕ = 0.025 +ϕ = 0.100 +ϕ = 1.000 +Figure 4: Temporal dependence of the density of empty spaces in simulations +of Fig. 2. The grey, orange, yellow, and brown lines show the dynamics of +empty sites in Simulations A, B, C, and D, respectively. +neighbourhood, they move randomly, independently of the risk +of being caught. This results in faster dynamics of species den- +sities, with organisms being destroyed and newborns appearing +at a high rate. Consequently, the species densities’ frequency +and amplitude are high, as shown in Fig. 3a. +In addition to the usual pattern formation process driven by +the cyclic game rules, the gregarious movement performed by +individuals under death risk promotes the formation of self- +protection clusters on the border that is attacked by enemies, +as shown in Figs. 2b and 2c. For example, the organisms of +species 2 aggregating (dark pink dots) are concentrated on the +border with spatial domains of species 1 (blue areas). The self- +preservation movement tactic produces a deformation of the +spiral patterns, with individuals concentrating in patches with +smaller sizes since they abdicate to explore extensive areas of +the territory to form clumps. Because of this, the population dy- +namics are decelerated, with reduced frequency and amplitude, +as depicted in Figs.3b and 3c. +Finally, the snapshot in Fig. 3d reveals what occurs in the +case of the non-adaptive aggregation strategy (ϕ = 0.0) - indi- +viduals move gregariously even if no enemy surrounds them. +In this scenario, the population dynamics are altered since the +individuals neglect the conquest of new territories to focus ex- +clusively on the survival movement strategy. This induces a +contraction of the spatial domains occupied by organisms of +a single species, since individuals do not advance in the terri- +tory even if they are not under death risk. Finally, Fig. 4 shows +the temporal dependence of the density of empty spaces, ρ0, in +Simulations A (grey line), B (orange line), C (green line), and +D (brown line). The results show that the density of empty +spaces decreases after an initial period of pattern formation. +Furthermore, the locally congregation reduces the organisms’ +death risk. Because of this, the lower the aggregation trigger, +the more the density of empty spaces is reduced. +4. Autocorrelation Function +Let us now quantify the scale of spatial domains in the pres- +ence of locally adaptive aggregation. For this, we compute the +spatial autocorrelation function. The autocorrelation function +is computed from the inverse Fourier transform of the spectral +density as +C(⃗r′) = F −1{S (⃗k)} +C(0) +, +(1) +where S (⃗k) is given by +S (⃗k) = +� +kx,ky +Φ(⃗κ), +(2) +4 + +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +5 +10 +15 +20 +25 +30 +35 +C +r +ϕ = 0.0 +ϕ = 0.1 +ϕ = 1.0 +Figure 5: Autocorrelation functions in terms of the radial coordinate. The grey, +orange, and brown lines depict the results for the standard model (ϕ = 1.0), +aggregation triggered when at least 10% of neighbours are enemies ϕ = 0.1, and +the non-adaptive aggregation (ϕ = 0.0), respectively. The error bars indicate +the standard deviation; the dashed black line shows the threshold assumed to +calculate the characteristic length scale. The interaction probabilities are r = +s = 0.25 and m = 0.5; the perception radius is R = 3. +and Φ(⃗κ) is Fourier transform +Φ(⃗κ) = F {φ(⃗r) − ⟨φ⟩}. +(3) +The function φ(⃗r) represents the spatial distribution of individ- +uals of species 1, with φ(⃗r) = 0 and φ(⃗r) = 1 indicating the +absence and the presence of an individual of species 1 in at the +position ⃗r in the lattice, respectively). The spatial autocorrela- +tion function is given by +C(r′) = +� +|⃗r′|=x+y +C(⃗r′) +min �2N − (x + y + 1), (x + y + 1)�. +(4) +Moreover, we compute the spatial domains’ scale for C(l) = +0.15, where l is the characteristic length. +We calculated the spatial autocorrelation function in terms +of the radial coordinate r for three cases: absence of group- +ing behaviour (ϕ = 1.0), aggregation triggered when the neigh- +bourhood is, at least, 10% hostile (ϕ = 0.1), and non-adaptive +aggregation (ϕ = 0.0). The outcomes were obtained by run- +ning sets of 100 simulations with different random initial con- +ditions in lattices with 5002 grid sites for a time span of 5000 +generations. To calculate the autocorrelation function, we used +the spatial configuration at the end of the simulation (t = 5000 +generations). Because organisms of every species can perform +the locally adaptive congregation, the autocorrelation function +is the same irrespective of the species; thus, we used the data +from species 1. In all simulations, we considered the interac- +tions probabilities s = r = 0.25 and m = 0.5; the perception +radius was set to R = 3. +The brown, orange, and grey lines in Figure 5 show C as a +function of the radial coordinate r for ϕ = 0.0, ϕ = 0.1, and ϕ = +0.0, respectively; the error bars indicate the standard deviation. +The horizontal dashed black line indicates the threshold used to +calculate the length scale: C(l) = 0.15. The results confirm +that once organisms move gregariously, the average size of the +spatial domains inhabited by a single species decreases. +Figure 6 shows the relative variation of the characteristic +length scale ˜l, defined as ˜l = (l − l0)/l0, where l0 is the value in +the absence of the adaptive aggregation (ϕ = 1.0). We repeated +the set of 100 simulations - starting from different initial condi- +tions - for 0 ≤ ϕ ≤ 0.4, with intervals of δϕ = 0.05. The error +−40 +−35 +−30 +−25 +−20 +−15 +−10 +−5 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +˜l(%) +ϕ +Figure 6: The relative change in the characteristic length scale of the typical +single-species spatial domain as a function of the threshold used to trigger the +gregarious movement compared with the standard model. The simulations ran +in lattices with 5002 grid sites, running until 5000 generations for r = s = 0.25 +and m = 0.5; the perception radius is R = 3. The outcomes were averaged from +sets of 100 simulations starting from different initial conditions; the error bars +show the standard deviation. We assumed the probabilities r = s = 0.25 and +m = 0.5. +bars show the standard deviation; the parameters are the same +used in the simulations in Fig. 5. The outcomes show that the +average group size decreases compared to the standard model, +with the reduction becoming significant for ϕ = 0.0. This hap- +pens because all individuals group themselves, independently +of what is happening in their surroundings, as we observed in +Fig. 2d. +5. The role of the locally adaptive aggregation in the organ- +isms’ death risk +We now investigate the effects of locally adaptive grouping +to reduce the organisms’ death risk. For this purpose, we in- +troduce the death risk, which is calculated as follows: i) it is +counted as the total number of individuals of species i at the +beginning of each generation; ii) the number of organisms of +species i killed by individuals of species i − 1 during the gen- +eration is computed; iii) the death risk, ζ is defined as the ratio +between the number of eliminated organisms and the amount +at the beginning of each generation. Due to the symmetry of +the rock-paper-scissors game rules, the average death risk is the +same for individuals of every species; thus, we choose the re- +sults for species 1 to represent the individuals’ death risk. +5.1. The influence of the aggregation trigger +First, we study the influence of the aggregation trigger ϕ in +the relative decrease of the individuals’ death risk by running +sets of 100 simulations starting from different initial conditions +for 0 ≤ ϕ ≤ 1.0 in intervals of δϕ = 0.1. This experiment was +conducted for two values of perception radius: R = 3 and R = 5; +the interaction probabilities are s = r = 0.25 and m = 0.5. To +guarantee the quality of the results, we remove the data from +the initial pattern formation stage, thus calculating the average +organisms’ death risk in the second half of each realisation. +The purple and red lines in Figure 7 show the organisms’ +death risk in terms of the aggregation trigger for R = 3 and +R = 5, respectively; the standard deviation is shown by error +bars. The outcomes reveal that for ϕ ≥ 0.6, the locally adap- +tive strategy is ineffective in reducing the organisms’ death risk +5 + +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +ζi +ϕ +R = 5 +R = 3 +Figure 7: Organisms’ death risk in terms of the aggregation trigger. The simu- +lations were performed in lattices with 5002 grid sites, running for a timespan +of 5000 generations. The red and purple lines show the outcomes for organisms +with perception radius R = 3 and R = 5, respectively. The results were aver- +aged from sets of 100 simulations starting from different initial conditions; the +standard deviation is depicted by error bars. The interaction probabilities are +s = r = 0.25 and m = 0.5. +−50 +−45 +−40 +−35 +−30 +−25 +−20 +−15 +−10 +0.05 +0.15 +0.25 +0.35 +0.45 +0.55 +0.65 +0.75 +0.85 +0.95 +˜ζ(%) +m +Figure 8: Relative change in the individuals’ death risk in terms of the mobility +probability in simulations running in lattices with 5002 grid sites, running for a +timespan of 5000 generations. We averaged the outcomes sets of 100 simula- +tions starting from different initial conditions; the standard deviation is shown +by error bars. The perception radius is R = 3; the interaction probabilities are +to s = r = (1 − m)/2). +compared with the standard model (ϕ = 1.0). This happens be- +cause most of organism of species i whose neighbourhood con- +tains 60% or more of organisms of species i − 1 is far from the +spatial domain dominated by their conspecifics; thus, grouping +may not be possible to be executed before the individual being +eliminated by enemies. +Our findings show that the locally adaptive aggregation jeop- +ardises the organisms’ safety for intermediate values of ϕ. As +shown in Fig. 7, for R = 3, the organisms’ death risk in- +creases for 0.4 ≤ ϕ < 0.6, while for R = 5, ζ increases for +0.4 ≤ ϕ < 0.3. Therefore, the adaptive is beneficial only if +the threshold assumed to move gregariously is in the interval +0 ≤ ϕ < 0.4 for R = 3 and 0 ≤ ϕ < 0.3 for R = 5, with the +relative reduction of ζ increasing as the ϕ is lowered. +The results in Fig. 7 show how the complexity of the spatial +interactions is influenced by the organism’s ability to make an +accurate decision, triggering the adaptive tactic correctly. Our +findings show that if organisms can perceive further distances, +they can more easily: i) identify the presence of invading en- +emies beyond the border of their territory; ii) distinguish the +direction with more conspecifics in case of need to move gre- +gariously. Because of this, the relative variation in the organ- +isms’ death risk is more accentuated for R = 5 than for R = 3 +in Fig. 7. +5.2. The interference of organisms’ mobility +The locally adaptive grouping is profitable for the organ- +isms because of the death risk reduction, as shown in Fig. 7 +for m = 0.5. +Now, we repeated the simulations to explore +how the benefits of the locally adaptive aggregation depend +on the organism’s mobility. For this purpose, we ran sets of +100 realisations starting from different initial conditions for +0.05 ≤ m ≤ 0.95, in intervals of δm = 0.05. The selection and +reproduction probabilities are set to s = r = (1 − m)/2; the per- +ception radius is R = 3, and the aggregation trigger is ϕ = 0.05. +We implemented the simulations in lattices with 5002 grid sites, +running until 5000 generations. +Figure 8 shows the relative change of the organisms’ death +risk: ˜ζ = (ζ − ζ0)/ζ0, where ζ0 is the death risk in the absence +of grouping behaviour (ϕ = 1.0). For 0.05 ≤ m ≤ 0.085, the +relative reduction in the organisms’ death risk is more signifi- +cant for individuals that explore greater fractions of the lattice +per time unit [39]. This happens because high-mobile individ- +uals are more vulnerable to being eliminated by enemies in the +cyclic game, thus, benefitting more from the self-preservation +movement strategy. However, if m > 0.085, the relative vari- +ation in ζ decreases because the selection probability becomes +very low, becoming the effect less significant. +6. Coexistence Probability +Now, we focus on the impact of locally adaptive flocking on +biodiversity in cyclic games. In this study, we ran sets of 1000 +simulations in lattices with 1002 grid points for 0.05 < m < +0.95 in intervals of δ m = 0.05; selection and reproduction +probabilities were set to s = r = (1 − m)/2. For each set +of simulations, each realisation began from different random +initial conditions, running until 10000 generations. If at least +one species is extinguished before the simulation ends, biodi- +versity is lost. Thus, the coexistence probability is the frac- +tion of the simulations where all species are present at the end. +We extended the investigation to quantify the impact of locally +adaptive aggregation in more complex systems by simulating +the generalised rock-paper-scissors models with five and seven +species. Figures 9a, 9b and 9c depict the coexistence probabil- +ity for ϕ = 0.0 (brown line), ϕ = 0.05 (green line), ϕ = 0.1 +(orange line), ϕ = 0.2 (blue line), and ϕ = 1.0 (grey line) for +the models with N = 3, N = 5, and N = 7 species, respectively. +Overall, species biodiversity is more threatened for systems +with highly mobile individuals, independent of the number of +species in the cyclic game. The outcomes also show the benefits +of the locally adaptive aggregation for biodiversity: the lower +the aggregation trigger, the higher is the coexistence proba- +bility. This conclusion holds independently of the number of +species in the cyclic game Furthermore, the outcomes show +that the more complex the system is, the more favourable it +is for biodiversity loss. By comparing the same color lines in +Fig. 9a, 9b and 9c, one observes that the coexistence probabil- +ity is lower for the system with N = 9 species, independently +of the organisms’ mobility. Finally, we observe that all simula- +tions resulted in coexistence when individuals agglomerate with +6 + +0 +0.2 +0.4 +0.6 +0.8 +1 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Coexistence Probability +m +ϕ = 0.00 +ϕ = 0.05 +ϕ = 0.10 +ϕ = 0.20 +ϕ = 1.00 +(a) +0 +0.2 +0.4 +0.6 +0.8 +1 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Coexistence Probability +m +ϕ = 0.00 +ϕ = 0.05 +ϕ = 0.10 +ϕ = 0.20 +ϕ = 1.00 +(b) +0 +0.2 +0.4 +0.6 +0.8 +1 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Coexistence Probability +m +ϕ = 0.00 +ϕ = 0.05 +ϕ = 0.10 +ϕ = 0.20 +ϕ = 1.00 +(c) +Figure 9: Coexistence probability as a function of the mobility probability +for the generalised rock-paper-scissors game with organisms’ locally adaptive +aggregation. Figures 9a, 9b, and 9c show the outcomes for the cyclic model +with three, five, and seven species, respectively. The results were obtained by +running 1000 simulations in lattices with 1002 grid points running until 10000 +generations for R = 3 and s = r = (1 − m)/2. The brown, green, orange, blue, +and grey lines depict the results for ϕ = 0.0, ϕ = 0.05, ϕ = 0.1, ϕ = 0.2, and +ϕ = 1.0, respectively. +their conspecifics irrespective of the local densities of enemies. +According to the brown lines in Figs. 9a, 9b and 9c. +7. Comments and Conclusions +Aggregation behaviour is found in many systems where or- +ganisms adapt their movement, grouping with their conspecifics +when in death risk. We investigate cyclic models described by +the rock-paper-scissors game rules, where individuals can scan +their environment and adapt their movement to environmental +cues. In our stochastic simulation, each organism freely ex- +plores the territory without precaution if there is no nearby en- +emy but prevents damage from enemy attack moving gregarious +to join the biggest group of conspecific in the neighbourhood. +To execute the locally adaptive grouping, each individual scans +their vicinity, thus triggering the gregarious movement if the lo- +cal density of enemies reaches a prefixed threshold. Running a +series of simulations, we investigate the role of adaptive aggre- +gation in transforming the organisms’ spatial organisation. The +results show that the characteristic length scale of the spatial +domains occupied by organisms of a single species is not ac- +centuated if the threshold is not inferior to 10%. Otherwise, the +typical group size decreases significantly, being minimal in the +case of organisms flock even when not under death risk pres- +sure. +We discover that the gregarious movement does not interfere +with organisms’ safety if the grouping is only triggered when +more than 70% neighbourhood is occupied by enemies. Coun- +terintuitively, if the self-preservation movement tactic is cali- +brated to be triggered if between 30% and 60% neighbours are +enemies, the individuals’ death risk increases instead of bene- +fiting the organisms. Our outcomes show that the behavioural +strategy is profitable only if each organism aggregates with con- +specifics when detecting the fraction of opponents in the vicin- +ity using a threshold inferior to 30%. In addition, we find that +if organisms can perceive further distances, they can accurately +scan and interpret the signals from the neighbourhood, increas- +ing the effects of the adaptive aggregation on the death risk. +Moreover, we study the impact of mobility on the benefits of +adaptive congregation considering low, intermediate and high- +mobile individuals. Our simulations provided evidence that lo- +cally adapting their movement to aggregate when under death +risk is more advantageous as the more mobile the organisms, +provided that the individuals’ mobility is not superior to 85%; +otherwise, the relative death risk reduction diminishes as the +mobility grows. +Finally, we study the influence of locally adaptive aggre- +gation on biodiversity maintenance. +Our findings show that +the coexistence probability increases independently of the or- +ganism’s mobility, being maximal in the case of non-adaptive +grouping, where the gregarious movement is executed even +when there is no local death risk for the individual. This re- +sult holds for more complex systems where an arbitrary odd +number of species participate in the cyclic game. Extending +our algorithm to implement the generalised rock-paper-scissors +model with five and seven species, we confirm that the gregari- +ous movement promotes biodiversity, being more beneficial for +low adaptive aggregation triggers. Our discoveries may be help- +ful to ecologists in understanding systems where organisms’ +self-defence behaviour adapts to local environmental cues. Our +results may also clarify the role of the local phenomena in com- +plex systems in other areas of nonlinear science. +Acknowledgments +We thank CNPq, ECT, Fapern, and IBED for financial and +technical support. +References +[1] M. Begon, C. R. Townsend, J. L. Harper, Ecology: from individuals to +ecosystems, Blackwell Publishing, Oxford, 2006. +7 + +[2] A. Purvis, A. Hector, Getting the measure of biodiversity, Nature 405 +(2000) 212–2019. +[3] R. Buchholz, Behavioural biology: an effective and relevant conservation +tool, Trends in Ecology & Evolution 22 (8) (2007) 401 – 407. +[4] J. A. Tobias, A. L. Pigot, Integrating behaviour and ecology into global +biodiversity conservation strategies, Philosophical Transactions of the +Royal Society B: Biological Sciences 374 (1781) (2019) 20190012. +[5] H. P. Riessen, Cost-benefit model for the induction of an antipredator +defense, American Natiralist 140 (1992) 349–362. +[6] P. Andersson, C. L¨ofstedt, P. A. Hamb¨ack, How insects sense olfac- +tory patches – the spatial scaling of olfactory information, Oikos 122 (7) +(2013) 1009–1016. +[7] S. Downes, A. M. Hoefer, Antipredatory behaviour in lizards: interactions +between group size and predation risk, Animal Behaviour 67 (2004) 485– +492. +[8] M. M. Dehn, Vigilance for predators: detection and dilution effects, Be- +hav. Ecol. Sociobiol. 26 (1990) 337–342. +[9] L. Dittmann, P. Schausberger, Adaptive aggregation by spider mites under +predation risk, Scientific Reports 7 (2017) 2045–2322. +[10] S. L. Lima, Motion extrapolation in catching, Animal Behaviour 49 +(1995) 11–20. +[11] F. J. Wrona, R. W. J. Dixon, Group size and predation risk: A field anal- +ysis of encounter and dilution effects, The American Naturalist 137 (2) +(1991) 186–201. +[12] G. F. Turner, T. J. Pitcher, Attack abatement: A model for group pro- +tection by combined avoidance and dilution, The American Naturalist +128 (2) (1986) 228–240. +[13] J. Berger, A group size, foraging, and antipredator ploys: An analysis of +bighorn sheep decisions, Behav. Ecol. Sociobiol. 4 (1978) 91–99. +[14] L. Dittmann, P. Schausberger, Adaptive aggregation by spider mites under +predation risk, Sci. Rep. 7 (2017) 10609. +[15] V. E. Brock, R. H. Riffenburgh, Fish schooling: A possible factor in re- +ducing predation, ICES Journal of Marine Science 25 (3) (1960) 307–317. +[16] A. Johannesen, A. M. Dunn, L. Morrell, Prey aggregation is an effective +olfactory predator avoidance strategy, Peer J. 2 (2014) e408. +[17] G. R. Hopkins, P. N. Lahanas, Aggregation behaviour in a neotropical +dendrobatid frog (allobates talamancae) in western panama, Behaviour +148 (3) (2011) 359–372. +[18] M. Spieler, K. E. Linsenmair, Aggregation behaviour of bufo maculatus +tadpoles as an antipredator mechanism, Ethology 105 (8) (1999) 665– +686. +[19] B. Guadin, +A. Gazzola, +A. Balestrieri, +G. Scribano, +J. Mart´ın, +D. Pellitteri-Rosa, Effects of a group-living experience on the antipredator +responses of individual tadpoles, Animal Behaviour 180 (2021) 93–99. +[20] P. P. Avelino, D. Bazeia, L. Losano, J. Menezes, B. F. Oliveira, Junctions +and spiral patterns in generalized rock-paper-scissors models, Phys. Rev. +E 86 (2012) 036112. +[21] J. Menezes, B. Moura, T. A. Pereira, Uneven rock-paper-scissors models: +Patterns and coexistence, Europhysics Letters 126 (1) (2019) 18003. +[22] B. Moura, J. Menezes, Behavioural movement strategies in cyclic models, +Scientific Reports 11 (2021) 6413. +[23] P. P. Avelino, J. Menezes, B. F. de Oliveira, T. A. Pereira, Expanding +spatial domains and transient scaling regimes in populations with local +cyclic competition, Phys. Rev. E 99 (2019) 052310. +[24] D. Bazeia, J. Menezes, B. F. de Oliveira, J. G. G. S. Ramos, Hamming +distance and mobility behavior in generalized rock-paper-scissors models, +Europhysics Letters 119 (5) (2017) 58003. +[25] J. Menezes, Antipredator behavior in the rock-paper-scissors model, +Phys. Rev. E 103 (2021) 052216. +[26] J. Menezes, B. Moura, Mobility-limiting antipredator response in the +rock-paper-scissors model, Phys. Rev. E 104 (2021) 054201. +[27] J. Menezes, S. Batista, M. Tenorio, E. Triaca, B. Moura, How local an- +tipredator response unbalances the rock-paper-scissors model, Chaos: An +Interdisciplinary Journal of Nonlinear Science 32 (12) (2022) 123142. +[28] J. Menezes, M. Tenorio, E. Rangel, Adaptive movement strategy may pro- +mote biodiversity in the rock-paper-scissors model, Europhysics Letters +139 (5) (2022) 57002. +[29] M. Tenorio, E. Rangel, J. Menezes, Adaptive movement strategy in rock- +paper-scissors models, Chaos, Solitons & Fractals 162 (2022) 112430. +[30] R. Durret, S. Levin, Allelopathy in spatially distributed populations, J. +Theor. Biol. 185 (1997) 165–171. +[31] B. Sinervo, C. M. Lively, The rock-scissors-paper game and the evolution +of alternative male strategies, Nature 380 (1996) 240–243. +[32] J. B. C. Jackson, L. Buss, The rock-scissors-paper game and the evolution +of alternative male strategies, Proc. Natl Acad. Sci. USA 72 (1975) 5160– +5163. +[33] I. Volkov, J. R. Banavar, S. P. Hubbell, A. Maritan, Patterns of relative +species abundance in rainforests and coral reefs, Nature 450 (2007) 45. +[34] A. Szolnoki, M. Perc, G. Szab´o, Defense mechanisms of empathetic play- +ers in the spatial ultimatum game, Phys. Rev. Lett. 109 (2012) 078701. +[35] A. Szolnoki, M. Perc, Correlation of positive and negative reciprocity +fails to confer an evolutionary advantage: Phase transitions to elementary +strategies. +[36] K. A. Kabir, J. Tanimoto, The role of pairwise nonlinear evolutionary dy- +namics in the rock–paper–scissors game with noise, Applied Mathematics +and Computation 394 (2021) 125767. +[37] A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A. M. Rucklidge, +M. Perc, Cyclic dominance in evolutionary games: a review, Journal of +The Royal Society Interface 11 (100) (2014) 20140735. +[38] A. Szolnoki, J. Vukov, M. c. v. Perc, From pairwise to group interactions +in games of cyclic dominance, Phys. Rev. E 89 (2014) 062125. +[39] T. Reichenbach, M. Mobilia, E. Frey, Mobility promotes and jeopardizes +biodiversity in rock-paper-scissors games, Nature 448 (2007) 1046–1049. +[40] M. Frean, E. R. Abraham, Rock–scissors–paper and the survival of the +weakest, Proc. R. Soc. Lond. B. 268 (2001) 1323–1327. +[41] A. J. Lotka, Undamped oscillations derived from the law of mass action., +Journal of the American Chemical Society 42 (8) (1920) 1595–1599. +[42] V. Volterra, Lecons dur la Theorie Mathematique de la Lutte pour la Vie, +1st Edition, Gauthier-Villars, Paris, (1931). +[43] M. Peltom¨aki, M. Alava, Three- and four-state rock-paper-scissors games +with diffusion, Phys. Rev. E 78 (2008) 031906. +[44] X. Ni, W.-X. Wang, Y.-C. Lai, C. Grebogi, Cyclic competition of mobile +species on continuous space: Pattern formation and coexistence, Phys. +Rev. E 82 (2010) 066211. +[45] J. Park, Effect of external migration on biodiversity in evolutionary dy- +namics of coupled cyclic competitions, Chaos, Solitons & Fractals 158 +(2022) 112019. +[46] J. Park, Correlation between the formation of new competing group and +spatial scale for biodiversity in the evolutionary dynamics of cyclic com- +petition, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (8) +(2022) 081101. +[47] L. Riotte-Lambert, J. Matthiopoulos, Environmental predictability as a +cause and consequence of animal movement, Trends in Ecology & Evo- +lution 35 (2) (2020) 163–174. +[48] A. Szolnoki, M. Perc, Vortices determine the dynamics of biodiversity in +cyclical interactions with protection spillovers, New Journal of Physics +17 (11) (2015) 113033. +[49] J. Menezes, E. Rangel, B. Moura, Aggregation as an antipredator strat- +egy in the rock-paper-scissors model, Ecological Informatics 69 (2022) +101606. +[50] R. M. May, W. J. Leonard, Nonlinear aspects of competition between +three species, SIAM J. Appl. Math. 29 (1975) 243–253. +[51] E. Rangel, B. Moura, J. Menezes, Combination of survival movement +strategies in cyclic game systems during an epidemic, Biosystems 217 +(2022) 104689. +8 + diff --git a/EdAzT4oBgHgl3EQfwv7y/content/tmp_files/load_file.txt b/EdAzT4oBgHgl3EQfwv7y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52ebb1d03e469acec2fea217182795c5716f0730 --- /dev/null +++ b/EdAzT4oBgHgl3EQfwv7y/content/tmp_files/load_file.txt @@ -0,0 +1,774 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf,len=773 +page_content='Locally adaptive aggregation of organisms under death risk in rock-paper-scissors models J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezesa,b, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rangelb aInstitute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands bSchool of Science and Technology, Federal University of Rio Grande do Norte Caixa Postal 1524, 59072-970, Natal, RN, Brazil cDepartment of Computer Engineering and Automation, Federal University of Rio Grande do Norte, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Senador Salgado Filho 300, Natal, 59078-970, Brazil Abstract We run stochastic simulations of the spatial version of the rock-paper-scissors game, considering that individuals use sensory abilities to scan the environment to detect the presence of enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' If the local dangerousness level is above a tolerable threshold, individuals aggregate instead of moving randomly on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We study the impact of the locally adaptive aggregation on the organisms’ spatial organisation by measuring the characteristic length scale of the spatial domains occupied by organisms of a single species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our results reveal that aggregation is beneficial if triggered when the local density of opponents does not exceed 30%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' otherwise, the behavioural strategy may harm individuals by increasing the average death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We show that if organisms can perceive further distances, they can accurately scan and interpret the signals from the neighbourhood, maximising the effects of the locally adaptive aggregation on the death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Finally, we show that the locally adaptive aggregation behaviour promotes biodiversity independently of the organism’s mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The coexistence probability rises if organisms join conspecifics, even in the presence of a small number of enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We verify that our conclusions hold for more complex systems by simulating the generalised rock-paper-scissors models with five and seven species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our discoveries may be helpful to ecologists in understanding systems where organisms’ self-defence behaviour adapts to local environmental cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Keywords: population dynamics, cyclic models, stochastic simulations, behavioural strategies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Introduction Behavioural biology has revealed the mechanisms that or- ganisms use to improve their fitness, being fundamental for the stability of the rich biodiversity in nature[1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' There is plenty of evidence that self-preservation strategies are properly exe- cuted because of the organism’s evolutionary ability to scan the environment cues, perceiving the presence of a nearby enemy and the energy expended in the action[5–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In this scenario, living in groups facilitates the defence action since individual protection against enemies is maximised by collective effort in surveillance and resistance, demanding less individual energy expenditure on defense against enemies [10–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Cyclic models of biodiversity have been studied using the rock-paper-scissors game rules, which successfully describe the nonhierarchical competition interactions found in many biolog- ical systems [20–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' However, experiments with bacteria Es- cherichia coli revealed that the cyclic dominance among three bacteria strains is insufficient to stabilise the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' It has been discovered that coexistence is ensured only if individuals inter- act locally [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This shows the central role of space in the sta- bility of biological systems, as it has been also observed in com- munities of lizards and systems of competing coral reefs [31– 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Furthermore, cyclic dominance has been shown to play a fundamental role in the spatial interactions in social systems, public good with punishment, and human bargaining [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' There is plenty of evidence that organisms’ mobility plays a central role in promoting or jeopardising biodiversity in struc- tured populations [36–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Evidence shows that organisms’ foraging behaviour may affect biodiversity in the spatial rock- paper-scissors game [22, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Organisms’ moving to escape their enemies and find natural resources to the species perpet- uation may unbalance the cyclic game or decelerate the pop- ulation dynamics, thus jeopardising or promoting biodiversity [28, 29, 47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Recently, it has been shown that aggregation behaviour is an efficient antipredator strategy in tritrophic predator-prey cyclic models [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Numerical simulations of the Lotka-Volterra ver- sion of the rock-paper-scissors game revealed that individu- als’ predation risk decreases if organisms execute a gregarious movement, instead of exploring the territory to found prey and reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In contrast with the standard model where organ- isms move in a random direction, the grouping strategy pro- duces spiral-type patterns with organisms of the same species living in spatial domains whose characteristic length depends on the the distance the individuals can scan their neighbour- hood, and their cognitive ability to perform the directional self- preservation movement tactic [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Although the revealing details of the complexity of the spa- tial interactions, the model in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [49] considers exclusively a non-adaptive aggregation tactic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=', individuals cannot smartly adapt their movement to trigger the grouping strategy only when pressured by an imminent enemy’ attack, as happens, for example, in spider mites communities [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In this case, the unnecessary expenditure is avoided since organisms can con- Preprint submitted to Journal of LATEX Templates January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='01729v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='PE] 4 Jan 2023 1 3 2 1 2 3 Figure 1: The rock-paper-scissors model rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The black arrows illustrate the dominance in the spatial game: individuals of species i eliminate organisms of species i+1, with i = 1, 2, 3 and i±3 = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Organisms of the same species aggre- gate when attacked and move randomly when not in danger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Dark blue, pink, and green represent individuals of species 1, 2, and 3 moving gregariously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' light blue, pink, and green indicate organisms of species 1, 2, and 3 moving randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' tinue freely advancing on the lattice to conquer territory, allow- ing the population growth [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In this work, we sophisticate the stochastic model to simulate a locally adaptive aggregation where organisms move gregariously only under death risk [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We also consider that the decision to aggregate is the individual competence, meaning that each organism acts autonomously according to its own local reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Therefore, each individual can decide if moving gregariously or randomly, with the con- gregation being triggered only if the local density of enemies is higher than a tolerable threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In addition, we implement the behavioural survival strategy using the May-Leonard imple- mentation of the spatial rock-paper-scissors game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This allows the generalisation of our results to systems where competition for natural resources is the goal of the cyclic game [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We aim to answer the following questions: i) how does the locally adaptive aggregation modify the spiral patterns, char- acteristic of the standard May-Leonard implementation of the rock-paper-scissors model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' ii) how does the aggregation trig- ger influence the organisms’ spatial organisation altering the size of the typical single-species domains?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' iii) how does adap- tive grouping benefit individuals by reducing the average death risk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' iv) how does the locally adaptive congregation behaviour impact species coexistence probability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2, we in- troduce our stochastic model and present the methods used to implement the locally adaptive grouping in our simulation algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 3, the changes in the spatial patterns are studied for various values of aggregation trigger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the autocorrelation function and characteristic length scales are addressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The reduction in the organisms’ average death risk is com- puted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 5 for a range of aggregation triggers and mobil- ity probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Finally, the coexistence probability in terms of the individual’s mobility is investigated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 6, while our comments and conclusions appear in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The Model We study a cyclic model of three species that outcompete each other according to the rock-paper-scissors game rules, il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This means that individuals of species i elim- inate organisms of species i + 1, with i = 1, 2, 3, with the cyclic identification i = i + 3 β, where β is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our model con- siders that organisms of the same species aggregate to minimize the probability of being killed in the spatial game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The gre- garious movement is locally adaptive, triggered whenever the density of enemies in the organisms’ neighbourhood is higher than a tolerable threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This means that each individual of species i can scan the environment to perceive the presence of organisms of species i + 1, thus, accurately deciding if the best strategy is to search for refuge joining their conspecifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' or continue moving randomly to explore the territory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The dark colours in Fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 stand for individuals executing the gregarious movement, whereas the light colours represent organisms mov- ing randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Numerical simulations To perform the numerical simulations, we use square lattices with periodic boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the number of grid sites is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We use the May-Leonard implementation, where the total number of individuals is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Therefore, as each grid point is occupied by at most one individual (or it is empty), the maximum number of organisms in the system is the total number of grid points N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Initially, the organisms are randomly distributed in the lat- tice: each individual is allocated at a random grid site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The ini- tial conditions are prepared so that the number of individuals is the same for every species is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We define the number of individuals of each species at the initial state as one-third of the total number of organisms: Ii(t = 0) ≈ N/3, with i = 1, 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the rest of grid sites are left empty in the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Once the random initial conditions are ready, the algorithm stochastically implements the interactions following the von Neumann neighbourhood, where each organism can interact with one of its four immediate neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The spatial inter- actions are: Selection: i j → i ⊗ , with j = i + 1, where ⊗ means an empty space: an individual of species i eliminates a neighbour of species i + 1 following the rules illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 - the grid site occupied by the eliminated individual is left empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Reproduction: i ⊗ → i i : an empty space is filled by a new organism of any species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Mobility: i ⊙ → ⊙ i , where ⊙ means either an individual of any species or an empty site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' An organism moves by switching positions with another individual of any species or an empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The interactions are implemented following a fixed set of probabilities which is the same for every species: s (selec- tion probability), r (reproduction probability), and m (mobility probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' During the interaction implementation, the code follows the steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' an active individual of any species is drawn among all or- ganisms in the lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' one interaction is randomly chosen following the set of probabilities rates (s, r, and m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2 (a) (b) (c) (d) Figure 2: Snapshots captured from simulations of the rock-paper-scissors game with individuals’ locally adaptive aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The realisations ran in lattice with 2002 grid points for a timespan of 2000 generations, with R = 3, r = s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Figures 2a, 2b, 2c, and 2d show the organisms’ spatial organisation at the end of Simulation A (ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0), B (ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1), (ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='025), and D (ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The colours follow the scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 1, with blue, pink, and green depicting individuals of species 1, 2, and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Dark and light colours distinguish organisms performing the congregation strategy and moving randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Yellow dots depict empty sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' one of the four immediate neighbours is drawn to suffer the action (selection, reproduction, and random mobility) the only exception is the adaptive gregarious movement, where the organism move towards the direction with more conspecifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Every time an interaction is implemented, one timestep is counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' After N timesteps, one generation is completed - our time unit is one generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' To understand the population dynamics during the simula- tions, we calculate the density of organisms of species i, ρi(t), with i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This is defined as the fraction of the lattice oc- cupied by individuals of the species i at time t, ρi(t) = Ii(t)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Also, the temporal dependence of the density of empty spaces is computed as ρ0 = 1 − ρ1 − ρ2 − ρ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Implementing the locally adaptive aggregation strategy To implement the locally adaptive grouping tactic, we define the perception radius, R, to represent the maximum distance an organism of species i can scan the environment to be aware of the presence of enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Thus, the local density of organisms of each species is computed within a circular area of radius R, cen- tred in the organism of species i [29, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In addition, we intro- duce the aggregation trigger, ϕ, to represent the minimum local density of individuals of species i − 1 (enemies) that stimulates the organism of species i to move gregariously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This means that if the local density of organisms of species i − 1 is lower than ϕ, the individual moves randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The numerical implementation of the gregarious movement is performed by dividing the observing disc into four circular sectors, each section in the directions of the one nearest neigh- bour of the von Neumann neighbourhood [22, 25, 26, 28, 49, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Next, it is computed how many individuals of species i exist within each circular sector, with organisms on the circu- lar sector borders assumed to be part of both circular sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Finally, the organism switches positions with the immediate neighbour in the direction with more conspecifics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' a draw in the event of a tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Spatial Patterns Our first goal is to understand the effects of the locally adap- tive congregation strategy in spatial patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Therefore, we ran a single simulation for four values of the aggregation trigger: Simulation A: ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 - the absence of organisms’ group- ing behaviour, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=', individuals do not aggregate even under death risk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Simulation B: ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 - organisms’ agglomeration occurs if, at least, 10% neighbours are enemies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Simulation C: ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='025 - an individual move gregariously if at least, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5% neighbours are enemies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Simulation D: ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 - the gregarious movement is not locally adaptive, with individuals always grouping inde- pendently of the presence of enemies surrounding them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The realisations were performed in lattices with 2002 grid sites, running for a timespan of 2000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We set the param- eters to s = r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5, and R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Figures 2a, 2b, 2c, and 2d show the individuals’ spatial or- ganisation at the end of Simulations A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' C, and D, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' To depict each organism, we use the same colours of the scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 1: blue, pink, and green dots show the in- dividuals of species 1, 2, and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The organisms performing the aggregation strategy are highlighted using dark colours, while the individuals moving randomly appear in light shades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We also quantified the dynamics of the species densities for Simulation A, B, C, and D, which are depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 3a, 3b, 3c, and 3d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1, blue, pink, and green lines shows the temporal dependence of densities of individuals of species 1, 2, and 3, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Let us first focus on Simulation A, where individuals do not aggregate to protect themselves against enemies (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Be- cause of the random initial conditions, selection interactions are frequent at the beginning of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' After that, spatial patterns are formed with organisms of the same species occu- pying departed patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Since organisms are unaware of the 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0 500 1000 1500 2000 ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 ρi t (generations) 1 2 3 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0 500 1000 1500 2000 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 ρi t (generations) 1 2 3 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0 500 1000 1500 2000 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='025 ρi t (generations) 1 2 3 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0 500 1000 1500 2000 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 ρi t (generations) 1 2 3 (d) Figure 3: Dynamics of species densities during the simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The blue, pink, and green lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 3a, 3b, 3c, and 3d depict the temporal dependence of the density of individuals of species 1, 2, and 3, in Simulations A, B, C, and D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='24 0 500 1000 1500 2000 ρ0 t (generations) ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='000 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='025 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='100 ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='000 Figure 4: Temporal dependence of the density of empty spaces in simulations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The grey, orange, yellow, and brown lines show the dynamics of empty sites in Simulations A, B, C, and D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' neighbourhood, they move randomly, independently of the risk of being caught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This results in faster dynamics of species den- sities, with organisms being destroyed and newborns appearing at a high rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Consequently, the species densities’ frequency and amplitude are high, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In addition to the usual pattern formation process driven by the cyclic game rules, the gregarious movement performed by individuals under death risk promotes the formation of self- protection clusters on the border that is attacked by enemies, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2b and 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' For example, the organisms of species 2 aggregating (dark pink dots) are concentrated on the border with spatial domains of species 1 (blue areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The self- preservation movement tactic produces a deformation of the spiral patterns, with individuals concentrating in patches with smaller sizes since they abdicate to explore extensive areas of the territory to form clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Because of this, the population dy- namics are decelerated, with reduced frequency and amplitude, as depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3b and 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Finally, the snapshot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 3d reveals what occurs in the case of the non-adaptive aggregation strategy (ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0) - indi- viduals move gregariously even if no enemy surrounds them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In this scenario, the population dynamics are altered since the individuals neglect the conquest of new territories to focus ex- clusively on the survival movement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This induces a contraction of the spatial domains occupied by organisms of a single species, since individuals do not advance in the terri- tory even if they are not under death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 4 shows the temporal dependence of the density of empty spaces, ρ0, in Simulations A (grey line), B (orange line), C (green line), and D (brown line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The results show that the density of empty spaces decreases after an initial period of pattern formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Furthermore, the locally congregation reduces the organisms’ death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Because of this, the lower the aggregation trigger, the more the density of empty spaces is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Autocorrelation Function Let us now quantify the scale of spatial domains in the pres- ence of locally adaptive aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' For this, we compute the spatial autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The autocorrelation function is computed from the inverse Fourier transform of the spectral density as C(⃗r′) = F −1{S (⃗k)} C(0) , (1) where S (⃗k) is given by S (⃗k) = � kx,ky Φ(⃗κ), (2) 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 1 0 5 10 15 20 25 30 35 C r ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 Figure 5: Autocorrelation functions in terms of the radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The grey, orange, and brown lines depict the results for the standard model (ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0), aggregation triggered when at least 10% of neighbours are enemies ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1, and the non-adaptive aggregation (ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The error bars indicate the standard deviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the dashed black line shows the threshold assumed to calculate the characteristic length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The interaction probabilities are r = s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the perception radius is R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' and Φ(⃗κ) is Fourier transform Φ(⃗κ) = F {φ(⃗r) − ⟨φ⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' (3) The function φ(⃗r) represents the spatial distribution of individ- uals of species 1, with φ(⃗r) = 0 and φ(⃗r) = 1 indicating the absence and the presence of an individual of species 1 in at the position ⃗r in the lattice, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The spatial autocorrela- tion function is given by C(r′) = � |⃗r′|=x+y C(⃗r′) min �2N − (x + y + 1), (x + y + 1)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' (4) Moreover, we compute the spatial domains’ scale for C(l) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='15, where l is the characteristic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We calculated the spatial autocorrelation function in terms of the radial coordinate r for three cases: absence of group- ing behaviour (ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0), aggregation triggered when the neigh- bourhood is, at least, 10% hostile (ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1), and non-adaptive aggregation (ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The outcomes were obtained by run- ning sets of 100 simulations with different random initial con- ditions in lattices with 5002 grid sites for a time span of 5000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' To calculate the autocorrelation function, we used the spatial configuration at the end of the simulation (t = 5000 generations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Because organisms of every species can perform the locally adaptive congregation, the autocorrelation function is the same irrespective of the species;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' thus, we used the data from species 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In all simulations, we considered the interac- tions probabilities s = r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the perception radius was set to R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The brown, orange, and grey lines in Figure 5 show C as a function of the radial coordinate r for ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1, and ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the error bars indicate the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The horizontal dashed black line indicates the threshold used to calculate the length scale: C(l) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The results confirm that once organisms move gregariously, the average size of the spatial domains inhabited by a single species decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Figure 6 shows the relative variation of the characteristic length scale ˜l, defined as ˜l = (l − l0)/l0, where l0 is the value in the absence of the adaptive aggregation (ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We repeated the set of 100 simulations - starting from different initial condi- tions - for 0 ≤ ϕ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4, with intervals of δϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The error −40 −35 −30 −25 −20 −15 −10 −5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 ˜l(%) ϕ Figure 6: The relative change in the characteristic length scale of the typical single-species spatial domain as a function of the threshold used to trigger the gregarious movement compared with the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The simulations ran in lattices with 5002 grid sites, running until 5000 generations for r = s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the perception radius is R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The outcomes were averaged from sets of 100 simulations starting from different initial conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the error bars show the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We assumed the probabilities r = s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' bars show the standard deviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the parameters are the same used in the simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The outcomes show that the average group size decreases compared to the standard model, with the reduction becoming significant for ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This hap- pens because all individuals group themselves, independently of what is happening in their surroundings, as we observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The role of the locally adaptive aggregation in the organ- isms’ death risk We now investigate the effects of locally adaptive grouping to reduce the organisms’ death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' For this purpose, we in- troduce the death risk, which is calculated as follows: i) it is counted as the total number of individuals of species i at the beginning of each generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' ii) the number of organisms of species i killed by individuals of species i − 1 during the gen- eration is computed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' iii) the death risk, ζ is defined as the ratio between the number of eliminated organisms and the amount at the beginning of each generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Due to the symmetry of the rock-paper-scissors game rules, the average death risk is the same for individuals of every species;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' thus, we choose the re- sults for species 1 to represent the individuals’ death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The influence of the aggregation trigger First, we study the influence of the aggregation trigger ϕ in the relative decrease of the individuals’ death risk by running sets of 100 simulations starting from different initial conditions for 0 ≤ ϕ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 in intervals of δϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This experiment was conducted for two values of perception radius: R = 3 and R = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the interaction probabilities are s = r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' To guarantee the quality of the results, we remove the data from the initial pattern formation stage, thus calculating the average organisms’ death risk in the second half of each realisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The purple and red lines in Figure 7 show the organisms’ death risk in terms of the aggregation trigger for R = 3 and R = 5, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the standard deviation is shown by error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The outcomes reveal that for ϕ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6, the locally adap- tive strategy is ineffective in reducing the organisms’ death risk 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='9 1 ζi ϕ R = 5 R = 3 Figure 7: Organisms’ death risk in terms of the aggregation trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The simu- lations were performed in lattices with 5002 grid sites, running for a timespan of 5000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The red and purple lines show the outcomes for organisms with perception radius R = 3 and R = 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The results were aver- aged from sets of 100 simulations starting from different initial conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the standard deviation is depicted by error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The interaction probabilities are s = r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 and m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' −50 −45 −40 −35 −30 −25 −20 −15 −10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='95 ˜ζ(%) m Figure 8: Relative change in the individuals’ death risk in terms of the mobility probability in simulations running in lattices with 5002 grid sites, running for a timespan of 5000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We averaged the outcomes sets of 100 simula- tions starting from different initial conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the standard deviation is shown by error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The perception radius is R = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the interaction probabilities are to s = r = (1 − m)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' compared with the standard model (ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This happens be- cause most of organism of species i whose neighbourhood con- tains 60% or more of organisms of species i − 1 is far from the spatial domain dominated by their conspecifics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' thus, grouping may not be possible to be executed before the individual being eliminated by enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our findings show that the locally adaptive aggregation jeop- ardises the organisms’ safety for intermediate values of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7, for R = 3, the organisms’ death risk in- creases for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 ≤ ϕ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6, while for R = 5, ζ increases for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 ≤ ϕ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Therefore, the adaptive is beneficial only if the threshold assumed to move gregariously is in the interval 0 ≤ ϕ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 for R = 3 and 0 ≤ ϕ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 for R = 5, with the relative reduction of ζ increasing as the ϕ is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7 show how the complexity of the spatial interactions is influenced by the organism’s ability to make an accurate decision, triggering the adaptive tactic correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our findings show that if organisms can perceive further distances, they can more easily: i) identify the presence of invading en- emies beyond the border of their territory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' ii) distinguish the direction with more conspecifics in case of need to move gre- gariously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Because of this, the relative variation in the organ- isms’ death risk is more accentuated for R = 5 than for R = 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The interference of organisms’ mobility The locally adaptive grouping is profitable for the organ- isms because of the death risk reduction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7 for m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Now, we repeated the simulations to explore how the benefits of the locally adaptive aggregation depend on the organism’s mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' For this purpose, we ran sets of 100 realisations starting from different initial conditions for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 ≤ m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='95, in intervals of δm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The selection and reproduction probabilities are set to s = r = (1 − m)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' the per- ception radius is R = 3, and the aggregation trigger is ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We implemented the simulations in lattices with 5002 grid sites, running until 5000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Figure 8 shows the relative change of the organisms’ death risk: ˜ζ = (ζ − ζ0)/ζ0, where ζ0 is the death risk in the absence of grouping behaviour (ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 ≤ m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='085, the relative reduction in the organisms’ death risk is more signifi- cant for individuals that explore greater fractions of the lattice per time unit [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This happens because high-mobile individ- uals are more vulnerable to being eliminated by enemies in the cyclic game, thus, benefitting more from the self-preservation movement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' However, if m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='085, the relative vari- ation in ζ decreases because the selection probability becomes very low, becoming the effect less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Coexistence Probability Now, we focus on the impact of locally adaptive flocking on biodiversity in cyclic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In this study, we ran sets of 1000 simulations in lattices with 1002 grid points for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 < m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='95 in intervals of δ m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' selection and reproduction probabilities were set to s = r = (1 − m)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' For each set of simulations, each realisation began from different random initial conditions, running until 10000 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' If at least one species is extinguished before the simulation ends, biodi- versity is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Thus, the coexistence probability is the frac- tion of the simulations where all species are present at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We extended the investigation to quantify the impact of locally adaptive aggregation in more complex systems by simulating the generalised rock-paper-scissors models with five and seven species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Figures 9a, 9b and 9c depict the coexistence probabil- ity for ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 (brown line), ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 (green line), ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 (orange line), ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 (blue line), and ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0 (grey line) for the models with N = 3, N = 5, and N = 7 species, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Overall, species biodiversity is more threatened for systems with highly mobile individuals, independent of the number of species in the cyclic game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The outcomes also show the benefits of the locally adaptive aggregation for biodiversity: the lower the aggregation trigger, the higher is the coexistence proba- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This conclusion holds independently of the number of species in the cyclic game Furthermore, the outcomes show that the more complex the system is, the more favourable it is for biodiversity loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' By comparing the same color lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 9a, 9b and 9c, one observes that the coexistence probabil- ity is lower for the system with N = 9 species, independently of the organisms’ mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Finally, we observe that all simula- tions resulted in coexistence when individuals agglomerate with 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='9 Coexistence Probability m ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='00 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='10 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='20 ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='00 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='9 Coexistence Probability m ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='00 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='10 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='20 ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='00 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='9 Coexistence Probability m ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='00 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='10 ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='20 ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='00 (c) Figure 9: Coexistence probability as a function of the mobility probability for the generalised rock-paper-scissors game with organisms’ locally adaptive aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Figures 9a, 9b, and 9c show the outcomes for the cyclic model with three, five, and seven species, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The results were obtained by running 1000 simulations in lattices with 1002 grid points running until 10000 generations for R = 3 and s = r = (1 − m)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The brown, green, orange, blue, and grey lines depict the results for ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='05, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='1, ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='2, and ϕ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' their conspecifics irrespective of the local densities of enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' According to the brown lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 9a, 9b and 9c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Comments and Conclusions Aggregation behaviour is found in many systems where or- ganisms adapt their movement, grouping with their conspecifics when in death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We investigate cyclic models described by the rock-paper-scissors game rules, where individuals can scan their environment and adapt their movement to environmental cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In our stochastic simulation, each organism freely ex- plores the territory without precaution if there is no nearby en- emy but prevents damage from enemy attack moving gregarious to join the biggest group of conspecific in the neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' To execute the locally adaptive grouping, each individual scans their vicinity, thus triggering the gregarious movement if the lo- cal density of enemies reaches a prefixed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Running a series of simulations, we investigate the role of adaptive aggre- gation in transforming the organisms’ spatial organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' The results show that the characteristic length scale of the spatial domains occupied by organisms of a single species is not ac- centuated if the threshold is not inferior to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Otherwise, the typical group size decreases significantly, being minimal in the case of organisms flock even when not under death risk pres- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' We discover that the gregarious movement does not interfere with organisms’ safety if the grouping is only triggered when more than 70% neighbourhood is occupied by enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Coun- terintuitively, if the self-preservation movement tactic is cali- brated to be triggered if between 30% and 60% neighbours are enemies, the individuals’ death risk increases instead of bene- fiting the organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our outcomes show that the behavioural strategy is profitable only if each organism aggregates with con- specifics when detecting the fraction of opponents in the vicin- ity using a threshold inferior to 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' In addition, we find that if organisms can perceive further distances, they can accurately scan and interpret the signals from the neighbourhood, increas- ing the effects of the adaptive aggregation on the death risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Moreover, we study the impact of mobility on the benefits of adaptive congregation considering low, intermediate and high- mobile individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our simulations provided evidence that lo- cally adapting their movement to aggregate when under death risk is more advantageous as the more mobile the organisms, provided that the individuals’ mobility is not superior to 85%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' otherwise, the relative death risk reduction diminishes as the mobility grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Finally, we study the influence of locally adaptive aggre- gation on biodiversity maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our findings show that the coexistence probability increases independently of the or- ganism’s mobility, being maximal in the case of non-adaptive grouping, where the gregarious movement is executed even when there is no local death risk for the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' This re- sult holds for more complex systems where an arbitrary odd number of species participate in the cyclic game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Extending our algorithm to implement the generalised rock-paper-scissors model with five and seven species, we confirm that the gregari- ous movement promotes biodiversity, being more beneficial for low adaptive aggregation triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our discoveries may be help- ful to ecologists in understanding systems where organisms’ self-defence behaviour adapts to local environmental cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Our results may also clarify the role of the local phenomena in com- plex systems in other areas of nonlinear science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Acknowledgments We thank CNPq, ECT, Fapern, and IBED for financial and technical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Begon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Townsend, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Harper, Ecology: from individuals to ecosystems, Blackwell Publishing, Oxford, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7 [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Purvis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Hector, Getting the measure of biodiversity, Nature 405 (2000) 212–2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Buchholz, Behavioural biology: an effective and relevant conservation tool, Trends in Ecology & Evolution 22 (8) (2007) 401 – 407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Tobias, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Pigot, Integrating behaviour and ecology into global biodiversity conservation strategies, Philosophical Transactions of the Royal Society B: Biological Sciences 374 (1781) (2019) 20190012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Riessen, Cost-benefit model for the induction of an antipredator defense, American Natiralist 140 (1992) 349–362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Andersson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' L¨ofstedt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Hamb¨ack, How insects sense olfac- tory patches – the spatial scaling of olfactory information, Oikos 122 (7) (2013) 1009–1016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Downes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Hoefer, Antipredatory behaviour in lizards: interactions between group size and predation risk, Animal Behaviour 67 (2004) 485– 492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Dehn, Vigilance for predators: detection and dilution effects, Be- hav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Sociobiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 26 (1990) 337–342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Dittmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Schausberger, Adaptive aggregation by spider mites under predation risk, Scientific Reports 7 (2017) 2045–2322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Lima, Motion extrapolation in catching, Animal Behaviour 49 (1995) 11–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Wrona, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Dixon, Group size and predation risk: A field anal- ysis of encounter and dilution effects, The American Naturalist 137 (2) (1991) 186–201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Turner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Pitcher, Attack abatement: A model for group pro- tection by combined avoidance and dilution, The American Naturalist 128 (2) (1986) 228–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Berger, A group size, foraging, and antipredator ploys: An analysis of bighorn sheep decisions, Behav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Sociobiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 4 (1978) 91–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Dittmann, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Schausberger, Adaptive aggregation by spider mites under predation risk, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 7 (2017) 10609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [15] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Brock, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Riffenburgh, Fish schooling: A possible factor in re- ducing predation, ICES Journal of Marine Science 25 (3) (1960) 307–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Johannesen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Dunn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Morrell, Prey aggregation is an effective olfactory predator avoidance strategy, Peer J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 2 (2014) e408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Hopkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Lahanas, Aggregation behaviour in a neotropical dendrobatid frog (allobates talamancae) in western panama, Behaviour 148 (3) (2011) 359–372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Spieler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Linsenmair, Aggregation behaviour of bufo maculatus tadpoles as an antipredator mechanism, Ethology 105 (8) (1999) 665– 686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Guadin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Gazzola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Balestrieri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Scribano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Mart´ın, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Pellitteri-Rosa, Effects of a group-living experience on the antipredator responses of individual tadpoles, Animal Behaviour 180 (2021) 93–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Avelino, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Bazeia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Losano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Oliveira, Junctions and spiral patterns in generalized rock-paper-scissors models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E 86 (2012) 036112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Moura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Pereira, Uneven rock-paper-scissors models: Patterns and coexistence, Europhysics Letters 126 (1) (2019) 18003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Moura, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, Behavioural movement strategies in cyclic models, Scientific Reports 11 (2021) 6413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Avelino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' de Oliveira, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Pereira, Expanding spatial domains and transient scaling regimes in populations with local cyclic competition, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E 99 (2019) 052310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Bazeia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' de Oliveira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Ramos, Hamming distance and mobility behavior in generalized rock-paper-scissors models, Europhysics Letters 119 (5) (2017) 58003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, Antipredator behavior in the rock-paper-scissors model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E 103 (2021) 052216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Moura, Mobility-limiting antipredator response in the rock-paper-scissors model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E 104 (2021) 054201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Batista, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Tenorio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Triaca, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Moura, How local an- tipredator response unbalances the rock-paper-scissors model, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (12) (2022) 123142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Tenorio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rangel, Adaptive movement strategy may pro- mote biodiversity in the rock-paper-scissors model, Europhysics Letters 139 (5) (2022) 57002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Tenorio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rangel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, Adaptive movement strategy in rock- paper-scissors models, Chaos, Solitons & Fractals 162 (2022) 112430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Durret, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Levin, Allelopathy in spatially distributed populations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 185 (1997) 165–171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [31] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Sinervo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Lively, The rock-scissors-paper game and the evolution of alternative male strategies, Nature 380 (1996) 240–243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Jackson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Buss, The rock-scissors-paper game and the evolution of alternative male strategies, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Natl Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' USA 72 (1975) 5160– 5163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [33] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Volkov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Banavar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Hubbell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Maritan, Patterns of relative species abundance in rainforests and coral reefs, Nature 450 (2007) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Perc, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Szab´o, Defense mechanisms of empathetic play- ers in the spatial ultimatum game, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 109 (2012) 078701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Perc, Correlation of positive and negative reciprocity fails to confer an evolutionary advantage: Phase transitions to elementary strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [36] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Kabir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Tanimoto, The role of pairwise nonlinear evolutionary dy- namics in the rock–paper–scissors game with noise, Applied Mathematics and Computation 394 (2021) 125767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Mobilia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Jiang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Szczesny, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rucklidge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Perc, Cyclic dominance in evolutionary games: a review, Journal of The Royal Society Interface 11 (100) (2014) 20140735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Szolnoki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Vukov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Perc, From pairwise to group interactions in games of cyclic dominance, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E 89 (2014) 062125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [39] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Reichenbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Mobilia, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Frey, Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games, Nature 448 (2007) 1046–1049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Frean, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Abraham, Rock–scissors–paper and the survival of the weakest, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 268 (2001) 1323–1327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Lotka, Undamped oscillations derived from the law of mass action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=', Journal of the American Chemical Society 42 (8) (1920) 1595–1599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [42] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Volterra, Lecons dur la Theorie Mathematique de la Lutte pour la Vie, 1st Edition, Gauthier-Villars, Paris, (1931).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Peltom¨aki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Alava, Three- and four-state rock-paper-scissors games with diffusion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E 78 (2008) 031906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [44] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Ni, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Lai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Grebogi, Cyclic competition of mobile species on continuous space: Pattern formation and coexistence, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' E 82 (2010) 066211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Park, Effect of external migration on biodiversity in evolutionary dy- namics of coupled cyclic competitions, Chaos, Solitons & Fractals 158 (2022) 112019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Park, Correlation between the formation of new competing group and spatial scale for biodiversity in the evolutionary dynamics of cyclic com- petition, Chaos: An Interdisciplinary Journal of Nonlinear Science 32 (8) (2022) 081101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [47] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Riotte-Lambert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Matthiopoulos, Environmental predictability as a cause and consequence of animal movement, Trends in Ecology & Evo- lution 35 (2) (2020) 163–174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [48] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Perc, Vortices determine the dynamics of biodiversity in cyclical interactions with protection spillovers, New Journal of Physics 17 (11) (2015) 113033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rangel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Moura, Aggregation as an antipredator strat- egy in the rock-paper-scissors model, Ecological Informatics 69 (2022) 101606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [50] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' May, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Leonard, Nonlinear aspects of competition between three species, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 29 (1975) 243–253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' [51] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Rangel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Moura, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' Menezes, Combination of survival movement strategies in cyclic game systems during an epidemic, Biosystems 217 (2022) 104689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAzT4oBgHgl3EQfwv7y/content/2301.01729v1.pdf'} diff --git a/EtE2T4oBgHgl3EQf-Ak-/content/tmp_files/2301.04233v1.pdf.txt b/EtE2T4oBgHgl3EQf-Ak-/content/tmp_files/2301.04233v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19bd003251a3ba91fe438267fe7d49e97ac8b71c --- /dev/null +++ b/EtE2T4oBgHgl3EQf-Ak-/content/tmp_files/2301.04233v1.pdf.txt @@ -0,0 +1,1471 @@ +Adapting to Skew: Imputing Spatiotemporal Urban Data +with 3D Partial Convolutions and Biased Masking +Bin Han +bh193@uw.edu +University of Washington +Seattle, USA +Bill Howe +billhowe@uw.edu +University of Washington +Seattle, USA +ABSTRACT +We adapt image inpainting techniques to impute large, irregular +missing regions in urban settings characterized by sparsity, variance +in both space and time, and anomalous events. Missing regions +in urban data can be caused by sensor or software failures, data +quality issues, interference from weather events, incomplete data +collection, or varying data use regulations; any missing data can +render the entire dataset unusable for downstream applications. To +ensure coverage and utility, we adapt computer vision techniques +for image inpainting to operate on 3D histograms (2D space + 1D +time) commonly used for data exchange in urban settings. +Adapting these techniques to the spatiotemporal setting requires +handling skew: urban data tend to follow population density pat- +terns (small dense regions surrounded by large sparse areas); these +patterns can dominate the learning process and fool the model into +ignoring local or transient effects. To combat skew, we 1) train +simultaneously in space and time, and 2) focus attention on dense +regions by biasing the masks used for training to the skew in the +data. We evaluate the core model and these two extensions using +the NYC taxi data and the NYC bikeshare data, simulating differ- +ent conditions for missing data. We show that the core model is +effective qualitatively and quantitatively, and that biased masking +during training reduces error in a variety of scenarios. We also ar- +ticulate a tradeoff in varying the number of timesteps per training +sample: too few timesteps and the model ignores transient events; +too many timesteps and the model is slow to train with limited +performance gain. +CCS CONCEPTS +• General and reference → Empirical studies; • Computing +methodologies → Computer vision; • Applied computing; +KEYWORDS +image inpainting, urban computing, spatial-temporal, missing data +ACM Reference Format: +Bin Han and Bill Howe. 2023. Adapting to Skew: Imputing Spatiotemporal +Urban Data with 3D Partial Convolutions and Biased Masking. In Proceedings +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2023 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +of ACM Conference (Conference’17). ACM, New York, NY, USA, 12 pages. +https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +High-quality, longitudinal, and freely available urban data, coupled +with advances in machine learning, improve our understanding +and management of urban environments. Although conventional +machine learning techniques are common in urban applications [35, +50, 55], neural architectures are opening new opportunities by +adapting convolutional, recurrent, and transformer architectures to +spatiotemporal settings [17, 27, 33, 43, 54, 60, 63, 64]; see Grekousis +2020 for a recent survey [14]. For example, spatio-temporal neural +architectures have been used in predictions of rideshare demand +[44, 53], traffic conditions [34, 56], and air quality [23, 32]. But these +models depend on access to complete, longitudinal datasets. Such +datasets are inconsistent in availability and quality, limiting the +opportunity for understanding cities as the complex systems they +are [2, 15, 22, 51]. +Figure 1: A histogram of taxi pickups in Manhattan. We +adapt imagine inpainting techniques to reconstruct missing +and corrupted data in urban settings: The improved model +(upper left) uses biased masking and temporal context to +capture local effects (red circle). The basic model (lower left) +uses ordinary masking and is insensitive to local effects. +Baseline methods that ignore space (lower middle) or time +(lower right) are not competitive. Classical linear methods +such as kriging and inverse-distance weighting (not shown) +cannot impute large irregular regions in dynamic settings. +arXiv:2301.04233v1 [cs.CV] 10 Jan 2023 + +L1 Error: 5.642 +L1 Error: 7.308 +L1 Error: 23.446 +L1 Error: 16.994Conference’17, July 2017, Washington, DC, USA +Bin Han and Bill Howe +This inconsistency persists despite significant investments in +open data. Over the last two decades, cities have increasingly re- +leased datasets publicly on the web, proactively, in response to +transparency regulation. For example, in the US, all 50 states and +the District of Columbia have passed some version of the federal +Freedom of Information (FOI) Act. While this first wave of open +data was driven by FOI laws and made national government data +available primarily to journalists, lawyers, and activists, a second +wave of open data, enabled by the advent of open source and web +2.0 technologies, was characterized by an attempt to make data +“open by default" to civic technologists, government agencies, and +corporations [49]. While open data has indeed made significant +data assets available online, their uptake and use has been weaker +than anticipated [49], an effect attributable to convenience sam- +pling effects [24]: We release what we can, even if portions are +missing, corrupt, or anomalous. +In this paper, we consider a neural data cleaning strategy based +on masking out corrupted regions and using a trained model to +reconstruct the masked region. These masks are necessarily large, +irregular, and extend in both time and space; they may represent po- +litical boundaries (municipal zoning, zip codes, city blocks), sensor +or software failures [26, 62, 65], varying legal restrictions [1, 39], +or unusual events (adverse weather). These missing patches can +destroy the utility of the entire dataset for applications that assume +coverage. By modeling missing or corrupted data by an arbitrary +mask, we afford user control: any areas can be masked and recon- +structed, regardless of the reason. We envision tools to improve the +coverage and quality of data for use in downstream urban learning +tasks [23, 32, 34, 44, 53, 56]. +Following the literature, we represent spatiotemporal event data +in a 2D or 3D raster form (e.g., a histogram). Our basic model uses +the partial convolution approach from Liu et al [29] to handle the +irregular boundaries of missing data (e.g., districts), which focuses +model attention on the valid regions while shrinking the masked +region, layer-by-layer, to obtain a complete prediction. More recent +approaches to image inpainting on the web emphasize eliminating +perceptual artifacts rather than numerical accuracy and are there- +fore less relevant to our setting. Our contribution is to extend the +basic model to the 3D spatiotemporal setting and propose a training +regime that adapts to the skewed distribution found in practice. +Spatiotemporal interpolation of missing data has been widely +studied in the earth sciences [38, 45], especially in remote sensing +where weather effects can obscure measurement [46, 65]. Con- +ventional statistical approaches to impute missing values, such as +global/local mean imputation, interpolation, and kriging, are essen- +tially linear, and therefore limited in their ability to capture the non- +linear dynamics needed to impute large irregular missing regions. +Neural image inpainting techniques [29, 57] can recover missing +patches via training on large datasets of independent images, such +that the reconstructed images appear realistic. These approaches +have shown promising results with global climate data [48], but +have not been adapted to the urban setting in which data are not +smooth functions of space and time, but are rather histograms of +events constrained by the built environment. +The goal of inpainting for natural images is to produce a subjec- +tively recognizable image free from perceptible artifacts. But the +goal in our setting is quantitative accuracy: we intend for our recon- +structed results to be used numerically in downstream applications. +The distribution is relatively stable, but exhibits skew and sparsity +that can obscure local, dynamic features (Figure 2). +The challenge for imputation in the urban setting is skew: urban +data tend to follow population density patterns — small dense +regions surrounded by large sparse areas. These population patterns +can dominate the learning process and fool the model into ignoring +numerical accuracy in dense regions, even while aggregate error +may remains low. To combat skew, we 1) bias the training process +to focus on populated regions by seeding the mask in non-zero +areas; (2) use 3D convolutions and vary the number of timesteps in +each 3D training sample to capture transient events. Together, these +two techniques complement each other: biased masking focuses +attention on dense regions, and 3D convolutions with a large chunk +size focus attention on sparse regions. +We evaluate these techniques on the NYC taxi data (a popular +dataset for its coverage and quality) and a NYC bikeshare dataset +(less dominated by the built environment). We find that the basic +model is effective for urban data imputation, while biased masking +reliably reduces error over random masking, both globally and +locally. Additionally, we find that the number of timesteps per +training sample exhibits a tradeoff: too few timesteps and the model +ignores transient patterns, while too many timesteps significantly +increases training time without enhancing the inpainting results. +We evaluate specific local scenarios (high-traffic locations, low- +traffic locations, high-variability locations, anomalous events) to +reflect the use cases distinct from image inpainting on the web +(where subjective quality is all that matters). +Figure 2: Urban data (bottom row) exhibits skewed, sparse, +yet stable distributions that can dominate learning, in con- +trast with the diversity of natural images (top row). +In summary, we make the following contributions: +• We evaluate a basic model adapting image inpainting techniques +to urban histograms characterized by skew and sparsity effects +due to constraints by the built environment, demonstrating qual- +itative and quantitative accuracy relative to classical methods. +• We improve on this basic model by extending to the 3D spatiotem- +poral setting to better recognize transient events; we analyze the +training time and performance tradeoffs of varying the number +of timesteps per training sample. +• We propose a self-supervised training process called biased mask- +ing to encourage the model to attend to dense population regions + +Adapting to Skew: Imputing Spatiotemporal Urban Data +with 3D Partial Convolutions and Biased Masking +Conference’17, July 2017, Washington, DC, USA +and thereby improve accuracy on the highly dynamic regions +typical in urban environments; we show that biased masking +reliably improves convergence. +• We evaluate these techniques on two real mobility datasets (NYC +taxi trips and NYC bikeshare trips), both globally and locally +in varying traffic conditions, weather events, and disruptions. +Finally, we show that the model can be used to remove or syn- +thesize anomalous events through targeted masking. +2 +RELATED WORK +Our work is informed by techniques in image inpainting and geospa- +tial interpolation. +Image Inpainting Image inpainting, or image completion, is a +task of synthesizing missing pixels in images, such that the recon- +structed images are visually credible and semantically realistic. In +computer vision, there are two broad categories of inpainting tech- +niques. The first category contains diffusion-based or patch-based +methods, which utilize low-level image features to recover the miss- +ing pixels. The second category contains learning-based methods +that generally involve the training of deep neural networks. +Diffusion-based methods [4, 6, 25] propagate information from +neighboring valid pixels to missing pixels, typically from border to +the center of the missing regions. Those techniques are convenient +to apply, but are limited to small missing regions. Recently, Saharia +et al. [42] developed an image-to-image translation framework +based on conditional diffusion models. The evaluation on inpainting +task outperformed several learning-based methods. Patch-based +inpainting techniques [7, 10, 12, 16] function by searching similar +patches from the valid regions of the same image or from other +images, and then paste the patches to the target missing region. +However, this process could induce high computational costs. A +milestone of patch-based approach, PatchMatch [5], speeds up the +search process with a new nearest neighbor algorithm. +Learning-based methods are trained to learn image patterns with +large volume of image data, thus being capable of recovering miss- +ing regions, as well preserving the semantics of the imagery. Pathak +et al.[36] proposed context encoder, which was the first work to +combine CNN with generative adversarial network. It applied the +encoder-decoder architecture and used both ℓ2 reconstruction loss +and generative adversarial loss in the objective function. Lizuka +et al. [18] improved on their work by incorporating global and +local discriminator, which improved content consistency between +the valid and missing region. Additionally, they replaced general +convolutional layers with dialated convolutional layers to better +capture information from distant pixels. Yu et al. [58] proposed +proposed a two-stage coarse-to-fine model architecture and incor- +porated contextual attention layer to attend to related features from +spatially distant regions. They also replaced general generative ad- +versarial loss with WGANS loss. Liu et al. [29] proposed partial +convolution, allowing inpainting models to be used on irregular +holes rather than just rectangular missing regions. On top the work +of partial convolution, Yu et al. [57] proposed gated convolutional +layers to automatically learn and update the masks as opposed to +rule-based update. To further address the problems of blurry tex- +tures and distorted structures in the inpainted images, Liu et al. [30] +proposed coherent semantic attention layer, which can both pre- +serve contextual structure and capture semantic relevance between +hole features. Zhou et al.[66] incorporated dual spatial attention +modules into the U-Net architecture, which can capture the corre- +lations between facial textures at different scales. Seven different +discriminators are utilized to ensure realistic local details as well +as global consistency. Yu et al. [59] designed spatial region-wise +normalization (RN) to overcome the problem of mean and variance +shifts. RN computes the mean and variances separately for the +missing and valid regions. Xu et al. [52] combined the paradigms +of both patch-based and learning-based methods, and inpainted +missing regions using textures of patch samples from unmasked +regions. Additionally, they proposed patch distribution loss to en- +sure the quality of synthesized missing regions. Zeng et al. [61] +introduced aggregated contextual transformation GAN, aiming to +improve content reasoning from distant pixels and enhance details +of synthesized textures. For more image inpainting works, we refer +reader to the following surveys [19, 31, 37]. +The recent trajectory in image inpainting involves reducing or +eliminating perceptual artifacts such as discontinuous edges and +blurred patches using new loss terms, image preprocessing, or train- +ing regimes that favor subjective quality over numerical accuracy. +For example, the work of Liu et al.[30], Yu et al. [59], and Xu et al. +[52] all propose extensions to partial convolutions to repair blurred +boundaries between missing and valid regions. Since our focus is +on numerical accuracy and downstream utility of the synthesized +data, we base our approach on partial convolutions from Liu et al. +[29]. Additionally, we aim to design and study architecture-agnostic +training regimes that can be used with newer models when appli- +cable. +Geospatial Missing Data Imputation Classical spatio-temporal +interpolation methods, generally variants of inverse-distance or +nearest-neighbor weighting [9, 41], kriging [3, 28], or matrix fac- +torization [13] are variations of linear methods that do not attempt +to (and cannot) interpolate within large, arbitrary, irregular re- +gions, and typically do not seamlessly consider both space and time. +Physics-based models based on computational fluid dynamics [8] or +agent-based models that directly encode human behavior [11, 47] +have been used to infer mobility dynamics, but must be designed +separately for each application rather than learned automatically +from data. Gong et al. [13] solve multi-variable non-negative ma- +trix factorization to impute urban data, but assume the availability +of multiple variables and do not consider arbitrary irregularities. +Zhang et al. [65] were concerned about the malfunction of satellites +and poor atmospheric conditions (e.g. thick cloud), which could +produce missing regions in remote sensing data. They proposed +unified spatial-temporal-spectral deep CNN architecture to recover +the missing information in satellite images. Kang et al. [21] mod- +ified the architecture from [58] to restore the missing patterns of +sea surface temperature (SST) from satellite images. Tasnim and +Mondal [48] also adopted the coarse-to-fine inpainting architecture +from [58] to restore satellite images. The innovation of their work is +the abandonment of coarse-inpainting pipeline. Instead, they used +another highly correlated temporal image as an auxiliary input to +go through the refinement pipeline. Additionally, Kadow, Hall and +Ulbrich [20] borrowed the architecture from [29] to reconstruct +missing climate information. In the geo-spatial domain, most of the + +Conference’17, July 2017, Washington, DC, USA +Bin Han and Bill Howe +literature that we found applied image inpainting techniques on +remote sensing data. As far as we acknowledge, there is no prior +work that has taken advantage of image inpainting methods to +reconstruct missing values in urban data. +3 +REPRESENTATIVE DATASETS +We worked with two mobility datasets: NYC taxi data and NYC +bikeshare data. Although potential applications of the proposed +model are widely available, datasets on which to evaluate the model +are rare: we need longitudinal coverage to provide ground truth, +sufficient complexity to study both global and local fidelity, and +accessibility to a general audience for expository purposes. Mobility +data achieves all three goals. +• NYC Taxi Data. NYC taxi trip data were collected from NYC +Open Data portal from 2011 to 20161. The year 2011 — 2015 cover +the trips throughout the entire year, while 2016 only covers the +first half of the year until June 30. The raw data are presented +in tabular format. Each record from the data summarizes the +information for one single taxi trip, which contains the longitude +and latitude of the location where the taxi took off. Each record +can be viewed as one taxi demand count. +• NYC Bikeshare Data: NYC bikeshare data were collected from +NYC DOT from 2019 to 2021 portal.2 All three datasets cover +the bike trips throughout the entire year. Similar to the taxi data, +the raw data are presented in tabular format. Each data point +summarizes the information for one single bike trip, including +the longitude and latitude of the location where the bike was +unlocked. Each record can be viewed as one bike demand count. +Figure 3: Left: Taxi pickups in 2011 overlaid on a regional +map. The distribution of taxi demand count is skewed — +high demand in Manhattan, and low demand in the sur- +rounding areas. Right: Taxi pickups aggregated into a 64×64 +histogram. +We aggregate both datasets into a 3D histogram by defining a +rectangular region, then binning mobility events into a regular grid +to create a 2D histogram amenable to image techniques. These +2D images are stacked to create 3D blocks. The temporal depth of +the block is a parameter we study in this paper. We defined the +NYC region as shown in Figure 3. At this resolution, the region +has a relatively balanced coverage of areas with different levels of +1https://opendata.cityofnewyork.us/data/ +2https://ride.citibikenyc.com/system-data +demand counts: high demand in Manhattan and near airports, and +low demand elsewhere. This skew is common in urban applications +and represents both an opportunity and a challenge for neural +prediction: the patterns are relatively stable, but the sparse regions +can dilute the learning process. +We then define a 64×64 grid over the region of interest. Then, +for each hour of each day, we count the number of taxi/bike trips +that began within each pixel. The value is commonly interpreted +as an estimate of demand. We do not consider multiple resolutions +in this paper. After processing, we have 30,648 training images and +3,620 test images in NYC taxi data, and we have 25,560 training +images and and 720 test images in NYC bikeshare data. Figure 3 +shows the defined region and an example of the corresponding taxi +demand histogram. +4 +INPAINTING MODEL +In this section, we describe the basic model for using partial con- +volutions for inpainting spatiotemporal urban histograms. Each +sample consists of a masked region with unknown, corrupted, or in- +accurate values to be reconstructed and a valid region with known +values. The task is to predict values in the masked region to match +the original image. Training is self-supervised by creating random +masks for any input image; we consider the manner in which the +masks are created in this paper. +4.1 +Model Architecture +We adapt the architecture from Liu et al.[29], which proposed par- +tial convolutional layers to accommodate irregular masks. Partial +convolutions ignore the masked region, but the mask is updated af- +ter each partial convolution layer: after several partial convolution +layers, all the values in the mask will be set to one such that the +entire output is considered valid. We use the U-Net architecture +with skip connections [40], with all the convolution layers replaced +with partial convolution layers. Web images only contain 2D in- +Figure 4: The model architecture is a U-Net extended to 3D, +partial convolutional layers [29] to ignore masked regions +during training. In the decoding branch, multiple 3D up- +convolutional layers are utilized and skip-connections are +applied. In total, there are six encoding layers and six decod- +ing layers. +formation (ignoring RGB channels), but urban histograms vary in +both space and time. As a result, our training data is essentially one +massive 3D block rather than a large number of independent train- +ing images. We therefore have a design choice of how to “shred” + +QUEENSUrban +3D +Data +Kernel +Block +3D ConV, +3D Up- +ReLU +Conv +Copy & +More 3D +Concaten +Conv +ate +LayersAdapting to Skew: Imputing Spatiotemporal Urban Data +with 3D Partial Convolutions and Biased Masking +Conference’17, July 2017, Washington, DC, USA +this block into training samples. In this paper, we consider only +the temporal extent in 3D; varying spatial resolution, bounds, or +overlap during rasterization of the source data is left for future +work. +If we slice the input into individual timesteps, the model cannot +exploit temporal consistency. We therefore extend all convolutional +layers, inputs, and masks, to 3D, and consider the effect of varying +the number of timesteps per training sample. The inputs are 3D +image blocks of dimension 𝑇 × 𝑊 × 𝐻, where 𝑇 represents the +temporal dimension. The masks are also in 3D blocks with the same +shape as the image block. The model architecture is illustrated in +Figure 4. The parameters of each convolutional layer appear in +Table 1. +Layers +Channel +Kernel Size +Stride +Padding +encoder 1 +64 +(1,3,3) +(1,2,2) +(0,1,1) +encoder 2 +128 +(1,3,3) +(1,2,2) +(0,1,1) +encoder 3 +256 +(1,3,3) +(1,2,2) +(0,1,1) +encoder 4 +512 +(1,3,3) +(1,2,2) +(0,1,1) +encoder 5 +512 +(T,3,3) +(2,2,2) +(2*((T-1)//4),1,1) +encoder 6 +512 +(T,3,3) +(2,2,2) +(2*((T-1)//4),1,1) +decoder 1 +512 +(1,3,3) +(1,1,1) +(0,1,1) +decoder 2 +512 +(1,3,3) +(1,1,1) +(0,1,1) +decoder 3 +256 +(1,3,3) +(1,1,1) +(0,1,1) +decoder 4 +128 +(1,3,3) +(1,1,1) +(0,1,1) +decoder 5 +64 +(1,3,3) +(1,1,1) +(0,1,1) +decoder 6 +1 +(1,3,3) +(1,1,1) +(0,1,1) +Table 1: Parameters of 3D convolutional layers. T represents +the temporal dimension of the image block. +4.2 +Loss function +We used ℓ1 loss as the objective function for pixel-wise reconstruc- +tion accuracy. The ℓ1 loss term bridges the absolute gap between the +reconstructed value and the ground truth. We adopt the following +notation +I𝑔𝑡 ∈ R𝑇×𝑊 ×𝐻: the block of ground truth images. 𝑇 represents +the temporal dimension of the block. +I𝑜𝑢𝑡 ∈ R𝑇×𝑊 ×𝐻 : the block of reconstructed images. +M ∈ R𝑇×𝑊 ×𝐻 : the block of binary masks. +𝑁I = 𝑇 ∗𝑊 ∗ 𝐻: the total number of pixels in the image block. +𝑁valid: the total number of valid pixels in the image block. +𝑁hole: the total number of missing pixels in the image block. +Following Liu, we separate the valid and hole regions in the +ℓ1 loss. Even though the valid region has available data and we +therefore typically would not use the predicted values in practice, +we want to include this loss during training to improve continuity +across mask boundaries (and therefore improve overall error). The +ℓ1 loss is calculated as +L𝑡𝑜𝑡𝑎𝑙 = L𝑣𝑎𝑙𝑖𝑑 + 𝜆Lℎ𝑜𝑙𝑒 +where +Lℎ𝑜𝑙𝑒 = +1 +𝑁hole +||(1 − M) ⊙ (I𝑜𝑢𝑡 − I𝑔𝑡)||1 +L𝑣𝑎𝑙𝑖𝑑 = +1 +𝑁valid +||M ⊙ (I𝑜𝑢𝑡 − I𝑔𝑡)||1 +4.3 +Biased Masking +By default, masks can be generated by randomly select a starting +point in the image and then conducting a random walk for a fixed +number of step. We call this process random masking. However, +since urban data is constrained by the built environment and is +therefore highly skewed toward populated areas, random masks +tend to include a large number of zero-valued cells, squandering +opportunities to learn from the steep gradients in dense, high-traffic +regions; Figure 5a illustrates an example. To focus attention on pop- +ulated areas, we use a biased masking approach: 1) Given an input +image, apply Gaussian blur to blend the pixel values and increase +the region of potential starting points. 2) Select a threshold (e.g., +90% percentile of the image values) to identify populous regions. +3) Randomly select a starting location from one of the detected +areas and generate masks via random walk. The probability of se- +lecting one of the detected areas is proportional to the size of the +area. These steps are illustrated in Figure 5b. The biased masking +approach makes the learning problem more challenging by increas- +ing “contrast”: ensuring that masks tend to include dense, dynamic +regions, but also include sparse, stable regions. To compare the +performance of the two masking approaches, we generated two +masks (one random and one biased) for each training sample. +(a) Random masking. For each image (left), randomly select a +starting point (orange dot, middle), then grow a mask via random +walk to generate a masked region (right). +(b) Biased masking. For each image (left), we first apply Gaussian +blur and then threshold the image (middle images), then select +a starting point at random in the thresholded region and grow a +mask via random walk (right). +Figure 5: Comparison of the random and biased masking +regimes. +5 +EXPERIMENTAL EVALUATION +We consider the following questions: +(Q1) Is the core 3D model qualitatively & quantitatively effective +at inpainting missing data? (Section 5.1, Figure 6, Table 2) +(Q2) Does increasing the number of timesteps per training sample +generally improve performance? (Section 5.2, Figure 7) +(Q3) Does biased masking improve performance overall, and in +specific regions? (Section 5.3, Figure 8) + +Conference’17, July 2017, Washington, DC, USA +Bin Han and Bill Howe +(Q4) Does varying the number of timesteps per training sample +influence the spatial distribution of error between sparse and +dense regions? (Section 5.2, Figure 9) +(Q5) Does the model faithfully reconstruct local, dynamic condi- +tions in specific areas of interest? (Section 5.5, Figure 11) +With NYC taxi data, we trained the models on both mask types +— random and biased, and with different temporal dimension T = +{1,2,3,5,7,10,15}. Based on initial experiments on both mask types +and at lower temporal chunk sizes, we found that 𝜆 = 12 offered +effective performance; we fix 𝜆 to be 12 for all experiments on the +taxi data. The batch size and initial learning rate are set to 16 and +0.01 respectively. Learning rate decays every 500 training iterations +at rate of 0.9. Unless otherwise stated, we evaluate the model on the +test set using ℓ1,ℎ𝑜𝑙𝑒, which is the sum of the absolute value of the +difference between the ground truth and predictions at the masked +positions only. +We compare our models with baseline statistical methods: +• Temporal Global Mean: On the training data, we calculate the +average taxi demand at each pixel, for each hour of the day. On +the test data, we assign each masked pixel the corresponding +global mean computed from the training data. +• Nearest Neighbor (NN) Interpolation: We assign each masked +pixel the value of the nearest unmasked pixel. We experimented +with both 2D and 3D implementations using scipy.3 +• RBF Interpolation We interpolate using radial basis functions +(RBF) on observations at points sampled outside the masked +region. We experimented with both 2D and 3D RBF interpolation +with RBF Python implementation.4 +We considered 3D kriging, but found the poor scalability to be +prohibitive: the estimated time to complete the computation for an +experiment with T=2 was about two weeks on a typical platform. +Moreover, kriging is a linear method, and we have no reason to +believe that it can reconstruct data across large, irregular regions. +Another approach, which we did not study, is to use physics- +based models based on computational fluid dynamics [8] or agent- +based models that directly encode human behavior [11, 47] to cap- +ture macro traffic dynamics. These approaches can potentially "fill" +large missing regions, but must be designed separately for each +application rather than learned automatically from data. +5.1 +Model Effectiveness (Q1) +We find that for both taxi and bikeshare datasets the proposed model +faithfully captures qualitative visual patterns and also significantly +outperforms baseline methods on multiple metrics. +5.1.1 +Qualitative Analysis. We first present some visual examples +of inpainting results on NYC taxi data in Figure 6. The left figure +shows taxi demand at four different hours of the day (8AM, 2PM, +8PM, and 2AM). From left to right, we show the ground truth, +the (biased) mask, the mask applied to the ground truth, and the +reconstructed image. The inpainting model was trained with 5 +timesteps per training sample and with biased masking. +3https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.griddata. +html#scipy.interpolate.griddata +4https://github.com/treverhines/RBF +For all hours and all masks, the model is effective at reconstruct- +ing missing data, even when the majority of the signal is obscured. +The reason is clear: the patterns are sufficiently stable from timestep +to timestep as to allow the model to infer missing values from tempo- +ral patterns as well as spatial patterns. The model is also responsive +to the time of day: We see fewer rides at 2AM than at 2PM, as +expected, suggesting that the model has learned temporally local +patterns as opposed to relying on global spatial patterns. The transi- +tion across the mask boundary is also smooth, suggesting the model +was able to consider local spatial patterns appropriately. Overall, +we find that the model is perceptually effective at reconstructing +missing values, even in challenging cases. +The right plot in Figure 6 visually shows corresponding results +for bikeshare data. The model was trained with bikeshare data using +T=3, biased masking and 𝜆 = 4. We observe similar observations as +the results from taxi data — at all times of day and for all masks, +the reconstructed images are visually similar to the ground truth +images, indicating the consistent effectiveness of our model. +5.1.2 +Quantitative Analysis. Table 2 contains quantitative results +of baseline models and our neural models in different evaluation +metrics. We observe that: 1) Our neural models, trained with either +masking type or with any temporal dimension, always outperform +the baseline models. The 2D baseline models that ignore the tempo- +ral dimension are especially ineffective. Global mean ignores spatial +effects and just models a function 𝑝𝑖𝑥𝑒𝑙,ℎ𝑜𝑢𝑟 → 𝑣𝑎𝑙𝑢𝑒. 2D- and +3D- nearest neighbor methods perform poorly when the nearest +neighbors may be far away; 2D- and 3D-RBF methods assume rela- +tively uniform sampling across the region, which is not possible in +our setting of wide-area missing data. 2) At T=5 and 7, our method +performs similarly and achieves the best performances — almost +50% lower ℓ1 error and 66% lower ℓ2 error than the best baseline. +3) SSIM does not significantly distinguish different models; while +popular in image inpainting, this metric is designed to capture per- +ceptual similarity of natural images, which are not relevant for the +spatiotemporal aggregations we study. 4) The model training time +increases by about 9 minutes for every additional hour included in +a chunk. At T=5, the model takes 55 minutes to train. The baseline +heuristic-based methods — global mean and 2D- and 3D-NN — are +very fast (completing in a few minutes) but very inaccurate given +that they do not attempt to model global dynamics. The 3D-RBF +method is inefficient: T=2 required over 24 hours to train. +5.2 +Temporal Dimension Tradeoff (Q2) +Figure 7 shows the prediction errors for NYC taxi data, evaluated +on random masks (top plot) and biased masks (bottom plot). The +y-axis is the ℓ1 loss considered for the masked region only ("Hole"). +The x-axis varies the number of timesteps included per training +sample (Temporal dimension), ranging from 1 to 15. (a) When tested +with random masks, the average mask covers the entire region, +concentrated at the center. Models trained with biased masking +reduces error at all sizes. The ℓ1 error decreases as the number +of timesteps increases up until T=7, then starts to increase again +(T=5 and T=7 have similar performances when trained with biased +masking.) At T=2, the model begins to make use of the temporal +dependency between the data by applying 3D convolutions. With +both biased and random masking, the ℓ1 loss decreases sharply + +Adapting to Skew: Imputing Spatiotemporal Urban Data +with 3D Partial Convolutions and Biased Masking +Conference’17, July 2017, Washington, DC, USA +Figure 6: Reconstructed results of taxi demand images (Left) and bike demand images (Right) at different hours time trained +with biased masking and 3D partial convolutions (T=5 for taxi data and T=3 for bikeshare). From left to right, each column +displays the ground truth image, mask, masked ground truth, and reconstructed data. From top to bottom, each row presents +the taxi demand at 8AM, 2PM, 8PM, and 2AM, respectively. +when T changes from 1 to 5. (b) When tested with biased masks, +the average masked cells are concentrated at the upper left due to +the bias toward populated regions. The plot has a similar U-shape +as that of random masking. +5.3 +Biased Masking is Effective (Q3) +Figure 7, as discussed, compares the effects of biased masking to +random masking at various value of T; we see that at all tested tem- +poral dimensions, models trained with biased masking outperform +those trained with random masking, indicated by smaller ℓ1 errors. +Model +Mask Type +ℓ1,ℎ𝑜𝑙𝑒 +ℓ2,ℎ𝑜𝑙𝑒 +SSIM +PSNR +Train (m) +Global Mean +- +1.2644 +55.3298 +0.9973 +61.4880 +<5 +2D-RBF +- +3.1442 +284.8807 +0.9890 +54.8346 +70 +2D-NN +- +3.1179 +318.6575 +0.9884 +54.0717 +<5 +3D-RBF +- +1.6653 +94.7708 +0.9956 +57.9921 +>24h +3D-NN +- +1.3632 +84.0529 +0.9964 +59.1652 +<5 +Ours, 𝑇 = 1 +biased +0.9081 +37.8468 +0.9984 +62.4268 +18 +random +0.9406 +40.3730 +0.9983 +62.4679 +18 +Ours, 𝑇 = 2 +biased +0.8551 +32.6429 +0.9986 +63.1815 +27 +random +0.8979 +35.2923 +0.9985 +63.1056 +27 +Ours, 𝑇 = 3 +biased +0.7847 +25.8374 +0.9987 +63.6445 +35 +random +0.7950 +26.4765 +0.9989 +63.7221 +35 +Ours, 𝑇 = 5 +biased +0.7196 +18.7080 +0.9991 +64.4028 +55 +random +0.7606 +20.6116 +0.9990 +64.1000 +55 +Ours, 𝑇 = 7 +biased +0.7185 +18.6746 +0.9990 +64.3407 +75 +random +0.7489 +20.0100 +0.9990 +64.2656 +75 +Ours, 𝑇 = 10 +biased +0.7537 +24.8383 +0.9986 +63.3329 +75 +random +0.7820 +26.1138 +0.9985 +63.1288 +75 +Ours, 𝑇 = 15 +biased +0.7729 +25.3386 +0.9985 +63.1885 +140 +random +0.7849 +21.9446 +0.9989 +63.8721 +140 +Table 2: Model training time and performance. +Figure 7: Evaluation of models trained with biased masking +against those trained with random masking, at seven tem- +poral dimensions, with two different masking scenarios — +random and biased masking. +In addition to the measurement of overall error, we also inspected +the convergence rates under both training regimes, as measured +by the validation set with our selected scenarios (Figure 8). The +scenario masks are chosen to evaluate local accuracy in high-traffic, +low-traffic, high-variability, and semantically important locations. +See 5.5 for masks of the scenarios and detailed evaluations. +Overall, when we tested with random and biased masks, the +model trained with biased masks converged faster and had smaller +errors, indicating that biased masking is beneficial to the imputation +task under skewed distributions (upper left). Evaluating the 5th +Avenue and Penn station scenarios, the model trained with biased + +Ground Truth, Time: 8AM +Mask +Masked Ground Truth +Prediction, Time: 8AM +Ground Truth, Time: 2PM +Mask +Masked Ground Truth +Prediction, Time: 2PM +Ground Truth, Time: 8PM +Mask +Masked Ground Truth +Prediction. Time: 8PM +Ground Truth. Time: 2AM +Mask +Masked Ground Truth +Prediction. Time: 2AMGround Truth, Time: 8AM +Mask +Masked Ground Truth +Prediction, Time: 8AM +Ground Truth. Time: 2PM +Mask +Masked Ground Truth +Prediction. Time: 2PM +Ground Truth. Time: 8PM +Mask +Masked Ground Truth +Prediction. Time: 8PM +Ground Truth, Time: 2AM +Mask +Masked Ground Truth +Prediction. Time: 2AMTrained With Biased Masking +Trained With Random Masking +Trained With Biased Masking +Trained With Random MaskingConference’17, July 2017, Washington, DC, USA +Bin Han and Bill Howe +masking displayed similar patterns — they converged faster and +achieved better results than the model trained with random masks. +Those two scenarios are representative of dense and busy areas. +We conjecture that biased masking avoids rewarding the model for +trivially predicting zero in sparse regions and ignoring the dynamics +in dense regions. We consider this result an initial foray: encoding +domain knowledge and data patterns into the masking strategy +appears to be a powerful, easy, and architecture-agnostic means of +improving model performance, aligned with emerging principles of +data-centric AI. The other three scenarios — airport, lower east side, +and Astoria, represent sparse regions with relatively light traffic. +The convergence lines for them are less stable, and no benefit of +biased masking is realized. We conjecture that variants of biased +masking to weight both dense and sparse (yet non-zero) areas may +further improve the model, as would specialized training on regions +of interest (though that approach could be considered data leakage +from training to test). +Figure 8: Convergence plots of the models trained with ei- +ther biased or random masking, and tested with random +masks, biased masks and other five additional scenarios +maskings. +5.4 +Spatial distribution of errors (Q4) +We hypothesized that the original 2D partial convolution archi- +tecture (corresponding to T=1, Figure 7(a)) would be insufficient +to capture transient events. For example, taxi rides occur in the +suburbs, but they are infrequent and less predictable; we expected +the model to be less capable of accurately predicting these events. +Increasing the temporal dimension is also expected to be helpful +with the dense region as well. +We can inspect the spatial distribution of the error for T=1 in +Figure 9 to check this hypothesis: Each map is the average of 3000 +timesteps, and is colored by the difference between the predicted +value and the ground truth: a blue cell indicates an underestimate +and a red cell represents an overestimate. We see that the suburban +regions are consistently underestimated, while the dense region is +overestimated. At T=5, we observe similar pattern, but with both +underestimation and overestimation errors significantly reduced. +The suburbs are still underestimated, but the dense regions are +Figure 9: Aggregated spatial errors between predicted and +ground truth values, from models trained with different +temporal dimensions. Red areas indicate overestimation, +while blue areas represent underestimation. +effectively improved when more temporal dimensions are incorpo- +rated. At T=15, the spatial error distribution is almost identical to +T=5, with slightly higher underestimation and lower overestima- +tion. However, T=15 requires prohibitive training time due to very +large training samples, so this approach is undesirable with just +slightly better performance. This tradeoff in temporal scope reflects +a subtle characteristic of the source data; we hypothesize that T=5 +corresponds to the window size needed to capture dynamic traffic +periods; e.g., morning and evening commutes. +5.5 +Scenario Based Evaluation (Q5) +Spatiotemporal patterns of missing data in practice are unlikely to +resemble random walks. Instead, outages will correlate with envi- +ronmental features: sensors may fail in certain weather conditions, +transient events may prevent data acquisition, or legal restrictions +on data availability may follow political boundaries. To demonstrate +the applicability of our inpainting models in real-world situations, +we evaluate the inpainting methods based on specific locations +representing varying conditions. We tested five different scenarios +to cover various spatial locations, temporal variances, and social +events. The five scenarios include the masking of 5th Avenue, Penn +Station, airport, lower east side, and Astoria. The masks are visual- +ized in Figure 10. +Figure 10: Scenario masks overlaid on NYC map. Annotation: +The ratio of masked-to-unmasked area. + +Trained With Biased Masks +Trained With Random MasksSpatial Error Distribution - T=l, Mask=biased +Spatial Error Distribution - T=3, Mask=biased +Total Overestimation Value: 93.57 +Total Overestimation Value: 101.0 +Total Underestimation Value: -249.85 +Total Underestimation Value: -194.18 +Total Absolute Error Value: 343.42 +Total Absolute Error Value: 295.18 +Spatial Error Distribution - T=5, Mask=biased +Spatial Error Distribution - T=15, Mask=biased +0 +-1 +-2 +Total Overestimation Value: 74.6 +Total Overestimation Value: 77.38 +Total Underestimation Value: -179.77 +Total Underestimation Value: -179.04 +Total Absolute Error Value: 254.37 +Total Absolute Error Value: 256.42Sth Avenue +Airport +Penn station +Lower East Side +Astoria +MaskingRatio:0.49% +MaskingRatio:0.8% +Masking Ratio:0.1% +MaskingRatio:0.42%MaskingRatio:2.17%Adapting to Skew: Imputing Spatiotemporal Urban Data +with 3D Partial Convolutions and Biased Masking +Conference’17, July 2017, Washington, DC, USA +As mentioned in Section 5.3, 5th Avenue and Penn station are rep- +resentative of busy and dense areas with heavy traffic. 5th Avenue +can also show the impacts of certain social events on traffic patterns: +The Pride Parade showed an anomalous intervention where traffic +was zero on the parade route. Lower East Side is away from central +Manhattan, with relatively lighter traffic than the first two cases. +The scenario of airport and Astoria represent the sparse regions +where traffic is light. +We chose two periods for those scenarios to cover temporal +variance – Feb. 1st to Feb. 15th, 2016, and June, 18th to June 29th, +2016. A snowstorm from Feb 5th to 8th in New York City is evident +in the data (Figure 11). On June 26th, 2016, the Pride Parade in +New York City started at 5th Avenue, and moved downtown to 8th +Street. The event blocked all traffic along the route and affected the +surrounding traffic as well. Therefore, testing in the selected June +period can help evaluate the model’s response to anomalies. +We test three inpainting models — our model trained with biased +masking at T=5, the same model but trained with random masking +at T=5, and the global mean approach. We plotted the ground truth +and predicted values at the average pixel level in the missing region, +for each hour during the selected periods. The visualizations are +provided in Figure 11. The average absolute errors between the +ground truth and predicted values, over the missing region and +during the evaluation periods, are reported in Table 3. We have the +following observations: +Scenarios +G.T.- Biased +G.T. - Random +G.T. - Mean +02/01/2016 — 02/15/2016 +5th Avenue +4.2 +6.2 +17.0 +Penn Station +19.3 +33.5 +30.0 +Lower East Side +2.5 +2.8 +8.2 +Airport +2.3 +1.6 +1.8 +Astoria +0.8 +0.7 +0.4 +06/18/2016 — 06/30/2016 +5th Avenue +3.6 +4.8 +22.53 +Penn Station +21.6 +37.5 +30.0 +Lower East Side +1.7 +2.1 +7.4 +Airport +2.4 +1.9 +2.0 +Astoria +0.8 +0.7 +0.4 +Table 3: Average absolute error between the predicted values +and ground truth, over the missing regions, and during the +selected evaluation periods. +• For three scenarios — 5th Avenue, Penn Station, and Lower East +Side, our models — whether trained with biased or random mask- +ing — have much smaller gaps between the predicted values and +the ground truth, compared with the temporal mean approach. +This benefit holds for both evaluated periods, as shown in both +Table 3 and Figure 11. For the airport and Astoria scenarios, the +temporal mean is slightly better, with much smaller magnitude +in comparison with other three cases. +• rom Table 3, we see that for both evaluation periods, the model +trained with biased masking has smaller average errors than the +model trained with random masking, other than the scenario of +airport during June. +• During the snow days (02/05-02/08/2016), it is expected that +the traffic in the dense regions would be significantly impacted, +which can be supported by the trough seen from the ground +truth line in the scenario of Penn Station (other scenarios are not +Figure 11: Temporal line plots of evaluations for five sce- +narios. In each plot, we visualize the ground truth, predic- +tion from model trained with biased masking and random +masking, and predictions from temporal mean method. Two +evaluation periods, Feb. and June are selected. The irregular +events, extreme snow days and pride parade, are annotated +with grey regions. +heavily impacted by the snow.) The model trained with biased +masking is responsive to the irregular traffic caused by extreme +weather, unlike the temporal mean baseline. +• During the event pride parade, the traffic on 5th Avenue was +all diverted to other routes, creating an anomaly in the traffic +patterns. Therefore, we saw a dip in the traffic counts. Similar +observation as the snow day, the temporal mean baseline does not +recover the missing values . However, even though the inpainting +results from our model are close to the ground truth values, they +slightly overestimate the results. +Overall, the reconstruction accuracy is compelling at specific +locations, but not perfect. For 5th Avenue scenario, the parade can +be seen as an anomaly, which is rare in the training stage and hard +to be detected. But this scenario represents another application + +Pride Parade +Ground Truth +Biased Prediction +Random Prediction +Temporal Mean +Snow Days +Ground Truth +Biased Prediction +Random Prediction +Temporal Mean +Snow Days +Ground Truth +Biased Prediction +Random Prediction +Temporal Mean +Snow Days +Ground Truth +Biased Prediction +Random Prediction +Mear +Snow Days +Ground Truth +Biased Prediction +Random Prediction +Temporal MeanConference’17, July 2017, Washington, DC, USA +Bin Han and Bill Howe +usage of our model: rather than assuming that ground truth data +is “correct". We use the masking to intentionally repair known bad +data, and reconstruct global patterns in a semantically reasonable +way. This “airbrushing” of flaws in the data can be used to improve +the quality of training sets for downstream applications, such as +biofouled or errant sensors and faulty telemetry. For example, from +the top visualization in Figure 12, we visualize the 5th Avenue +scenario: The first column shows the taxi counts along 5th Avenue +during parade day, zoomed in on the Manhattan region. Several +locations of missing data (white dots) can be seen on the avenue. +We masked out the 5th Avenue altogether and used our inpainting +model to reconstruct the missing values. The use case is to enable +policymakers and researchers to conduct counterfactual studies: +what would have taxi demand been like were it not for the parade? +The results, as shown in the forth column, recover the missing +regions in a realistic way. +Alternatively, the model might be used to synthesize parade-day +traffic rather than removing its effects. By masking the surrounding +area and retaining the parade disruption, the model can attempt to +represent the influence of the disruption elsewhere in the city. As +shown from the bottom visualization in Figure 12, the generated +results are smaller in magnitude, but overall the pattern is matched +faithfully, suggesting this use case is viable for synthesizing scenar- +ios that may not be present in the data record (natural disasters, +proposed construction, accidents, etc.). Penn Station is a train sta- +Figure 12: Top: “Airbrushing” the parade event (white pix- +els) to remove its effect on the data. Bottom: Inferring traf- +fic effects of the parade by reconstructing data everywhere +except 5th Avenue to produce qualitatively realistic results. +tion and represents a high-demand area for taxis. Our model tends +to underestimate the high demand at this location, though biased +masking improves the prediction. For Lower East Side, there are a +few anomalous spikes, to which the proposed models are respon- +sive. For airport and Astoria, our models are no better than the +temporal mean approach. We conjecture that for airport, the highly +variable rides in and out of the airport confound the model. For +Astoria, the much lower demand is harder to predict; note the lower +scale of the y-axis. +6 +DISCUSSION +Our study is motivated by the inconsistent availability of urban data +caused by missing, corrupt, or inaccurate data, which hinders their +use in downstream tasks, especially learning tasks, that require +coverage and accuracy. We designed and implemented a model +based on partial convolutions that can tolerate irregular missing +regions — zip codes, geographical boundaries, congrssional districts, +or other regions that may correlated with data absence or quality. +To capture the temporal dependency in urban data, we replaced 2D +convolutional layers in the model with 3D convolutional layers and +experimented with varying the number of timesteps per training +sample, finding non-trivial tradeoffs and a local optimum around +T=5 for taxis and T=3 for bikeshare, potentially interpretable as the +autocorrelation period of traffic (i.e., about 5 hours of rush hour). +To address the spatial skew in human activity, we proposed a +masking approach that can reflect the skew in the distribution. By +encouraging the model to attend to dense, dynamic regions (via a +percentile threshold), the model learns faster and is not rewarded +for accurate predictions in trivially inactive areas. Biased mask- +ing showed improved performance across all values of 𝑇, multiple +global evaluation strategies, and most local evaluation scenarios. +This approach suggests a broader family of related masking strate- +gies to help users encode domain knowledge about the data and +setting. For example, encoding correlations between high-traffic +areas (e.g., subway stops and train stations during lunch time) as +masks may help the model learn these correlations with less data. +Qualitatively, we confirmed from the visual examples that im- +age inpainting techniques can be used to reconstruct data in large, +irregular regions in space and time. Quantitatively, we confirmed +that extending the model architecture to 3D benefits improves per- +formance, as supported by the sharp decrease in ℓ1 when T changes +from 1 to 2. Second, we observe that increasing the temporal di- +mension to a certain threshold improves performance in general, +regardless of masking strategy; ignoring the temporal dimension +in this setting is untenable. +Additionally, we evaluated performance in local settings, demon- +strating that the model is not just learning an average value, but is +responsive to subtle spatial variation. The model captures irregular +traffic patterns caused by transient events, such as extreme weather +and the Pride Parade, and showed that biased masking can improve +performance in local settings. Additionally, the scenario evaluations +also showcased the better results introduced by the biased masking +than the random masking. +7 +LIMITATIONS & FUTURE WORK +There are several limitations of our study that represent directions +for future work. First, our results on mobility data may extend +to other urban activity (e.g., 311 calls, crowd movement, business +permits, public safety events, housing events, and more). We do not +consider the generalizability of these methods to multiple variables, +or variables that do not follow the same spatial patterns; there are +opportunities to exploit correlations between variables to improve +performance. Additionally, the taxi dataset is exceptionally large +and complete; understanding how these techniques behave in low- +data regimes is important for practical applications. Integration +of masked multi-variate data may be an opportunity: given the +shared built environment, models trained on one variable may +transfer to predictions of other variables. Second, rasterizing event +data to a form amenable to computer vision techniques involves a +number of design choices we did not study: resolution, overlap, and + +Adapting to Skew: Imputing Spatiotemporal Urban Data +with 3D Partial Convolutions and Biased Masking +Conference’17, July 2017, Washington, DC, USA +irregular boundaries may present opportunities or challenges. In +particular, data associated with census blocks, tracts, or individual +trajectories lose information when regridded as histograms. In +these cases, graph neural networks may be more appropriate to +represent the spatial adjacency relationships. Third, even with the +best model configuration, we consistently overestimate in the city +region and underestimate in the sparse suburban region. Some +model architectures (attention mechanism, multi-view learning) or +loss functions may improve performance, as may more specialized +masking and training regimes. +8 +CODE AVAILABLITY +Our code is available at [anonymized for review]. +REFERENCES +[1] Sebastian Abt. 2013. The Missing Data Problem in Cyber Security Research. In 8. +GI FG SIDAR Graduierten-Workshop über Reaktive Sicherheit. , , 8. +[2] Peter M Allen. 2012. Cities and regions as self-organizing systems: models of +complexity. Routledge, . +[3] Bumjoon Bae, Hyun Kim, Hyeonsup Lim, Yuandong Liu, Lee D. Han, and Phillip B. +Freeze. 2018. Missing data imputation for traffic flow speed using spatio-temporal +cokriging. Transportation Research Part C: Emerging Technologies 88 (2018), 124– +139. https://doi.org/10.1016/j.trc.2018.01.015 +[4] C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro, and J. Verdera. 2001. Filling-in +by joint interpolation of vector fields and gray levels. IEEE Transactions on Image +Processing 10, 8 (2001), 1200–1211. https://doi.org/10.1109/83.935036 +[5] Connelly Barnes, Eli Shechtman, Adam Finkelstein, and Dan B Goldman. 2009. +PatchMatch: A Randomized Correspondence Algorithm for Structural Image +Editing. ACM Transactions on Graphics (Proc. SIGGRAPH) 28, 3 (Aug. 2009), . +[6] Marcelo Bertalmío, Guillermo Sapiro, Vicent Caselles, and Coloma Ballester. 2000. +Image inpainting. Proceedings of the 27th annual conference on Computer graphics +and interactive techniques , (2000), . +[7] M. Bertalmio, L. Vese, G. Sapiro, and S. Osher. 2003. Simultaneous structure and +texture image inpainting. IEEE Transactions on Image Processing 12, 8 (2003), +882–889. https://doi.org/10.1109/TIP.2003.815261 +[8] Bert Blocken. 2015. Computational Fluid Dynamics for urban physics: Importance, +scales, possibilities, limitations and ten tips and tricks towards accurate and +reliable simulations. Building and Environment 91 (2015), 219–245. +https:// +doi.org/10.1016/j.buildenv.2015.02.015 Fifty Year Anniversary for Building and +Environment. +[9] Sebastian Daberdaku, Erica Tavazzi, and Barbara Di Camillo. 2019. Interpolation +and K-Nearest Neighbours Combined Imputation for Longitudinal ICU Labora- +tory Data. In 2019 IEEE International Conference on Healthcare Informatics (ICHI). +1–3. https://doi.org/10.1109/ICHI.2019.8904624 +[10] Soheil Darabi, Eli Shechtman, Connelly Barnes, Dan B. Goldman, and Pradeep +Sen. 2012. Image Melding: Combining Inconsistent Images Using Patch-Based +Synthesis. ACM Trans. Graph. 31, 4, Article 82 (jul 2012), 10 pages. +https: +//doi.org/10.1145/2185520.2185578 +[11] Felipe de Souza, Omer Verbas, and Joshua Auld. 2019. Mesoscopic Traffic Flow +Model for Agent-Based Simulation. Procedia Computer Science 151 (2019), 858–863. +https://doi.org/10.1016/j.procs.2019.04.118 The 10th International Conference on +Ambient Systems, Networks and Technologies (ANT 2019) / The 2nd Interna- +tional Conference on Emerging Data and Industry 4.0 (EDI40 2019) / Affiliated +Workshops. +[12] Iddo Drori, Daniel Cohen-Or, and Hezy Yeshurun. 2003. Fragment-Based Image +Completion. ACM Trans. Graph. 22, 3 (jul 2003), 303–312. +https://doi.org/10. +1145/882262.882267 +[13] Yongshun Gong, Zhibin Li, Jian Zhang, Wei Liu, Bei Chen, and Xiangjun Dong. +2021. A spatial missing value imputation method for multi-view urban statistical +data. In Proceedings of the Twenty-Ninth International Conference on International +Joint Conferences on Artificial Intelligence. 1310–1316. +[14] George Grekousis. 2019. Artificial neural networks and deep learning in urban +geography: A systematic review and meta-analysis. Computers, Environment and +Urban Systems 74 (2019), 244–256. +[15] Seungwoong Ha and Hawoong Jeong. 2021. Unraveling hidden interactions in +complex systems with deep learning. Scientific reports 11, 1 (2021), 1–13. +[16] James Hays and Alexei A. Efros. 2007. Scene Completion Using Millions of +Photographs. ACM Trans. Graph. 26, 3 (jul 2007), 4–es. https://doi.org/10.1145/ +1276377.1276382 +[17] Bill Howe, Jackson Maxfield Brown, Bin Han, Bernease Herman, Nic Weber, An +Yan, Sean Yang, and Yiwei Yang. 2022. Integrative urban AI to expand coverage, +access, and equity of urban data. The European physical journal. Special topics +231, 9 (2022), 1741—1752. https://doi.org/10.1140/epjs/s11734-022-00475-z +[18] Satoshi Iizuka, Edgar Simo-Serra, and Hiroshi Ishikawa. 2017. Globally and +Locally Consistent Image Completion. ACM Trans. Graph. 36, 4, Article 107 (July +2017), 14 pages. https://doi.org/10.1145/3072959.3073659 +[19] Jireh Jam, Connah Kendrick, Kevin Walker, Vincent Drouard, Jison Gee-Sern +Hsu, and Moi Hoon Yap. 2021. A comprehensive review of past and present +image inpainting methods. Computer vision and image understanding 203 (2021), +103147. +[20] Christopher Kadow, David Matthew Hall, and Uwe Ulbrich. 2020. Artificial +intelligence reconstructs missing climate information. Nature Geoscience 13, 6 +(June 2020), 408–413. https://doi.org/10.1038/s41561-020-0582-5 +[21] Song-Hee Kang, Youngjin Choi, and Jae Young Choi. 2021. Restoration of Missing +Patterns on Satellite Infrared Sea Surface Temperature Images Due to Cloud +Coverage Using Deep Generative Inpainting Network. Journal of Marine Science +and Engineering 9, 3 (2021), . https://doi.org/10.3390/jmse9030310 +[22] Christopher P Kempes and Geoffrey B West. 2020. The Simplicity and Complexity +of Cities. The Bridge 50 (2020), . +[23] Van-Duc Le, Tien-Cuong Bui, and Sang-Kyun Cha. 2020. Spatiotemporal Deep +Learning Model for Citywide Air Pollution Interpolation and Prediction. In 2020 +IEEE International Conference on Big Data and Smart Computing (BigComp). , , +55–62. https://doi.org/10.1109/BigComp48618.2020.00-99 +[24] Melissa Lee, Esteve Almirall, and Jonathan Wareham. 2015. Open data and civic +apps: first-generation failures, second-generation improvements. Commun. ACM +59, 1 (2015), 82–89. +[25] Levin, Zomet, and Weiss. 2003. Learning how to inpaint from global image +statistics. In Proceedings Ninth IEEE International Conference on Computer Vision. +, , 305–312 vol.1. https://doi.org/10.1109/ICCV.2003.1238360 +[26] YuanYuan Li and Lynne E. Parker. 2014. Nearest neighbor imputation using +spatial–temporal correlations in wireless sensor networks. Information Fusion 15 +(2014), 64–79. https://doi.org/10.1016/j.inffus.2012.08.007 Special Issue: Resource +Constrained Networks. +[27] Binbing Liao, Jingqing Zhang, Chao Wu, Douglas McIlwraith, Tong Chen, Sheng- +wen Yang, Yike Guo, and Fei Wu. 2018. Deep sequence learning with auxiliary +information for traffic prediction. In Proceedings of the 24th ACM SIGKDD Inter- +national Conference on Knowledge Discovery & Data Mining. ACM, , , 537–546. +[28] Qiushuang Lin and Chunxiang Li. 2020. Kriging based sequence interpolation and +probability distribution correction for gaussian wind field data reconstruction. +Journal of Wind Engineering and Industrial Aerodynamics 205 (2020), 104340. +https://doi.org/10.1016/j.jweia.2020.104340 +[29] Guilin Liu, Fitsum A. Reda, Kevin J. Shih, Ting-Chun Wang, Andrew Tao, and +Bryan Catanzaro. 2018. +Image Inpainting for Irregular Holes Using Partial +Convolutions. arXiv:1804.07723 [cs.CV] +[30] Hongyu Liu, Bin Jiang, Yi Xiao, and Chao Yang. 2019. Coherent Semantic Atten- +tion for Image Inpainting. In 2019 IEEE/CVF International Conference on Computer +Vision (ICCV). , , 4169–4178. https://doi.org/10.1109/ICCV.2019.00427 +[31] Kai Liu, Junke Li, and Syed Sabahat Hussain Bukhari. 2022. Overview of Image +Inpainting and Forensic Technology. Security and Communication Networks 2022 +(2022), . +[32] Eric Hsueh-Chan Lu and Chia-Yu Liu. 2021. A Spatial-Temporal Approach for +Air Quality Forecast in Urban Areas. Applied Sciences 11, 11 (2021), . +https: +//doi.org/10.3390/app11114971 +[33] Xiaolei Ma, Zhuang Dai, Zhengbing He, Jihui Ma, Yong Wang, and Yunpeng +Wang. 2017. Learning traffic as images: a deep convolutional neural network for +large-scale transportation network speed prediction. Sensors 17, 4 (2017), 818. +[34] Xiaolei Ma, Zhuang Dai, Zhengbing He, Jihui Na, Yong Wang, and Yunpeng Wang. +2017. Learning Traffic as Images: A Deep Convolutional Neural Network for +Large-Scale Transportation Network Speed Prediction. arXiv:1701.04245 [cs.LG] +[35] Luis Moreira-Matias, Joao Gama, Michel Ferreira, Joao Mendes-Moreira, and Luis +Damas. 2013. Predicting taxi–passenger demand using streaming data. IEEE +Transactions on Intelligent Transportation Systems 14, 3 (2013), 1393–1402. +[36] Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and +Alexei A. Efros. 2016. +Context Encoders: Feature Learning by Inpainting. +arXiv:1604.07379 [cs.CV] +[37] Zhen Qin, Qingliang Zeng, Yixin Zong, and Fan Xu. 2021. Image inpainting based +on deep learning: A review. Displays 69 (2021), 102028. +[38] K. R. Ridgway, J. R. Dunn, and J. L. Wilkin. 2002. Ocean Interpolation by Four- +Dimensional Weighted Least Squares—Application to the Waters around Aus- +tralasia. Journal of Atmospheric and Oceanic Technology 19, 9 (2002), 1357 – 1375. +https://doi.org/10.1175/1520-0426(2002)019<1357:OIBFDW>2.0.CO;2 +[39] Marc Riedel and Wendy C. Regoeczi. 2004. Missing Data in Homicide Research. +Homicide Studies 8, 3 (2004), 163–192. https://doi.org/10.1177/1088767904265447 +arXiv:https://doi.org/10.1177/1088767904265447 +[40] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. 2015. U-Net: Convolutional +Networks for Biomedical Image Segmentation. CoRR abs/1505.04597 (2015), . +arXiv:1505.04597 http://arxiv.org/abs/1505.04597 + +Conference’17, July 2017, Washington, DC, USA +Bin Han and Bill Howe +[41] Parisa Saeipourdizaj, Parvin Sarbakhsh, and Akbar and Gholampour. 2021. Appli- +cation of imputation methods for missing values of PM10 and O3 data: Interpola- +tion, moving average and K-nearest neighbor methods. Environmental Health +Engineering and Management Journal 8, 3 (2021). https://doi.org/10.34172/EHEM. +2021.25 arXiv:http://ehemj.com/article-1-815-en.pdf +[42] Chitwan Saharia, William Chan, Huiwen Chang, Chris Lee, Jonathan Ho, Tim +Salimans, David Fleet, and Mohammad Norouzi. 2022. Palette: Image-to-image +diffusion models. In ACM SIGGRAPH 2022 Conference Proceedings. , , 1–10. +[43] Bilong Shen, Xiaodan Liang, Yufeng Ouyang, Miaofeng Liu, Weimin Zheng, and +Kathleen M Carley. 2018. StepDeep: A Novel Spatial-temporal Mobility Event +Prediction Framework based on Deep Neural Network. In Proceedings of the 24th +ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. +ACM, , , 724–733. +[44] Bilong Shen, Xiaodan Liang, Yufeng Ouyang, Miaofeng Liu, Weimin Zheng, and +Kathleen M. Carley. 2018. StepDeep: A Novel Spatial-Temporal Mobility Event +Prediction Framework Based on Deep Neural Network. In Proceedings of the 24th +ACM SIGKDD International Conference on Knowledge Discovery & Data Min- +ing (London, United Kingdom) (KDD ’18). Association for Computing Machinery, +New York, NY, USA, 724–733. https://doi.org/10.1145/3219819.3219931 +[45] Raymond Sluiter. 2009. Interpolation methods for climate data. KNMI , (01 +2009). +[46] Andy Stock, Ajit Subramaniam, Gert Dijken, Lisa Wedding, Kevin Arrigo, +Matthew Mills, Mary Cameron, and Fiorenza Micheli. 2020. Comparison of +Cloud-Filling Algorithms for Marine Satellite Data. Remote Sensing 12 (10 2020), +3313. https://doi.org/10.3390/rs12203313 +[47] Dazhi Sun, Jinpeng Lv, and S Waller. 2013. In-depth analysis of traffic congestion +using computational fluid dynamics (CFD) modeling method. Journal of Modern +Transportation 19 (03 2013). https://doi.org/10.1007/BF03325741 +[48] Jarin Tasnim and Debajyoti Mondal. 2020. Data Reduction and Deep-Learning +Based Recovery for Geospatial Visualization and Satellite Imagery. In 2020 IEEE +International Conference on Big Data (Big Data). , , 5276–5285. https://doi.org/10. +1109/BigData50022.2020.9378008 +[49] Stefaan Verhulst, Andrew Young, Andrew Zahuranec, Susan Ariel Aaronson, +Ania Calderon, and Matt Gee. 2020. The emergence of a third wave of open data. +, (2020), . +[50] Patrick Vogel, Torsten Greiser, and Dirk Christian Mattfeld. 2011. Understanding +bike-sharing systems using data mining: Exploring activity patterns. Procedia- +Social and Behavioral Sciences 20 (2011), 514–523. +[51] Geoffrey B West. 2017. Scale: the universal laws of growth, innovation, sustainabil- +ity, and the pace of life in organisms, cities, economies, and companies. Penguin, +. +[52] Rui Xu, Minghao Guo, Jiaqi Wang, Xiaoxiao Li, Bolei Zhou, and Chen Change +Loy. 2021. Texture memory-augmented deep patch-based image inpainting. IEEE +Transactions on Image Processing 30 (2021), 9112–9124. +[53] An Yan and Bill Howe. 2019. FairST: Equitable Spatial and Temporal Demand +Prediction for New Mobility Systems. arXiv:1907.03827 [cs.CY] +[54] Huaxiu Yao, Xianfeng Tang, Hua Wei, Guanjie Zheng, Yanwei Yu, and Zhenhui Li. +2018. Modeling spatial-temporal dynamics for traffic prediction. arXiv preprint +arXiv:1803.01254 , (2018), . +[55] Ji Won Yoon, Fabio Pinelli, and Francesco Calabrese. 2012. Cityride: a predictive +bike sharing journey advisor. In 2012 IEEE 13th International Conference on Mobile +Data Management. IEEE, , , 306–311. +[56] Haiyang Yu, Zhihai Wu, Shuqin Wang, Yunpeng Wang, and Xiaolei Ma. 2017. +Spatiotemporal Recurrent Convolutional Networks for Traffic Prediction in Trans- +portation Networks. arXiv:1705.02699 [cs.LG] +[57] Jiahui Yu, Zhe Lin, Jimei Yang, Xiaohui Shen, Xin Lu, and Thomas Huang. 2019. +Free-Form Image Inpainting with Gated Convolution. arXiv:1806.03589 [cs.CV] +[58] Jiahui Yu, Zhe Lin, Jimei Yang, Xiaohui Shen, Xin Lu, and Thomas S. +Huang. 2018. +Generative Image Inpainting with Contextual Attention. +arXiv:1801.07892 [cs.CV] +[59] Tao Yu, Zongyu Guo, Xin Jin, Shilin Wu, Zhibo Chen, Weiping Li, Zhizheng Zhang, +and Sen Liu. 2020. Region normalization for image inpainting. In Proceedings of +the AAAI Conference on Artificial Intelligence, Vol. 34. , , 12733–12740. +[60] Zhuoning Yuan, Xun Zhou, and Tianbao Yang. 2018. Hetero-ConvLSTM: A +deep learning approach to traffic accident prediction on heterogeneous spatio- +temporal data. In Proceedings of the 24th ACM SIGKDD International Conference +on Knowledge Discovery & Data Mining. ACM, , , 984–992. +[61] Yanhong Zeng, Jianlong Fu, Hongyang Chao, and Baining Guo. 2022. Aggre- +gated contextual transformations for high-resolution image inpainting. IEEE +Transactions on Visualization and Computer Graphics , (2022), . +[62] Han Zhang, Peng Chen, Jianfeng Zheng, Jingqing Zhu, Guizhen Yu, Yunpeng +Wang, and Henry X. Liu. 2019. Missing data detection and imputation for urban +ANPR system using an iterative tensor decomposition approach. Transportation +Research Part C: Emerging Technologies 107 (2019), 337–355. https://doi.org/10. +1016/j.trc.2019.08.013 +[63] Junbo Zhang, Yu Zheng, Dekang Qi, Ruiyuan Li, and Xiuwen Yi. 2016. DNN- +based prediction model for spatio-temporal data. In Proceedings of the 24th ACM +SIGSPATIAL International Conference on Advances in Geographic Information +Systems. ACM, , , 92. +[64] Junbo Zhang, Yu Zheng, Dekang Qi, Ruiyuan Li, Xiuwen Yi, and Tianrui Li. 2018. +Predicting citywide crowd flows using deep spatio-temporal residual networks. +Artificial Intelligence 259 (2018), 147–166. +[65] Qiang Zhang, Qiangqiang Yuan, Chao Zeng, Xinghua Li, and Yancong Wei. +2018. +Missing Data Reconstruction in Remote Sensing Image With a Uni- +fied Spatial–Temporal–Spectral Deep Convolutional Neural Network. +IEEE +Transactions on Geoscience and Remote Sensing 56, 8 (2018), 4274–4288. https: +//doi.org/10.1109/TGRS.2018.2810208 +[66] Tong Zhou, Changxing Ding, Shaowen Lin, Xinchao Wang, and Dacheng Tao. +2020. Learning oracle attention for high-fidelity face completion. In Proceedings +of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. , , 7680– +7689. + diff --git a/EtE2T4oBgHgl3EQf-Ak-/content/tmp_files/load_file.txt b/EtE2T4oBgHgl3EQf-Ak-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a9a4f12c68a09a95d9ddb5b514e4c376007a00f --- /dev/null +++ b/EtE2T4oBgHgl3EQf-Ak-/content/tmp_files/load_file.txt @@ -0,0 +1,1132 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf,len=1131 +page_content='Adapting to Skew: Imputing Spatiotemporal Urban Data with 3D Partial Convolutions and Biased Masking Bin Han bh193@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='edu University of Washington Seattle, USA Bill Howe billhowe@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='edu University of Washington Seattle, USA ABSTRACT We adapt image inpainting techniques to impute large, irregular missing regions in urban settings characterized by sparsity, variance in both space and time, and anomalous events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Missing regions in urban data can be caused by sensor or software failures, data quality issues, interference from weather events, incomplete data collection, or varying data use regulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' any missing data can render the entire dataset unusable for downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To ensure coverage and utility, we adapt computer vision techniques for image inpainting to operate on 3D histograms (2D space + 1D time) commonly used for data exchange in urban settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Adapting these techniques to the spatiotemporal setting requires handling skew: urban data tend to follow population density pat- terns (small dense regions surrounded by large sparse areas);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' these patterns can dominate the learning process and fool the model into ignoring local or transient effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To combat skew, we 1) train simultaneously in space and time, and 2) focus attention on dense regions by biasing the masks used for training to the skew in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We evaluate the core model and these two extensions using the NYC taxi data and the NYC bikeshare data, simulating differ- ent conditions for missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We show that the core model is effective qualitatively and quantitatively, and that biased masking during training reduces error in a variety of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We also ar- ticulate a tradeoff in varying the number of timesteps per training sample: too few timesteps and the model ignores transient events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' too many timesteps and the model is slow to train with limited performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' CCS CONCEPTS General and reference → Empirical studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' • Computing methodologies → Computer vision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' • Applied computing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' KEYWORDS image inpainting, urban computing, spatial-temporal, missing data ACM Reference Format: Bin Han and Bill Howe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Adapting to Skew: Imputing Spatiotemporal Urban Data with 3D Partial Convolutions and Biased Masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='nnnnnnn of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM, New York, NY, USA, 12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION High-quality, longitudinal, and freely available urban data, coupled with advances in machine learning, improve our understanding and management of urban environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Although conventional machine learning techniques are common in urban applications [35, 50, 55], neural architectures are opening new opportunities by adapting convolutional, recurrent, and transformer architectures to spatiotemporal settings [17, 27, 33, 43, 54, 60, 63, 64];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' see Grekousis 2020 for a recent survey [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For example, spatio-temporal neural architectures have been used in predictions of rideshare demand [44, 53], traffic conditions [34, 56], and air quality [23, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' But these models depend on access to complete, longitudinal datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Such datasets are inconsistent in availability and quality, limiting the opportunity for understanding cities as the complex systems they are [2, 15, 22, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 1: A histogram of taxi pickups in Manhattan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We adapt imagine inpainting techniques to reconstruct missing and corrupted data in urban settings: The improved model (upper left) uses biased masking and temporal context to capture local effects (red circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The basic model (lower left) uses ordinary masking and is insensitive to local effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Baseline methods that ignore space (lower middle) or time (lower right) are not competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Classical linear methods such as kriging and inverse-distance weighting (not shown) cannot impute large irregular regions in dynamic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='04233v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='CV] 10 Jan 2023 L1 Error: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='642 L1 Error: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='308 L1 Error: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='446 L1 Error: 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='994Conference’17, July 2017, Washington, DC, USA Bin Han and Bill Howe This inconsistency persists despite significant investments in open data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Over the last two decades, cities have increasingly re- leased datasets publicly on the web, proactively, in response to transparency regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For example, in the US, all 50 states and the District of Columbia have passed some version of the federal Freedom of Information (FOI) Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' While this first wave of open data was driven by FOI laws and made national government data available primarily to journalists, lawyers, and activists, a second wave of open data, enabled by the advent of open source and web 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0 technologies, was characterized by an attempt to make data “open by default" to civic technologists, government agencies, and corporations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' While open data has indeed made significant data assets available online, their uptake and use has been weaker than anticipated [49], an effect attributable to convenience sam- pling effects [24]: We release what we can, even if portions are missing, corrupt, or anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In this paper, we consider a neural data cleaning strategy based on masking out corrupted regions and using a trained model to reconstruct the masked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' These masks are necessarily large, irregular, and extend in both time and space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' they may represent po- litical boundaries (municipal zoning, zip codes, city blocks), sensor or software failures [26, 62, 65], varying legal restrictions [1, 39], or unusual events (adverse weather).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' These missing patches can destroy the utility of the entire dataset for applications that assume coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' By modeling missing or corrupted data by an arbitrary mask, we afford user control: any areas can be masked and recon- structed, regardless of the reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We envision tools to improve the coverage and quality of data for use in downstream urban learning tasks [23, 32, 34, 44, 53, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Following the literature, we represent spatiotemporal event data in a 2D or 3D raster form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=', a histogram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Our basic model uses the partial convolution approach from Liu et al [29] to handle the irregular boundaries of missing data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=', districts), which focuses model attention on the valid regions while shrinking the masked region, layer-by-layer, to obtain a complete prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' More recent approaches to image inpainting on the web emphasize eliminating perceptual artifacts rather than numerical accuracy and are there- fore less relevant to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Our contribution is to extend the basic model to the 3D spatiotemporal setting and propose a training regime that adapts to the skewed distribution found in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Spatiotemporal interpolation of missing data has been widely studied in the earth sciences [38, 45], especially in remote sensing where weather effects can obscure measurement [46, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Con- ventional statistical approaches to impute missing values, such as global/local mean imputation, interpolation, and kriging, are essen- tially linear, and therefore limited in their ability to capture the non- linear dynamics needed to impute large irregular missing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Neural image inpainting techniques [29, 57] can recover missing patches via training on large datasets of independent images, such that the reconstructed images appear realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' These approaches have shown promising results with global climate data [48], but have not been adapted to the urban setting in which data are not smooth functions of space and time, but are rather histograms of events constrained by the built environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The goal of inpainting for natural images is to produce a subjec- tively recognizable image free from perceptible artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' But the goal in our setting is quantitative accuracy: we intend for our recon- structed results to be used numerically in downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The distribution is relatively stable, but exhibits skew and sparsity that can obscure local, dynamic features (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The challenge for imputation in the urban setting is skew: urban data tend to follow population density patterns — small dense regions surrounded by large sparse areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' These population patterns can dominate the learning process and fool the model into ignoring numerical accuracy in dense regions, even while aggregate error may remains low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To combat skew, we 1) bias the training process to focus on populated regions by seeding the mask in non-zero areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (2) use 3D convolutions and vary the number of timesteps in each 3D training sample to capture transient events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Together, these two techniques complement each other: biased masking focuses attention on dense regions, and 3D convolutions with a large chunk size focus attention on sparse regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We evaluate these techniques on the NYC taxi data (a popular dataset for its coverage and quality) and a NYC bikeshare dataset (less dominated by the built environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We find that the basic model is effective for urban data imputation, while biased masking reliably reduces error over random masking, both globally and locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, we find that the number of timesteps per training sample exhibits a tradeoff: too few timesteps and the model ignores transient patterns, while too many timesteps significantly increases training time without enhancing the inpainting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We evaluate specific local scenarios (high-traffic locations, low- traffic locations, high-variability locations, anomalous events) to reflect the use cases distinct from image inpainting on the web (where subjective quality is all that matters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 2: Urban data (bottom row) exhibits skewed, sparse, yet stable distributions that can dominate learning, in con- trast with the diversity of natural images (top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In summary, we make the following contributions: We evaluate a basic model adapting image inpainting techniques to urban histograms characterized by skew and sparsity effects due to constraints by the built environment, demonstrating qual- itative and quantitative accuracy relative to classical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We improve on this basic model by extending to the 3D spatiotem- poral setting to better recognize transient events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' we analyze the training time and performance tradeoffs of varying the number of timesteps per training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We propose a self-supervised training process called biased mask- ing to encourage the model to attend to dense population regions Adapting to Skew: Imputing Spatiotemporal Urban Data with 3D Partial Convolutions and Biased Masking Conference’17, July 2017, Washington, DC, USA and thereby improve accuracy on the highly dynamic regions typical in urban environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' we show that biased masking reliably improves convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We evaluate these techniques on two real mobility datasets (NYC taxi trips and NYC bikeshare trips), both globally and locally in varying traffic conditions, weather events, and disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Finally, we show that the model can be used to remove or syn- thesize anomalous events through targeted masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2 RELATED WORK Our work is informed by techniques in image inpainting and geospa- tial interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Image Inpainting Image inpainting, or image completion, is a task of synthesizing missing pixels in images, such that the recon- structed images are visually credible and semantically realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In computer vision, there are two broad categories of inpainting tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The first category contains diffusion-based or patch-based methods, which utilize low-level image features to recover the miss- ing pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The second category contains learning-based methods that generally involve the training of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Diffusion-based methods [4, 6, 25] propagate information from neighboring valid pixels to missing pixels, typically from border to the center of the missing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Those techniques are convenient to apply, but are limited to small missing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Recently, Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [42] developed an image-to-image translation framework based on conditional diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The evaluation on inpainting task outperformed several learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Patch-based inpainting techniques [7, 10, 12, 16] function by searching similar patches from the valid regions of the same image or from other images, and then paste the patches to the target missing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' However, this process could induce high computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' A milestone of patch-based approach, PatchMatch [5], speeds up the search process with a new nearest neighbor algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Learning-based methods are trained to learn image patterns with large volume of image data, thus being capable of recovering miss- ing regions, as well preserving the semantics of the imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [36] proposed context encoder, which was the first work to combine CNN with generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' It applied the encoder-decoder architecture and used both ℓ2 reconstruction loss and generative adversarial loss in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Lizuka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [18] improved on their work by incorporating global and local discriminator, which improved content consistency between the valid and missing region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, they replaced general convolutional layers with dialated convolutional layers to better capture information from distant pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [58] proposed proposed a two-stage coarse-to-fine model architecture and incor- porated contextual attention layer to attend to related features from spatially distant regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' They also replaced general generative ad- versarial loss with WGANS loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [29] proposed partial convolution, allowing inpainting models to be used on irregular holes rather than just rectangular missing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' On top the work of partial convolution, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [57] proposed gated convolutional layers to automatically learn and update the masks as opposed to rule-based update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To further address the problems of blurry tex- tures and distorted structures in the inpainted images, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [30] proposed coherent semantic attention layer, which can both pre- serve contextual structure and capture semantic relevance between hole features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [66] incorporated dual spatial attention modules into the U-Net architecture, which can capture the corre- lations between facial textures at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Seven different discriminators are utilized to ensure realistic local details as well as global consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [59] designed spatial region-wise normalization (RN) to overcome the problem of mean and variance shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' RN computes the mean and variances separately for the missing and valid regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [52] combined the paradigms of both patch-based and learning-based methods, and inpainted missing regions using textures of patch samples from unmasked regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, they proposed patch distribution loss to en- sure the quality of synthesized missing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [61] introduced aggregated contextual transformation GAN, aiming to improve content reasoning from distant pixels and enhance details of synthesized textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For more image inpainting works, we refer reader to the following surveys [19, 31, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The recent trajectory in image inpainting involves reducing or eliminating perceptual artifacts such as discontinuous edges and blurred patches using new loss terms, image preprocessing, or train- ing regimes that favor subjective quality over numerical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For example, the work of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [30], Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [59], and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [52] all propose extensions to partial convolutions to repair blurred boundaries between missing and valid regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Since our focus is on numerical accuracy and downstream utility of the synthesized data, we base our approach on partial convolutions from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, we aim to design and study architecture-agnostic training regimes that can be used with newer models when appli- cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Geospatial Missing Data Imputation Classical spatio-temporal interpolation methods, generally variants of inverse-distance or nearest-neighbor weighting [9, 41], kriging [3, 28], or matrix fac- torization [13] are variations of linear methods that do not attempt to (and cannot) interpolate within large, arbitrary, irregular re- gions, and typically do not seamlessly consider both space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Physics-based models based on computational fluid dynamics [8] or agent-based models that directly encode human behavior [11, 47] have been used to infer mobility dynamics, but must be designed separately for each application rather than learned automatically from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [13] solve multi-variable non-negative ma- trix factorization to impute urban data, but assume the availability of multiple variables and do not consider arbitrary irregularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [65] were concerned about the malfunction of satellites and poor atmospheric conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' thick cloud), which could produce missing regions in remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' They proposed unified spatial-temporal-spectral deep CNN architecture to recover the missing information in satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [21] mod- ified the architecture from [58] to restore the missing patterns of sea surface temperature (SST) from satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Tasnim and Mondal [48] also adopted the coarse-to-fine inpainting architecture from [58] to restore satellite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The innovation of their work is the abandonment of coarse-inpainting pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Instead, they used another highly correlated temporal image as an auxiliary input to go through the refinement pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, Kadow, Hall and Ulbrich [20] borrowed the architecture from [29] to reconstruct missing climate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In the geo-spatial domain, most of the Conference’17, July 2017, Washington, DC, USA Bin Han and Bill Howe literature that we found applied image inpainting techniques on remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' As far as we acknowledge, there is no prior work that has taken advantage of image inpainting methods to reconstruct missing values in urban data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 3 REPRESENTATIVE DATASETS We worked with two mobility datasets: NYC taxi data and NYC bikeshare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Although potential applications of the proposed model are widely available, datasets on which to evaluate the model are rare: we need longitudinal coverage to provide ground truth, sufficient complexity to study both global and local fidelity, and accessibility to a general audience for expository purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Mobility data achieves all three goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' NYC Taxi Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' NYC taxi trip data were collected from NYC Open Data portal from 2011 to 20161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The year 2011 — 2015 cover the trips throughout the entire year, while 2016 only covers the first half of the year until June 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The raw data are presented in tabular format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Each record from the data summarizes the information for one single taxi trip, which contains the longitude and latitude of the location where the taxi took off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Each record can be viewed as one taxi demand count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' NYC Bikeshare Data: NYC bikeshare data were collected from NYC DOT from 2019 to 2021 portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 All three datasets cover the bike trips throughout the entire year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Similar to the taxi data, the raw data are presented in tabular format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Each data point summarizes the information for one single bike trip, including the longitude and latitude of the location where the bike was unlocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Each record can be viewed as one bike demand count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 3: Left: Taxi pickups in 2011 overlaid on a regional map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The distribution of taxi demand count is skewed — high demand in Manhattan, and low demand in the sur- rounding areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Right: Taxi pickups aggregated into a 64×64 histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We aggregate both datasets into a 3D histogram by defining a rectangular region, then binning mobility events into a regular grid to create a 2D histogram amenable to image techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' These 2D images are stacked to create 3D blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The temporal depth of the block is a parameter we study in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We defined the NYC region as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' At this resolution, the region has a relatively balanced coverage of areas with different levels of 1https://opendata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='cityofnewyork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='us/data/ 2https://ride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='citibikenyc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='com/system-data demand counts: high demand in Manhattan and near airports, and low demand elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' This skew is common in urban applications and represents both an opportunity and a challenge for neural prediction: the patterns are relatively stable, but the sparse regions can dilute the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We then define a 64×64 grid over the region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Then, for each hour of each day, we count the number of taxi/bike trips that began within each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The value is commonly interpreted as an estimate of demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We do not consider multiple resolutions in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' After processing, we have 30,648 training images and 3,620 test images in NYC taxi data, and we have 25,560 training images and and 720 test images in NYC bikeshare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 3 shows the defined region and an example of the corresponding taxi demand histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 4 INPAINTING MODEL In this section, we describe the basic model for using partial con- volutions for inpainting spatiotemporal urban histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Each sample consists of a masked region with unknown, corrupted, or in- accurate values to be reconstructed and a valid region with known values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The task is to predict values in the masked region to match the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Training is self-supervised by creating random masks for any input image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' we consider the manner in which the masks are created in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1 Model Architecture We adapt the architecture from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [29], which proposed par- tial convolutional layers to accommodate irregular masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Partial convolutions ignore the masked region, but the mask is updated af- ter each partial convolution layer: after several partial convolution layers, all the values in the mask will be set to one such that the entire output is considered valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We use the U-Net architecture with skip connections [40], with all the convolution layers replaced with partial convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Web images only contain 2D in- Figure 4: The model architecture is a U-Net extended to 3D, partial convolutional layers [29] to ignore masked regions during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In the decoding branch, multiple 3D up- convolutional layers are utilized and skip-connections are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In total, there are six encoding layers and six decod- ing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' formation (ignoring RGB channels), but urban histograms vary in both space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' As a result, our training data is essentially one massive 3D block rather than a large number of independent train- ing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We therefore have a design choice of how to “shred” QUEENSUrban 3D Data Kernel Block 3D ConV, 3D Up- ReLU Conv Copy & More 3D Concaten Conv ate LayersAdapting to Skew: Imputing Spatiotemporal Urban Data with 3D Partial Convolutions and Biased Masking Conference’17, July 2017, Washington, DC, USA this block into training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In this paper, we consider only the temporal extent in 3D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' varying spatial resolution, bounds, or overlap during rasterization of the source data is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' If we slice the input into individual timesteps, the model cannot exploit temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We therefore extend all convolutional layers, inputs, and masks, to 3D, and consider the effect of varying the number of timesteps per training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The inputs are 3D image blocks of dimension 𝑇 × 𝑊 × 𝐻, where 𝑇 represents the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The masks are also in 3D blocks with the same shape as the image block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The model architecture is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The parameters of each convolutional layer appear in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Layers Channel Kernel Size Stride Padding encoder 1 64 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) encoder 2 128 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) encoder 3 256 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) encoder 4 512 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) encoder 5 512 (T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2) (2*((T-1)//4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) encoder 6 512 (T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2) (2*((T-1)//4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) decoder 1 512 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) decoder 2 512 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) decoder 3 256 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) decoder 4 128 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) decoder 5 64 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) decoder 6 1 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1) Table 1: Parameters of 3D convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' T represents the temporal dimension of the image block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 Loss function We used ℓ1 loss as the objective function for pixel-wise reconstruc- tion accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The ℓ1 loss term bridges the absolute gap between the reconstructed value and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We adopt the following notation I𝑔𝑡 ∈ R𝑇×𝑊 ×𝐻: the block of ground truth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 𝑇 represents the temporal dimension of the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' I𝑜𝑢𝑡 ∈ R𝑇×𝑊 ×𝐻 : the block of reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' M ∈ R𝑇×𝑊 ×𝐻 : the block of binary masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 𝑁I = 𝑇 ∗𝑊 ∗ 𝐻: the total number of pixels in the image block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 𝑁valid: the total number of valid pixels in the image block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 𝑁hole: the total number of missing pixels in the image block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Following Liu, we separate the valid and hole regions in the ℓ1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Even though the valid region has available data and we therefore typically would not use the predicted values in practice, we want to include this loss during training to improve continuity across mask boundaries (and therefore improve overall error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The ℓ1 loss is calculated as L𝑡𝑜𝑡𝑎𝑙 = L𝑣𝑎𝑙𝑖𝑑 + 𝜆Lℎ𝑜𝑙𝑒 where Lℎ𝑜𝑙𝑒 = 1 𝑁hole ||(1 − M) ⊙ (I𝑜𝑢𝑡 − I𝑔𝑡)||1 L𝑣𝑎𝑙𝑖𝑑 = 1 𝑁valid ||M ⊙ (I𝑜𝑢𝑡 − I𝑔𝑡)||1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3 Biased Masking By default, masks can be generated by randomly select a starting point in the image and then conducting a random walk for a fixed number of step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We call this process random masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' However, since urban data is constrained by the built environment and is therefore highly skewed toward populated areas, random masks tend to include a large number of zero-valued cells, squandering opportunities to learn from the steep gradients in dense, high-traffic regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 5a illustrates an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To focus attention on pop- ulated areas, we use a biased masking approach: 1) Given an input image, apply Gaussian blur to blend the pixel values and increase the region of potential starting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2) Select a threshold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=', 90% percentile of the image values) to identify populous regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 3) Randomly select a starting location from one of the detected areas and generate masks via random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The probability of se- lecting one of the detected areas is proportional to the size of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' These steps are illustrated in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The biased masking approach makes the learning problem more challenging by increas- ing “contrast”: ensuring that masks tend to include dense, dynamic regions, but also include sparse, stable regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To compare the performance of the two masking approaches, we generated two masks (one random and one biased) for each training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (a) Random masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For each image (left), randomly select a starting point (orange dot, middle), then grow a mask via random walk to generate a masked region (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (b) Biased masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For each image (left), we first apply Gaussian blur and then threshold the image (middle images), then select a starting point at random in the thresholded region and grow a mask via random walk (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 5: Comparison of the random and biased masking regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5 EXPERIMENTAL EVALUATION We consider the following questions: (Q1) Is the core 3D model qualitatively & quantitatively effective at inpainting missing data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1, Figure 6, Table 2) (Q2) Does increasing the number of timesteps per training sample generally improve performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2, Figure 7) (Q3) Does biased masking improve performance overall, and in specific regions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3, Figure 8) Conference’17, July 2017, Washington, DC, USA Bin Han and Bill Howe (Q4) Does varying the number of timesteps per training sample influence the spatial distribution of error between sparse and dense regions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2, Figure 9) (Q5) Does the model faithfully reconstruct local, dynamic condi- tions in specific areas of interest?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='5, Figure 11) With NYC taxi data, we trained the models on both mask types — random and biased, and with different temporal dimension T = {1,2,3,5,7,10,15}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Based on initial experiments on both mask types and at lower temporal chunk sizes, we found that 𝜆 = 12 offered effective performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' we fix 𝜆 to be 12 for all experiments on the taxi data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The batch size and initial learning rate are set to 16 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='01 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Learning rate decays every 500 training iterations at rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Unless otherwise stated, we evaluate the model on the test set using ℓ1,ℎ𝑜𝑙𝑒, which is the sum of the absolute value of the difference between the ground truth and predictions at the masked positions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We compare our models with baseline statistical methods: Temporal Global Mean: On the training data, we calculate the average taxi demand at each pixel, for each hour of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' On the test data, we assign each masked pixel the corresponding global mean computed from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Nearest Neighbor (NN) Interpolation: We assign each masked pixel the value of the nearest unmasked pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We experimented with both 2D and 3D implementations using scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3 RBF Interpolation We interpolate using radial basis functions (RBF) on observations at points sampled outside the masked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We experimented with both 2D and 3D RBF interpolation with RBF Python implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4 We considered 3D kriging, but found the poor scalability to be prohibitive: the estimated time to complete the computation for an experiment with T=2 was about two weeks on a typical platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Moreover, kriging is a linear method, and we have no reason to believe that it can reconstruct data across large, irregular regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Another approach, which we did not study, is to use physics- based models based on computational fluid dynamics [8] or agent- based models that directly encode human behavior [11, 47] to cap- ture macro traffic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' These approaches can potentially "fill" large missing regions, but must be designed separately for each application rather than learned automatically from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1 Model Effectiveness (Q1) We find that for both taxi and bikeshare datasets the proposed model faithfully captures qualitative visual patterns and also significantly outperforms baseline methods on multiple metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1 Qualitative Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We first present some visual examples of inpainting results on NYC taxi data in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The left figure shows taxi demand at four different hours of the day (8AM, 2PM, 8PM, and 2AM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' From left to right, we show the ground truth, the (biased) mask, the mask applied to the ground truth, and the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The inpainting model was trained with 5 timesteps per training sample and with biased masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 3https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/doc/scipy/reference/generated/scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='interpolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='griddata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' html#scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='interpolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='griddata 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='com/treverhines/RBF For all hours and all masks, the model is effective at reconstruct- ing missing data, even when the majority of the signal is obscured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The reason is clear: the patterns are sufficiently stable from timestep to timestep as to allow the model to infer missing values from tempo- ral patterns as well as spatial patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The model is also responsive to the time of day: We see fewer rides at 2AM than at 2PM, as expected, suggesting that the model has learned temporally local patterns as opposed to relying on global spatial patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The transi- tion across the mask boundary is also smooth, suggesting the model was able to consider local spatial patterns appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Overall, we find that the model is perceptually effective at reconstructing missing values, even in challenging cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The right plot in Figure 6 visually shows corresponding results for bikeshare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The model was trained with bikeshare data using T=3, biased masking and 𝜆 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We observe similar observations as the results from taxi data — at all times of day and for all masks, the reconstructed images are visually similar to the ground truth images, indicating the consistent effectiveness of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 Quantitative Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Table 2 contains quantitative results of baseline models and our neural models in different evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We observe that: 1) Our neural models, trained with either masking type or with any temporal dimension, always outperform the baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The 2D baseline models that ignore the tempo- ral dimension are especially ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Global mean ignores spatial effects and just models a function 𝑝𝑖𝑥𝑒𝑙,ℎ𝑜𝑢𝑟 → 𝑣𝑎𝑙𝑢𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2D- and 3D- nearest neighbor methods perform poorly when the nearest neighbors may be far away;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2D- and 3D-RBF methods assume rela- tively uniform sampling across the region, which is not possible in our setting of wide-area missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2) At T=5 and 7, our method performs similarly and achieves the best performances — almost 50% lower ℓ1 error and 66% lower ℓ2 error than the best baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 3) SSIM does not significantly distinguish different models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' while popular in image inpainting, this metric is designed to capture per- ceptual similarity of natural images, which are not relevant for the spatiotemporal aggregations we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 4) The model training time increases by about 9 minutes for every additional hour included in a chunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' At T=5, the model takes 55 minutes to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The baseline heuristic-based methods — global mean and 2D- and 3D-NN — are very fast (completing in a few minutes) but very inaccurate given that they do not attempt to model global dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The 3D-RBF method is inefficient: T=2 required over 24 hours to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 Temporal Dimension Tradeoff (Q2) Figure 7 shows the prediction errors for NYC taxi data, evaluated on random masks (top plot) and biased masks (bottom plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The y-axis is the ℓ1 loss considered for the masked region only ("Hole").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The x-axis varies the number of timesteps included per training sample (Temporal dimension), ranging from 1 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (a) When tested with random masks, the average mask covers the entire region, concentrated at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Models trained with biased masking reduces error at all sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The ℓ1 error decreases as the number of timesteps increases up until T=7, then starts to increase again (T=5 and T=7 have similar performances when trained with biased masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=') At T=2, the model begins to make use of the temporal dependency between the data by applying 3D convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' With both biased and random masking, the ℓ1 loss decreases sharply Adapting to Skew: Imputing Spatiotemporal Urban Data with 3D Partial Convolutions and Biased Masking Conference’17, July 2017, Washington, DC, USA Figure 6: Reconstructed results of taxi demand images (Left) and bike demand images (Right) at different hours time trained with biased masking and 3D partial convolutions (T=5 for taxi data and T=3 for bikeshare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' From left to right, each column displays the ground truth image, mask, masked ground truth, and reconstructed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' From top to bottom, each row presents the taxi demand at 8AM, 2PM, 8PM, and 2AM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' when T changes from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' (b) When tested with biased masks, the average masked cells are concentrated at the upper left due to the bias toward populated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The plot has a similar U-shape as that of random masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3 Biased Masking is Effective (Q3) Figure 7, as discussed, compares the effects of biased masking to random masking at various value of T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' we see that at all tested tem- poral dimensions, models trained with biased masking outperform those trained with random masking, indicated by smaller ℓ1 errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Model Mask Type ℓ1,ℎ𝑜𝑙𝑒 ℓ2,ℎ𝑜𝑙𝑒 SSIM PSNR Train (m) Global Mean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2644 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9973 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4880 <5 2D-RBF 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1442 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9890 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8346 70 2D-NN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1179 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9884 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0717 <5 3D-RBF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6653 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9956 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9921 >24h 3D-NN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3632 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9964 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1652 <5 Ours, 𝑇 = 1 biased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9081 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9984 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4268 18 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9406 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3730 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9983 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4679 18 Ours, 𝑇 = 2 biased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8551 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9986 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1815 27 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8979 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9985 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1056 27 Ours, 𝑇 = 3 biased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7847 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9987 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6445 35 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7950 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9989 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7221 35 Ours, 𝑇 = 5 biased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7196 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9991 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4028 55 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7606 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9990 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1000 55 Ours, 𝑇 = 7 biased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7185 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9990 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3407 75 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7489 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9990 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2656 75 Ours, 𝑇 = 10 biased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7537 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9986 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3329 75 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7820 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9985 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1288 75 Ours, 𝑇 = 15 biased 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7729 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9985 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1885 140 random 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7849 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9989 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8721 140 Table 2: Model training time and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 7: Evaluation of models trained with biased masking against those trained with random masking, at seven tem- poral dimensions, with two different masking scenarios — random and biased masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In addition to the measurement of overall error, we also inspected the convergence rates under both training regimes, as measured by the validation set with our selected scenarios (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The scenario masks are chosen to evaluate local accuracy in high-traffic, low-traffic, high-variability, and semantically important locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' See 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='5 for masks of the scenarios and detailed evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Overall, when we tested with random and biased masks, the model trained with biased masks converged faster and had smaller errors, indicating that biased masking is beneficial to the imputation task under skewed distributions (upper left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Evaluating the 5th Avenue and Penn station scenarios, the model trained with biased Ground Truth, Time: 8AM Mask Masked Ground Truth Prediction, Time: 8AM Ground Truth, Time: 2PM Mask Masked Ground Truth Prediction, Time: 2PM Ground Truth, Time: 8PM Mask Masked Ground Truth Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 8PM Ground Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 2AM Mask Masked Ground Truth Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 2AMGround Truth, Time: 8AM Mask Masked Ground Truth Prediction, Time: 8AM Ground Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 2PM Mask Masked Ground Truth Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 2PM Ground Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 8PM Mask Masked Ground Truth Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 8PM Ground Truth, Time: 2AM Mask Masked Ground Truth Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Time: 2AMTrained With Biased Masking Trained With Random Masking Trained With Biased Masking Trained With Random MaskingConference’17, July 2017, Washington, DC, USA Bin Han and Bill Howe masking displayed similar patterns — they converged faster and achieved better results than the model trained with random masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Those two scenarios are representative of dense and busy areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We conjecture that biased masking avoids rewarding the model for trivially predicting zero in sparse regions and ignoring the dynamics in dense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We consider this result an initial foray: encoding domain knowledge and data patterns into the masking strategy appears to be a powerful, easy, and architecture-agnostic means of improving model performance, aligned with emerging principles of data-centric AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The other three scenarios — airport, lower east side, and Astoria, represent sparse regions with relatively light traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The convergence lines for them are less stable, and no benefit of biased masking is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We conjecture that variants of biased masking to weight both dense and sparse (yet non-zero) areas may further improve the model, as would specialized training on regions of interest (though that approach could be considered data leakage from training to test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 8: Convergence plots of the models trained with ei- ther biased or random masking, and tested with random masks, biased masks and other five additional scenarios maskings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4 Spatial distribution of errors (Q4) We hypothesized that the original 2D partial convolution archi- tecture (corresponding to T=1, Figure 7(a)) would be insufficient to capture transient events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For example, taxi rides occur in the suburbs, but they are infrequent and less predictable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' we expected the model to be less capable of accurately predicting these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Increasing the temporal dimension is also expected to be helpful with the dense region as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We can inspect the spatial distribution of the error for T=1 in Figure 9 to check this hypothesis: Each map is the average of 3000 timesteps, and is colored by the difference between the predicted value and the ground truth: a blue cell indicates an underestimate and a red cell represents an overestimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We see that the suburban regions are consistently underestimated, while the dense region is overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' At T=5, we observe similar pattern, but with both underestimation and overestimation errors significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The suburbs are still underestimated, but the dense regions are Figure 9: Aggregated spatial errors between predicted and ground truth values, from models trained with different temporal dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Red areas indicate overestimation, while blue areas represent underestimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' effectively improved when more temporal dimensions are incorpo- rated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' At T=15, the spatial error distribution is almost identical to T=5, with slightly higher underestimation and lower overestima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' However, T=15 requires prohibitive training time due to very large training samples, so this approach is undesirable with just slightly better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' This tradeoff in temporal scope reflects a subtle characteristic of the source data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' we hypothesize that T=5 corresponds to the window size needed to capture dynamic traffic periods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=', morning and evening commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='5 Scenario Based Evaluation (Q5) Spatiotemporal patterns of missing data in practice are unlikely to resemble random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Instead, outages will correlate with envi- ronmental features: sensors may fail in certain weather conditions, transient events may prevent data acquisition, or legal restrictions on data availability may follow political boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To demonstrate the applicability of our inpainting models in real-world situations, we evaluate the inpainting methods based on specific locations representing varying conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We tested five different scenarios to cover various spatial locations, temporal variances, and social events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The five scenarios include the masking of 5th Avenue, Penn Station, airport, lower east side, and Astoria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The masks are visual- ized in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Figure 10: Scenario masks overlaid on NYC map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Annotation: The ratio of masked-to-unmasked area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Trained With Biased Masks Trained With Random MasksSpatial Error Distribution - T=l, Mask=biased Spatial Error Distribution - T=3, Mask=biased Total Overestimation Value: 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='57 Total Overestimation Value: 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0 Total Underestimation Value: -249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='85 Total Underestimation Value: -194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='18 Total Absolute Error Value: 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='42 Total Absolute Error Value: 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='18 Spatial Error Distribution - T=5, Mask=biased Spatial Error Distribution - T=15, Mask=biased 0 1 2 Total Overestimation Value: 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6 Total Overestimation Value: 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='38 Total Underestimation Value: -179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='77 Total Underestimation Value: -179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='04 Total Absolute Error Value: 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='37 Total Absolute Error Value: 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='42Sth Avenue Airport Penn station Lower East Side Astoria MaskingRatio:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='49% MaskingRatio:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8% Masking Ratio:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1% MaskingRatio:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='42%MaskingRatio:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='17%Adapting to Skew: Imputing Spatiotemporal Urban Data with 3D Partial Convolutions and Biased Masking Conference’17, July 2017, Washington, DC, USA As mentioned in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3, 5th Avenue and Penn station are rep- resentative of busy and dense areas with heavy traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 5th Avenue can also show the impacts of certain social events on traffic patterns: The Pride Parade showed an anomalous intervention where traffic was zero on the parade route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Lower East Side is away from central Manhattan, with relatively lighter traffic than the first two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The scenario of airport and Astoria represent the sparse regions where traffic is light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We chose two periods for those scenarios to cover temporal variance – Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 1st to Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 15th, 2016, and June, 18th to June 29th, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' A snowstorm from Feb 5th to 8th in New York City is evident in the data (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' On June 26th, 2016, the Pride Parade in New York City started at 5th Avenue, and moved downtown to 8th Street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The event blocked all traffic along the route and affected the surrounding traffic as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Therefore, testing in the selected June period can help evaluate the model’s response to anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We test three inpainting models — our model trained with biased masking at T=5, the same model but trained with random masking at T=5, and the global mean approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We plotted the ground truth and predicted values at the average pixel level in the missing region, for each hour during the selected periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The visualizations are provided in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The average absolute errors between the ground truth and predicted values, over the missing region and during the evaluation periods, are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We have the following observations: Scenarios G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='- Biased G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' - Random G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' - Mean 02/01/2016 — 02/15/2016 5th Avenue 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0 Penn Station 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0 Lower East Side 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 Airport 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8 Astoria 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4 06/18/2016 — 06/30/2016 5th Avenue 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='53 Penn Station 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0 Lower East Side 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4 Airport 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0 Astoria 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='4 Table 3: Average absolute error between the predicted values and ground truth, over the missing regions, and during the selected evaluation periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For three scenarios — 5th Avenue, Penn Station, and Lower East Side, our models — whether trained with biased or random mask- ing — have much smaller gaps between the predicted values and the ground truth, compared with the temporal mean approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' This benefit holds for both evaluated periods, as shown in both Table 3 and Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For the airport and Astoria scenarios, the temporal mean is slightly better, with much smaller magnitude in comparison with other three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' rom Table 3, we see that for both evaluation periods, the model trained with biased masking has smaller average errors than the model trained with random masking, other than the scenario of airport during June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' During the snow days (02/05-02/08/2016), it is expected that the traffic in the dense regions would be significantly impacted, which can be supported by the trough seen from the ground truth line in the scenario of Penn Station (other scenarios are not Figure 11: Temporal line plots of evaluations for five sce- narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In each plot, we visualize the ground truth, predic- tion from model trained with biased masking and random masking, and predictions from temporal mean method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Two evaluation periods, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' and June are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The irregular events, extreme snow days and pride parade, are annotated with grey regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' heavily impacted by the snow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=') The model trained with biased masking is responsive to the irregular traffic caused by extreme weather, unlike the temporal mean baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' During the event pride parade, the traffic on 5th Avenue was all diverted to other routes, creating an anomaly in the traffic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Therefore, we saw a dip in the traffic counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Similar observation as the snow day, the temporal mean baseline does not recover the missing values .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' However, even though the inpainting results from our model are close to the ground truth values, they slightly overestimate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Overall, the reconstruction accuracy is compelling at specific locations, but not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For 5th Avenue scenario, the parade can be seen as an anomaly, which is rare in the training stage and hard to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' But this scenario represents another application Pride Parade Ground Truth Biased Prediction Random Prediction Temporal Mean Snow Days Ground Truth Biased Prediction Random Prediction Temporal Mean Snow Days Ground Truth Biased Prediction Random Prediction Temporal Mean Snow Days Ground Truth Biased Prediction Random Prediction Mear Snow Days Ground Truth Biased Prediction Random Prediction Temporal MeanConference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' USA Bin Han and Bill Howe usage of our model: rather than assuming that ground truth data is “correct".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We use the masking to intentionally repair known bad data, and reconstruct global patterns in a semantically reasonable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' This “airbrushing” of flaws in the data can be used to improve the quality of training sets for downstream applications, such as biofouled or errant sensors and faulty telemetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For example, from the top visualization in Figure 12, we visualize the 5th Avenue scenario: The first column shows the taxi counts along 5th Avenue during parade day, zoomed in on the Manhattan region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Several locations of missing data (white dots) can be seen on the avenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We masked out the 5th Avenue altogether and used our inpainting model to reconstruct the missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The use case is to enable policymakers and researchers to conduct counterfactual studies: what would have taxi demand been like were it not for the parade?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The results, as shown in the forth column, recover the missing regions in a realistic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Alternatively, the model might be used to synthesize parade-day traffic rather than removing its effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' By masking the surrounding area and retaining the parade disruption, the model can attempt to represent the influence of the disruption elsewhere in the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' As shown from the bottom visualization in Figure 12, the generated results are smaller in magnitude, but overall the pattern is matched faithfully, suggesting this use case is viable for synthesizing scenar- ios that may not be present in the data record (natural disasters, proposed construction, accidents, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Penn Station is a train sta- Figure 12: Top: “Airbrushing” the parade event (white pix- els) to remove its effect on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Bottom: Inferring traf- fic effects of the parade by reconstructing data everywhere except 5th Avenue to produce qualitatively realistic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' tion and represents a high-demand area for taxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Our model tends to underestimate the high demand at this location, though biased masking improves the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For Lower East Side, there are a few anomalous spikes, to which the proposed models are respon- sive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For airport and Astoria, our models are no better than the temporal mean approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We conjecture that for airport, the highly variable rides in and out of the airport confound the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For Astoria, the much lower demand is harder to predict;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' note the lower scale of the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 6 DISCUSSION Our study is motivated by the inconsistent availability of urban data caused by missing, corrupt, or inaccurate data, which hinders their use in downstream tasks, especially learning tasks, that require coverage and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We designed and implemented a model based on partial convolutions that can tolerate irregular missing regions — zip codes, geographical boundaries, congrssional districts, or other regions that may correlated with data absence or quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To capture the temporal dependency in urban data, we replaced 2D convolutional layers in the model with 3D convolutional layers and experimented with varying the number of timesteps per training sample, finding non-trivial tradeoffs and a local optimum around T=5 for taxis and T=3 for bikeshare, potentially interpretable as the autocorrelation period of traffic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=', about 5 hours of rush hour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' To address the spatial skew in human activity, we proposed a masking approach that can reflect the skew in the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' By encouraging the model to attend to dense, dynamic regions (via a percentile threshold), the model learns faster and is not rewarded for accurate predictions in trivially inactive areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Biased mask- ing showed improved performance across all values of 𝑇, multiple global evaluation strategies, and most local evaluation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' This approach suggests a broader family of related masking strate- gies to help users encode domain knowledge about the data and setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' For example, encoding correlations between high-traffic areas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=', subway stops and train stations during lunch time) as masks may help the model learn these correlations with less data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Qualitatively, we confirmed from the visual examples that im- age inpainting techniques can be used to reconstruct data in large, irregular regions in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Quantitatively, we confirmed that extending the model architecture to 3D benefits improves per- formance, as supported by the sharp decrease in ℓ1 when T changes from 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Second, we observe that increasing the temporal di- mension to a certain threshold improves performance in general, regardless of masking strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ignoring the temporal dimension in this setting is untenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, we evaluated performance in local settings, demon- strating that the model is not just learning an average value, but is responsive to subtle spatial variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The model captures irregular traffic patterns caused by transient events, such as extreme weather and the Pride Parade, and showed that biased masking can improve performance in local settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, the scenario evaluations also showcased the better results introduced by the biased masking than the random masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 7 LIMITATIONS & FUTURE WORK There are several limitations of our study that represent directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' First, our results on mobility data may extend to other urban activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=', 311 calls, crowd movement, business permits, public safety events, housing events, and more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' We do not consider the generalizability of these methods to multiple variables, or variables that do not follow the same spatial patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' there are opportunities to exploit correlations between variables to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Additionally, the taxi dataset is exceptionally large and complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' understanding how these techniques behave in low- data regimes is important for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Integration of masked multi-variate data may be an opportunity: given the shared built environment, models trained on one variable may transfer to predictions of other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Second, rasterizing event data to a form amenable to computer vision techniques involves a number of design choices we did not study: resolution, overlap, and Adapting to Skew: Imputing Spatiotemporal Urban Data with 3D Partial Convolutions and Biased Masking Conference’17, July 2017, Washington, DC, USA irregular boundaries may present opportunities or challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In particular, data associated with census blocks, tracts, or individual trajectories lose information when regridded as histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In these cases, graph neural networks may be more appropriate to represent the spatial adjacency relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Third, even with the best model configuration, we consistently overestimate in the city region and underestimate in the sparse suburban region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Some model architectures (attention mechanism, multi-view learning) or loss functions may improve performance, as may more specialized masking and training regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 8 CODE AVAILABLITY Our code is available at [anonymized for review].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' REFERENCES [1] Sebastian Abt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The Missing Data Problem in Cyber Security Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' GI FG SIDAR Graduierten-Workshop über Reaktive Sicherheit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [2] Peter M Allen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Cities and regions as self-organizing systems: models of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Routledge, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [3] Bumjoon Bae, Hyun Kim, Hyeonsup Lim, Yuandong Liu, Lee D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Han, and Phillip B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Freeze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Missing data imputation for traffic flow speed using spatio-temporal cokriging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies 88 (2018), 124– 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='trc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='015 [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Ballester, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Bertalmio, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Caselles, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Sapiro, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Verdera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Filling-in by joint interpolation of vector fields and gray levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' IEEE Transactions on Image Processing 10, 8 (2001), 1200–1211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1109/83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='935036 [5] Connelly Barnes, Eli Shechtman, Adam Finkelstein, and Dan B Goldman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' PatchMatch: A Randomized Correspondence Algorithm for Structural Image Editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM Transactions on Graphics (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' SIGGRAPH) 28, 3 (Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2009), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [6] Marcelo Bertalmío, Guillermo Sapiro, Vicent Caselles, and Coloma Ballester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Proceedings of the 27th annual conference on Computer graphics and interactive techniques , (2000), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Bertalmio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Vese, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Sapiro, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Osher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Simultaneous structure and texture image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' IEEE Transactions on Image Processing 12, 8 (2003), 882–889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1109/TIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='815261 [8] Bert Blocken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Computational Fluid Dynamics for urban physics: Importance, scales, possibilities, limitations and ten tips and tricks towards accurate and reliable simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Building and Environment 91 (2015), 219–245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https:// doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='buildenv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='015 Fifty Year Anniversary for Building and Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [9] Sebastian Daberdaku, Erica Tavazzi, and Barbara Di Camillo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Interpolation and K-Nearest Neighbours Combined Imputation for Longitudinal ICU Labora- tory Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In 2019 IEEE International Conference on Healthcare Informatics (ICHI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1109/ICHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='8904624 [10] Soheil Darabi, Eli Shechtman, Connelly Barnes, Dan B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Goldman, and Pradeep Sen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Image Melding: Combining Inconsistent Images Using Patch-Based Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 31, 4, Article 82 (jul 2012), 10 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1145/2185520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2185578 [11] Felipe de Souza, Omer Verbas, and Joshua Auld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Mesoscopic Traffic Flow Model for Agent-Based Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Procedia Computer Science 151 (2019), 858–863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='procs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='118 The 10th International Conference on Ambient Systems, Networks and Technologies (ANT 2019) / The 2nd Interna- tional Conference on Emerging Data and Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0 (EDI40 2019) / Affiliated Workshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [12] Iddo Drori, Daniel Cohen-Or, and Hezy Yeshurun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Fragment-Based Image Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 22, 3 (jul 2003), 303–312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 1145/882262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='882267 [13] Yongshun Gong, Zhibin Li, Jian Zhang, Wei Liu, Bei Chen, and Xiangjun Dong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' A spatial missing value imputation method for multi-view urban statistical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 1310–1316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [14] George Grekousis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Artificial neural networks and deep learning in urban geography: A systematic review and meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Computers, Environment and Urban Systems 74 (2019), 244–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [15] Seungwoong Ha and Hawoong Jeong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Unraveling hidden interactions in complex systems with deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Scientific reports 11, 1 (2021), 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [16] James Hays and Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Scene Completion Using Millions of Photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 26, 3 (jul 2007), 4–es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1145/ 1276377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1276382 [17] Bill Howe, Jackson Maxfield Brown, Bin Han, Bernease Herman, Nic Weber, An Yan, Sean Yang, and Yiwei Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Integrative urban AI to expand coverage, access, and equity of urban data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The European physical journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Special topics 231, 9 (2022), 1741—1752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1140/epjs/s11734-022-00475-z [18] Satoshi Iizuka, Edgar Simo-Serra, and Hiroshi Ishikawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Globally and Locally Consistent Image Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 36, 4, Article 107 (July 2017), 14 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1145/3072959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3073659 [19] Jireh Jam, Connah Kendrick, Kevin Walker, Vincent Drouard, Jison Gee-Sern Hsu, and Moi Hoon Yap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' A comprehensive review of past and present image inpainting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Computer vision and image understanding 203 (2021), 103147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [20] Christopher Kadow, David Matthew Hall, and Uwe Ulbrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Artificial intelligence reconstructs missing climate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Nature Geoscience 13, 6 (June 2020), 408–413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1038/s41561-020-0582-5 [21] Song-Hee Kang, Youngjin Choi, and Jae Young Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Restoration of Missing Patterns on Satellite Infrared Sea Surface Temperature Images Due to Cloud Coverage Using Deep Generative Inpainting Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Journal of Marine Science and Engineering 9, 3 (2021), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3390/jmse9030310 [22] Christopher P Kempes and Geoffrey B West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The Simplicity and Complexity of Cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The Bridge 50 (2020), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [23] Van-Duc Le, Tien-Cuong Bui, and Sang-Kyun Cha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Spatiotemporal Deep Learning Model for Citywide Air Pollution Interpolation and Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In 2020 IEEE International Conference on Big Data and Smart Computing (BigComp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 55–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1109/BigComp48618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='00-99 [24] Melissa Lee, Esteve Almirall, and Jonathan Wareham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Open data and civic apps: first-generation failures, second-generation improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM 59, 1 (2015), 82–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [25] Levin, Zomet, and Weiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Learning how to inpaint from global image statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings Ninth IEEE International Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 305–312 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1109/ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1238360 [26] YuanYuan Li and Lynne E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Parker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Nearest neighbor imputation using spatial–temporal correlations in wireless sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Information Fusion 15 (2014), 64–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='inffus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='007 Special Issue: Resource Constrained Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [27] Binbing Liao, Jingqing Zhang, Chao Wu, Douglas McIlwraith, Tong Chen, Sheng- wen Yang, Yike Guo, and Fei Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Deep sequence learning with auxiliary information for traffic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the 24th ACM SIGKDD Inter- national Conference on Knowledge Discovery & Data Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM, , , 537–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [28] Qiushuang Lin and Chunxiang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Kriging based sequence interpolation and probability distribution correction for gaussian wind field data reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Journal of Wind Engineering and Industrial Aerodynamics 205 (2020), 104340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='jweia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='104340 [29] Guilin Liu, Fitsum A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Reda, Kevin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Shih, Ting-Chun Wang, Andrew Tao, and Bryan Catanzaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Image Inpainting for Irregular Holes Using Partial Convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='07723 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='CV] [30] Hongyu Liu, Bin Jiang, Yi Xiao, and Chao Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Coherent Semantic Atten- tion for Image Inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In 2019 IEEE/CVF International Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 4169–4178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1109/ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='00427 [31] Kai Liu, Junke Li, and Syed Sabahat Hussain Bukhari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Overview of Image Inpainting and Forensic Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Security and Communication Networks 2022 (2022), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [32] Eric Hsueh-Chan Lu and Chia-Yu Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' A Spatial-Temporal Approach for Air Quality Forecast in Urban Areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Applied Sciences 11, 11 (2021), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3390/app11114971 [33] Xiaolei Ma, Zhuang Dai, Zhengbing He, Jihui Ma, Yong Wang, and Yunpeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Learning traffic as images: a deep convolutional neural network for large-scale transportation network speed prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Sensors 17, 4 (2017), 818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [34] Xiaolei Ma, Zhuang Dai, Zhengbing He, Jihui Na, Yong Wang, and Yunpeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Learning Traffic as Images: A Deep Convolutional Neural Network for Large-Scale Transportation Network Speed Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='04245 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='LG] [35] Luis Moreira-Matias, Joao Gama, Michel Ferreira, Joao Mendes-Moreira, and Luis Damas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Predicting taxi–passenger demand using streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' IEEE Transactions on Intelligent Transportation Systems 14, 3 (2013), 1393–1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [36] Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Context Encoders: Feature Learning by Inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='07379 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='CV] [37] Zhen Qin, Qingliang Zeng, Yixin Zong, and Fan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Image inpainting based on deep learning: A review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Displays 69 (2021), 102028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [38] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Ridgway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Dunn, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Wilkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Ocean Interpolation by Four- Dimensional Weighted Least Squares—Application to the Waters around Aus- tralasia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Journal of Atmospheric and Oceanic Technology 19, 9 (2002), 1357 – 1375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1175/1520-0426(2002)019<1357:OIBFDW>2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='CO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2 [39] Marc Riedel and Wendy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Regoeczi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Missing Data in Homicide Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Homicide Studies 8, 3 (2004), 163–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1177/1088767904265447 arXiv:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1177/1088767904265447 [40] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' U-Net: Convolutional Networks for Biomedical Image Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' CoRR abs/1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='04597 (2015), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='04597 http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/abs/1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='04597 Conference’17, July 2017, Washington, DC, USA Bin Han and Bill Howe [41] Parisa Saeipourdizaj, Parvin Sarbakhsh, and Akbar and Gholampour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Appli- cation of imputation methods for missing values of PM10 and O3 data: Interpola- tion, moving average and K-nearest neighbor methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Environmental Health Engineering and Management Journal 8, 3 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='34172/EHEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='25 arXiv:http://ehemj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='com/article-1-815-en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='pdf [42] Chitwan Saharia, William Chan, Huiwen Chang, Chris Lee, Jonathan Ho, Tim Salimans, David Fleet, and Mohammad Norouzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Palette: Image-to-image diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In ACM SIGGRAPH 2022 Conference Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [43] Bilong Shen, Xiaodan Liang, Yufeng Ouyang, Miaofeng Liu, Weimin Zheng, and Kathleen M Carley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' StepDeep: A Novel Spatial-temporal Mobility Event Prediction Framework based on Deep Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM, , , 724–733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [44] Bilong Shen, Xiaodan Liang, Yufeng Ouyang, Miaofeng Liu, Weimin Zheng, and Kathleen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Carley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' StepDeep: A Novel Spatial-Temporal Mobility Event Prediction Framework Based on Deep Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery &' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Data Min- ing (London, United Kingdom) (KDD ’18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 724–733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1145/3219819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3219931 [45] Raymond Sluiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Interpolation methods for climate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' KNMI , (01 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [46] Andy Stock, Ajit Subramaniam, Gert Dijken, Lisa Wedding, Kevin Arrigo, Matthew Mills, Mary Cameron, and Fiorenza Micheli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Comparison of Cloud-Filling Algorithms for Marine Satellite Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Remote Sensing 12 (10 2020), 3313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='3390/rs12203313 [47] Dazhi Sun, Jinpeng Lv, and S Waller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In-depth analysis of traffic congestion using computational fluid dynamics (CFD) modeling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Journal of Modern Transportation 19 (03 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1007/BF03325741 [48] Jarin Tasnim and Debajyoti Mondal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Data Reduction and Deep-Learning Based Recovery for Geospatial Visualization and Satellite Imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In 2020 IEEE International Conference on Big Data (Big Data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 5276–5285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 1109/BigData50022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='9378008 [49] Stefaan Verhulst, Andrew Young, Andrew Zahuranec, Susan Ariel Aaronson, Ania Calderon, and Matt Gee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' The emergence of a third wave of open data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , (2020), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [50] Patrick Vogel, Torsten Greiser, and Dirk Christian Mattfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Understanding bike-sharing systems using data mining: Exploring activity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Procedia- Social and Behavioral Sciences 20 (2011), 514–523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [51] Geoffrey B West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Scale: the universal laws of growth, innovation, sustainabil- ity, and the pace of life in organisms, cities, economies, and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Penguin, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [52] Rui Xu, Minghao Guo, Jiaqi Wang, Xiaoxiao Li, Bolei Zhou, and Chen Change Loy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Texture memory-augmented deep patch-based image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' IEEE Transactions on Image Processing 30 (2021), 9112–9124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [53] An Yan and Bill Howe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' FairST: Equitable Spatial and Temporal Demand Prediction for New Mobility Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='03827 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='CY] [54] Huaxiu Yao, Xianfeng Tang, Hua Wei, Guanjie Zheng, Yanwei Yu, and Zhenhui Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Modeling spatial-temporal dynamics for traffic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='01254 , (2018), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [55] Ji Won Yoon, Fabio Pinelli, and Francesco Calabrese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Cityride: a predictive bike sharing journey advisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In 2012 IEEE 13th International Conference on Mobile Data Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' IEEE, , , 306–311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [56] Haiyang Yu, Zhihai Wu, Shuqin Wang, Yunpeng Wang, and Xiaolei Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Spatiotemporal Recurrent Convolutional Networks for Traffic Prediction in Trans- portation Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='02699 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='LG] [57] Jiahui Yu, Zhe Lin, Jimei Yang, Xiaohui Shen, Xin Lu, and Thomas Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Free-Form Image Inpainting with Gated Convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='03589 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='CV] [58] Jiahui Yu, Zhe Lin, Jimei Yang, Xiaohui Shen, Xin Lu, and Thomas S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Generative Image Inpainting with Contextual Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='07892 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='CV] [59] Tao Yu, Zongyu Guo, Xin Jin, Shilin Wu, Zhibo Chen, Weiping Li, Zhizheng Zhang, and Sen Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Region normalization for image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 12733–12740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [60] Zhuoning Yuan, Xun Zhou, and Tianbao Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Hetero-ConvLSTM: A deep learning approach to traffic accident prediction on heterogeneous spatio- temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM, , , 984–992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [61] Yanhong Zeng, Jianlong Fu, Hongyang Chao, and Baining Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Aggre- gated contextual transformations for high-resolution image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' IEEE Transactions on Visualization and Computer Graphics , (2022), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [62] Han Zhang, Peng Chen, Jianfeng Zheng, Jingqing Zhu, Guizhen Yu, Yunpeng Wang, and Henry X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Missing data detection and imputation for urban ANPR system using an iterative tensor decomposition approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Transportation Research Part C: Emerging Technologies 107 (2019), 337–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='trc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='013 [63] Junbo Zhang, Yu Zheng, Dekang Qi, Ruiyuan Li, and Xiuwen Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' DNN- based prediction model for spatio-temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the 24th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' ACM, , , 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [64] Junbo Zhang, Yu Zheng, Dekang Qi, Ruiyuan Li, Xiuwen Yi, and Tianrui Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Predicting citywide crowd flows using deep spatio-temporal residual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Artificial Intelligence 259 (2018), 147–166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' [65] Qiang Zhang, Qiangqiang Yuan, Chao Zeng, Xinghua Li, and Yancong Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Missing Data Reconstruction in Remote Sensing Image With a Uni- fied Spatial–Temporal–Spectral Deep Convolutional Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' IEEE Transactions on Geoscience and Remote Sensing 56, 8 (2018), 4274–4288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='1109/TGRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content='2810208 [66] Tong Zhou, Changxing Ding, Shaowen Lin, Xinchao Wang, and Dacheng Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' Learning oracle attention for high-fidelity face completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} +page_content=' , , 7680– 7689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE2T4oBgHgl3EQf-Ak-/content/2301.04233v1.pdf'} diff --git a/FdFJT4oBgHgl3EQfDCyN/content/tmp_files/2301.11432v1.pdf.txt b/FdFJT4oBgHgl3EQfDCyN/content/tmp_files/2301.11432v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dfa8f54df2f90677dc93d4df7d585771a4f9b807 --- /dev/null +++ b/FdFJT4oBgHgl3EQfDCyN/content/tmp_files/2301.11432v1.pdf.txt @@ -0,0 +1,413 @@ +Emotional Interaction Qualities: Vocabulary, Modalities, Actions, And Mapping +Albrecht Kurze +Chemnitz University of Technology, Albrecht.Kurze@informatik.tu-chemnitz.de +Have you ever typed particularly powerful on your keyboard, maybe even harsh, to write and send a message with some emphasis of +your emotional state or message? Did it work? Probably not. It didn't affect how you typed or interacted with your mouse. But what +if you had other, connected devices, with other modalities for inputs and outputs? Which would you have chosen, and how would +you characterize your interactions with them? We researched with our multisensory and multimodal tool, the Loaded Dice, in co- +design workshops the design space of IoT usage scenarios: what interaction qualities users want, characterized using an interaction +vocabulary, and how they might map them to a selection of sensors and actuators. We discuss based on our experience some +thoughts of such a mapping. +CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI) +Additional Keywords and Phrases: interaction, vocabulary, emotions, qualities, design, ideation, tools, methods, IoT +ACM Reference Format: +Albrecht Kurze. 2022. Emotional Interaction Qualities: Vocabulary, Modalities, Actions, And Mapping. In Workshop +The Future of Emotion in Human-Computer Interaction (CHI’22). April 13-14, 2022. 4 pages. +1 +INTRODUCTION +Some years ago we designed and developed the Loaded Dice [9,10], a multisensory and multimodal hybrid toolkit to +ideate IoT devices and scenarios, e.g. for the ‘smart’ home, and with different groups of co-designers [3,8,9]. The +Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.g. KnowCards [1], and functional but +tinkering based tools, e.g. littleBits [2], for multisensory and multimodal exploration, ideation and prototyping. +We will introduce a) our adapted and extended interaction vocabulary, b) how we use it in a method to explore and +the describe the WHY and HOW of interactions with and through connected devices; and c) we introduce the Loaded +Dice, what sensing and actuating functions they have, and how we use the to let participants map interaction qualities +to modalities to align the WHY and the HOW of interactions. +This brings us to our core question: Is it possible to derive a (somehow universal) mapping between certain interaction +qualities, i.e. emotional ones, and specific modalities and actions? + +1st +2nd +Goals +Actors +Spaces +3rd +precise +frendly +.. +4publie +private +covered +apparent +deterministic +chaotic +close +distant +fluent +stepwise +prosaic +poetic +papeas +binary +precise +approximate +metaphorical +factual +spotial +spatial +proximity +separation +direct +mediated +familiar +strange +targeted +incidental +constant +inconstant +srabbing +powerful +gentle +instant +delayed +harsh +tender +diverging +uniform +fast +slow +angn +friendlySENSORDIE +Temperature Sensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +FanINPUTFigure 1: left: the types of cards of our co-design method, 1st setting a goal (why), 2nd context of interaction (actors and space), +3rd defining desired interaction qualities; right: cards with terms of the extended interaction vocabulary +2 +ADAPTED AND EXTENDED INTERACTION VOCABULARY +Diefenbach [4] introduced in 2013 a first interaction vocabulary to describe interaction qualities in a user +perspective. The original vocabulary consisted of 11 pairs of adjectives and antonyms, e.g. fast and slow. An +example of how the vocabulary was intended to describe interaction qualities: When we switch the light in a +room, this happens in a binary way at the switch (on/off) with an instant effect in the same way at the lamp +distant to the switch. With a dimmer the input and output are graded in a fluent or stepwise manner (vocabulary +terms in italics). The original vocabulary was intended for use as a semantic differential on a graded scale, e.g. in a +questionnaire. +Our intention was to not only characterize interactions with a single object but also in complex connected +interaction scenarios. In scenarios as the IoT allows them for smart connected things, across multiple devices and +shared between multiple involved actors (typically human users but not limited to them). +While Diefenbach’s intention for the original interaction vocabulary was first to describe “the HOW of +interaction” [4] they also had drawn a first conclusion between HOW and WHY of interaction. We put this first. It +became clear to us that it is often not meaningful to isolate the HOW from the WHY of interaction. Therefore we +embedded the interaction vocabulary methodically in a goal-actors-properties driven scenario creation to match +the IoT design space [3]. We adapted and extended the original vocabulary. We did this in the same way as the +original vocabulary was constructed – as pairs of adjectives and antonyms. Additionally, we introduced to the +vocabulary an extension with some more emotional terms to grasp interaction qualities often beyond a non- +judgmental dimension [4]. We discuss their role following on with mappings to specific modalities. We have +already iterated the actual terms based on what we have learned in co-design workshops using the vocabulary. +We see the vocabulary still as work in progress. +We created based on the vocabulary a set of cards for the use in co-design workshops. On the front face an +adjective and on the back the antonym (fig. 1b). We introduced a subtle differentiation between the two +categories. The non-judgmental terms (including the original terms) are in black letters on colored background +while the slightly more emotional terms are in white letters. In contrast to Diefenbach’s graded semantic +differential we decided with the cards only for the extremes - an ‘either or’. However, this stimulates in the co- +design workshops a verbalization how something is meant – often not in the extremes but then user defined +graded. +2 + +1st +2nd +Goals +Actors +Spaces +3rd +precise +frendly +.. +4publie +private +covered +apparent +deterministic +chaotic +close +distant +fluent +stepwise +prosaic +poetic +papeas +binary +precise +approximate +metaphorical +factual +spotial +spatial +proximity +separation +direct +mediated +familiar +strange +targeted +incidental +constant +inconstant +srabbing +powerful +gentle +instant +delayed +harsh +tender +diverging +uniform +fast +slow +angn +friendlySENSORDIE +Temperature Sensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +FanINPUTFigure 2: left: faces and functions of the Loaded Dice – sensors and actuators [9]; right: an example of using the cards to +characterize and map input and output qualities of an ideated connected product with help of the Loaded Dice [6] +3 +INTERACTION MODALITIES +The Loaded Dice are a set of two cubical devices wirelessly connected (fig. 2a). Each cube has six sides, offering +in one cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and +multimodal environmental and user interactions. The sensor cube normalizes a raw sensor value meaningfully, +transmits it, and then the other cube actuates it mapped on an output. The cubical shape communicates the +intuitive reading that the top side is active, like a die, offering an easy and spontaneous way to re-combine sensors +and actuators. Every sensor-in and actuator-out combination is possible resulting in 36 combinations in total. [6] +New multisensory interaction modalities, not yet implemented, e.g. smell, have the potential to broaden +interaction qualities even further and especially in an emotional way [7]. +4 +MAPPING INTERACTION QUALITIES +Last step in our co-design method is a mapping of desired interaction qualities by participants to interaction +modalities represented by the Loaded Dice (fig. 2b). The extended interaction vocabulary allows characterizing the +intended interactions very well while the Loaded Dice allow participants to try them out up to a certain degree. +This way our workshops often brought up a number of unconventional ideas of multisensory interactions with +devices, often far beyond ordinary inputs and especially outputs. In our experience sensory sensations and +modalities do not need to be perfect, at least for ideation of interactions and scenarios as well as for mapping +interaction qualities. It is about bringing the idea and the core concept behind it to the co-design activities. A +demonstration of a technical possibility for sensing and actuating as a stimulus is often enough to trigger further +thinking and verbalization of how something might be used. +We found some repeating themes when it comes to a mapping between certain interaction characteristics and +suitable sensing as well as actuating possibilities. For example, the thermo-element was not only associated with +slow and warmth literally but also with ‘love’ and tender in a poetic way, while loud sound and bright light were +selected for powerful, attention-grabbing and sometimes even harsh interactions etc. Participants often chose non- +visible and non-audible modalities for private interactions, covered and not easily perceivable by others, only +noticeable to a mentioned one, e.g. using heat or vibration in ideated wearable devices. In another case +participants mapped the vibration motors and the associated sound caused when having the Loaded Dice placed +on a wooden table to attention-grabbing and harsh, associated to feelings of being alarmed and named it +“electronic rattlesnake”. +We also found similar patterns for inputs. The distance sensor can detect a hand in proximity in different ways. +In a graded kind, if done slowly, allowing for gentle gestures, e.g. swiping with the hand through the air above the +sensor, without touching something, without any force. As these gestures can be very similar to petting +3 + +1st +2nd +Goals +Actors +Spaces +3rd +precise +frendly +.. +4publie +private +covered +apparent +deterministic +chaotic +close +distant +fluent +stepwise +prosaic +poetic +papeas +binary +precise +approximate +metaphorical +factual +spotial +spatial +proximity +separation +direct +mediated +familiar +strange +targeted +incidental +constant +inconstant +srabbing +powerful +gentle +instant +delayed +harsh +tender +diverging +uniform +fast +slow +angn +friendlySENSORDIE +Temperature Sensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +FanINPUTsomething they were associated with this action in a poetic and very tender way. On the other hand, a fast and +sudden movement is also detectable, like a punch, being very powerful, targeted and harsh. While the distance +sensor allows for such a differentiation based on the speed of hand movement the PIR movement detection sensor +allows not – what also might be wanted, e.g. for an only binary type of input. +Both ways of using the distance sensor for proximity-based hand gestures are possible and meaningful. +However, the HOW of the interaction is then depending on the emotional state of the user and the WHY of +interaction. Therefore, it makes a difference what a user tries to express and in an end-to-end view of interactions +‘through’ devices [6], from one device to another device, as communication to another actor. Has the user the +intention to send a message with a positive emotion, a non-verbal equivalent of “I love you tender”, or to send a +message associated with a negative emotion, e.g. with an equivalent in “Turn the damn music down”? In terms of +Hassenzahl’s model of interactions for experience design [5], as a hierarchy of WHY, WHAT, and HOW of +interactions, it is clear that this WHY, the motive, defines the WHAT and the HOW. Therefore, it will not be a +simple one-dimensional mapping between an emotional interaction quality and specific sensor / actuator +modalities. The motive of use (as higher-level use goal) and the expressed or implied associate actions (e.g. petting +or punching) must also be considered. +5 +CONCLUSION +While we see especially big potential in the use of an interaction vocabulary and different modalities for +intending or expressing emotional interaction qualities, it still needs further exploration to identify certain +patterns for a mapping. +ACKNOWLEDGMENTS +This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116. +References +[1] +Tina Aspiala and Alexandra Deschamps-Sonsino. 2016. Know Cards: Learn. Play. Collect. Know Cards. Retrieved December 6, 2016 from +http://know-cards.myshopify.com/ +[2] +Ayah Bdeir. 2009. Electronics As Material: LittleBits. In Proceedings of the 3rd International Conference on Tangible and Embedded Interaction +(TEI ’09), 397–400. https://doi.org/10.1145/1517664.1517743 +[3] +Arne Berger, William Odom, Michael Storz, Andreas Bischof, Albrecht Kurze, and Eva Hornecker. 2019. The Inflatable Cat: Idiosyncratic +Ideation Of Smart Objects For The Home. In CHI Conference on Human Factors in Computing Systems Proceedings. +https://doi.org/10.1145/3290605.3300631 +[4] +Sarah Diefenbach, Eva Lenz, and Marc Hassenzahl. 2013. An Interaction Vocabulary. Describing the How of Interaction. In CHI ’13 Extended +Abstracts on Human Factors in Computing Systems (CHI EA ’13), 607–612. https://doi.org/10.1145/2468356.2468463 +[5] +Marc Hassenzahl. 2010. Experience Design: Technology for All the Right Reasons. Synthesis Lectures on Human-Centered Informatics 3, 1: 1– +95. https://doi.org/10.2200/S00261ED1V01Y201003HCI008 +[6] +Albrecht Kurze. 2021. Interaction Qualities For Interactions With, Between, And Through IoT Devices. +https://doi.org/10.1145/3494322.3494348 +[7] +Albrecht Kurze. 2021. Scented Dice: New interaction qualities for ideating connected devices. In Workshop Smell, Taste, and Temperature +Interfaces at Conference on Human Factors in Computing Systems (CHI ’21). Retrieved from https://arxiv.org/abs/2201.10484 +[8] +Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger. 2016. Explorative Co-Design-Werkzeuge +zum Entwerfen von Smart Connected Things am Beispiel eines Workshops mit Blinden und Sehbehinderten. In Technische +Unterstützungssysteme, die die Menschen wirklich wollen, 395–400. Retrieved January 19, 2017 from http://tinyurl.com/janya26 +[9] +Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger. 2016. Loaded Dice: +Exploring the Design Space of Connected Devices with Blind and Visually Impaired People. In Proceedings of the 9th Nordic Conference on +Human-Computer Interaction (NordiCHI ’16), 31:1-31:10. https://doi.org/10.1145/2971485.2971524 +[10] +Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Michael Storz, Albrecht Kurze, and Arne Berger. 2017. Loaded Dice: How to cheat your +way to creativity. In Proceedings of the 3rd Biennial Research Through Design Conference. https://doi.org/10.6084/m9.figshare.4746976.v1 +4 + +1st +2nd +Goals +Actors +Spaces +3rd +precise +frendly +.. +4publie +private +covered +apparent +deterministic +chaotic +close +distant +fluent +stepwise +prosaic +poetic +papeas +binary +precise +approximate +metaphorical +factual +spotial +spatial +proximity +separation +direct +mediated +familiar +strange +targeted +incidental +constant +inconstant +srabbing +powerful +gentle +instant +delayed +harsh +tender +diverging +uniform +fast +slow +angn +friendlySENSORDIE +Temperature Sensor +Light Sensor +Microphone +MovementSensor +Potentiometer +Distance Sensor +ACTUATORDIE +Vibration +Heating Surface +LED-Bargraph +Loudspeaker +Power-LEDs +FanINPUT \ No newline at end of file diff --git a/FdFJT4oBgHgl3EQfDCyN/content/tmp_files/load_file.txt b/FdFJT4oBgHgl3EQfDCyN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ebc713cc9cae93414128971c39b2c7ab4728768 --- /dev/null +++ b/FdFJT4oBgHgl3EQfDCyN/content/tmp_files/load_file.txt @@ -0,0 +1,411 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf,len=410 +page_content='Emotional Interaction Qualities: Vocabulary, Modalities, Actions, And Mapping Albrecht Kurze Chemnitz University of Technology, Albrecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Kurze@informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='de Have you ever typed particularly powerful on your keyboard, maybe even harsh, to write and send a message with some emphasis of your emotional state or message?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Did it work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Probably not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=" It didn't affect how you typed or interacted with your mouse." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' But what if you had other, connected devices, with other modalities for inputs and outputs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Which would you have chosen, and how would you characterize your interactions with them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We researched with our multisensory and multimodal tool, the Loaded Dice, in co- design workshops the design space of IoT usage scenarios: what interaction qualities users want, characterized using an interaction vocabulary, and how they might map them to a selection of sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We discuss based on our experience some thoughts of such a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI) Additional Keywords and Phrases: interaction, vocabulary, emotions, qualities, design, ideation, tools, methods, IoT ACM Reference Format: Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Emotional Interaction Qualities: Vocabulary, Modalities, Actions, And Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In Workshop The Future of Emotion in Human-Computer Interaction (CHI’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' April 13-14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 1 INTRODUCTION Some years ago we designed and developed the Loaded Dice [9,10], a multisensory and multimodal hybrid toolkit to ideate IoT devices and scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' for the ‘smart’ home, and with different groups of co-designers [3,8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' KnowCards [1], and functional but tinkering based tools, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' littleBits [2], for multisensory and multimodal exploration, ideation and prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We will introduce a) our adapted and extended interaction vocabulary, b) how we use it in a method to explore and the describe the WHY and HOW of interactions with and through connected devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' and c) we introduce the Loaded Dice, what sensing and actuating functions they have, and how we use the to let participants map interaction qualities to modalities to align the WHY and the HOW of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' This brings us to our core question: Is it possible to derive a (somehow universal) mapping between certain interaction qualities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' emotional ones, and specific modalities and actions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 1st 2nd Goals Actors Spaces 3rd precise frendly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='4publie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='covered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='apparent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='deterministic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='chaotic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='close ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='distant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fluent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='stepwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='prosaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='poetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='papeas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='binary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='precise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='approximate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='metaphorical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='factual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spotial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spatial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='proximity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='separation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='mediated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='familiar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='strange ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='targeted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='incidental ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='inconstant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='srabbing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='powerful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='gentle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='instant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='delayed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='harsh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='tender ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='diverging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='uniform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='slow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='angn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='friendlySENSORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Temperature Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Light Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Microphone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='MovementSensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Potentiometer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Distance Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='ACTUATORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Vibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Heating Surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='LED-Bargraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Loudspeaker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Power-LEDs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='FanINPUTFigure 1: left: the types of cards of our co-design method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 1st setting a goal (why),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2nd context of interaction (actors and space),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 3rd defining desired interaction qualities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' right: cards with terms of the extended interaction vocabulary 2 ADAPTED AND EXTENDED INTERACTION VOCABULARY Diefenbach [4] introduced in 2013 a first interaction vocabulary to describe interaction qualities in a user perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The original vocabulary consisted of 11 pairs of adjectives and antonyms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' fast and slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' An example of how the vocabulary was intended to describe interaction qualities: When we switch the light in a room, this happens in a binary way at the switch (on/off) with an instant effect in the same way at the lamp distant to the switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' With a dimmer the input and output are graded in a fluent or stepwise manner (vocabulary terms in italics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The original vocabulary was intended for use as a semantic differential on a graded scale, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' in a questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Our intention was to not only characterize interactions with a single object but also in complex connected interaction scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In scenarios as the IoT allows them for smart connected things, across multiple devices and shared between multiple involved actors (typically human users but not limited to them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' While Diefenbach’s intention for the original interaction vocabulary was first to describe “the HOW of interaction” [4] they also had drawn a first conclusion between HOW and WHY of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We put this first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' It became clear to us that it is often not meaningful to isolate the HOW from the WHY of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Therefore we embedded the interaction vocabulary methodically in a goal-actors-properties driven scenario creation to match the IoT design space [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We adapted and extended the original vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We did this in the same way as the original vocabulary was constructed – as pairs of adjectives and antonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Additionally, we introduced to the vocabulary an extension with some more emotional terms to grasp interaction qualities often beyond a non- judgmental dimension [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We discuss their role following on with mappings to specific modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We have already iterated the actual terms based on what we have learned in co-design workshops using the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We see the vocabulary still as work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We created based on the vocabulary a set of cards for the use in co-design workshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' On the front face an adjective and on the back the antonym (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We introduced a subtle differentiation between the two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The non-judgmental terms (including the original terms) are in black letters on colored background while the slightly more emotional terms are in white letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In contrast to Diefenbach’s graded semantic differential we decided with the cards only for the extremes - an ‘either or’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' However, this stimulates in the co- design workshops a verbalization how something is meant – often not in the extremes but then user defined graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2 1st 2nd Goals Actors Spaces 3rd precise frendly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='4publie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='covered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='apparent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='deterministic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='chaotic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='close ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='distant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fluent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='stepwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='prosaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='poetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='papeas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='binary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='precise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='approximate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='metaphorical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='factual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spotial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spatial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='proximity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='separation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='mediated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='familiar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='strange ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='targeted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='incidental ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='inconstant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='srabbing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='powerful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='gentle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='instant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='delayed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='harsh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='tender ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='diverging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='uniform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='slow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='angn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='friendlySENSORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Temperature Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Light Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Microphone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='MovementSensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Potentiometer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Distance Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='ACTUATORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Vibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Heating Surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='LED-Bargraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Loudspeaker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Power-LEDs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='FanINPUTFigure 2: left: faces and functions of the Loaded Dice – sensors and actuators [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' right: an example of using the cards to characterize and map input and output qualities of an ideated connected product with help of the Loaded Dice [6] 3 INTERACTION MODALITIES The Loaded Dice are a set of two cubical devices wirelessly connected (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Each cube has six sides, offering in one cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and multimodal environmental and user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The sensor cube normalizes a raw sensor value meaningfully, transmits it, and then the other cube actuates it mapped on an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The cubical shape communicates the intuitive reading that the top side is active, like a die, offering an easy and spontaneous way to re-combine sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Every sensor-in and actuator-out combination is possible resulting in 36 combinations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' [6] New multisensory interaction modalities, not yet implemented, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' smell, have the potential to broaden interaction qualities even further and especially in an emotional way [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 4 MAPPING INTERACTION QUALITIES Last step in our co-design method is a mapping of desired interaction qualities by participants to interaction modalities represented by the Loaded Dice (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The extended interaction vocabulary allows characterizing the intended interactions very well while the Loaded Dice allow participants to try them out up to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' This way our workshops often brought up a number of unconventional ideas of multisensory interactions with devices, often far beyond ordinary inputs and especially outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In our experience sensory sensations and modalities do not need to be perfect, at least for ideation of interactions and scenarios as well as for mapping interaction qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' It is about bringing the idea and the core concept behind it to the co-design activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' A demonstration of a technical possibility for sensing and actuating as a stimulus is often enough to trigger further thinking and verbalization of how something might be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We found some repeating themes when it comes to a mapping between certain interaction characteristics and suitable sensing as well as actuating possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' For example, the thermo-element was not only associated with slow and warmth literally but also with ‘love’ and tender in a poetic way, while loud sound and bright light were selected for powerful, attention-grabbing and sometimes even harsh interactions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Participants often chose non- visible and non-audible modalities for private interactions, covered and not easily perceivable by others, only noticeable to a mentioned one, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' using heat or vibration in ideated wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In another case participants mapped the vibration motors and the associated sound caused when having the Loaded Dice placed on a wooden table to attention-grabbing and harsh, associated to feelings of being alarmed and named it “electronic rattlesnake”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' We also found similar patterns for inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The distance sensor can detect a hand in proximity in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In a graded kind, if done slowly, allowing for gentle gestures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' swiping with the hand through the air above the sensor, without touching something, without any force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' As these gestures can be very similar to petting 3 1st 2nd Goals Actors Spaces 3rd precise frendly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='4publie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='covered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='apparent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='deterministic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='chaotic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='close ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='distant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fluent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='stepwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='prosaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='poetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='papeas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='binary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='precise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='approximate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='metaphorical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='factual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spotial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spatial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='proximity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='separation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='mediated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='familiar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='strange ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='targeted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='incidental ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='inconstant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='srabbing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='powerful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='gentle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='instant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='delayed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='harsh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='tender ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='diverging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='uniform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='slow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='angn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='friendlySENSORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Temperature Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Light Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Microphone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='MovementSensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Potentiometer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Distance Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='ACTUATORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Vibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Heating Surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='LED-Bargraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Loudspeaker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Power-LEDs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='FanINPUTsomething they were associated with this action in a poetic and very tender way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' On the other hand, a fast and sudden movement is also detectable, like a punch, being very powerful, targeted and harsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' While the distance sensor allows for such a differentiation based on the speed of hand movement the PIR movement detection sensor allows not – what also might be wanted, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' for an only binary type of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Both ways of using the distance sensor for proximity-based hand gestures are possible and meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' However, the HOW of the interaction is then depending on the emotional state of the user and the WHY of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Therefore, it makes a difference what a user tries to express and in an end-to-end view of interactions ‘through’ devices [6], from one device to another device, as communication to another actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Has the user the intention to send a message with a positive emotion, a non-verbal equivalent of “I love you tender”, or to send a message associated with a negative emotion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' with an equivalent in “Turn the damn music down”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In terms of Hassenzahl’s model of interactions for experience design [5], as a hierarchy of WHY, WHAT, and HOW of interactions, it is clear that this WHY, the motive, defines the WHAT and the HOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Therefore, it will not be a simple one-dimensional mapping between an emotional interaction quality and specific sensor / actuator modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The motive of use (as higher-level use goal) and the expressed or implied associate actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' petting or punching) must also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 5 CONCLUSION While we see especially big potential in the use of an interaction vocabulary and different modalities for intending or expressing emotional interaction qualities, it still needs further exploration to identify certain patterns for a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' ACKNOWLEDGMENTS This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' References [1] Tina Aspiala and Alexandra Deschamps-Sonsino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Know Cards: Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Know Cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Retrieved December 6, 2016 from http://know-cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='myshopify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='com/ [2] Ayah Bdeir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Electronics As Material: LittleBits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In Proceedings of the 3rd International Conference on Tangible and Embedded Interaction (TEI ’09), 397–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='1145/1517664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='1517743 [3] Arne Berger, William Odom, Michael Storz, Andreas Bischof, Albrecht Kurze, and Eva Hornecker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' The Inflatable Cat: Idiosyncratic Ideation Of Smart Objects For The Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In CHI Conference on Human Factors in Computing Systems Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='1145/3290605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='3300631 [4] Sarah Diefenbach, Eva Lenz, and Marc Hassenzahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' An Interaction Vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Describing the How of Interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In CHI ’13 Extended Abstracts on Human Factors in Computing Systems (CHI EA ’13), 607–612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='1145/2468356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='2468463 [5] Marc Hassenzahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Experience Design: Technology for All the Right Reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Synthesis Lectures on Human-Centered Informatics 3, 1: 1– 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='2200/S00261ED1V01Y201003HCI008 [6] Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Interaction Qualities For Interactions With, Between, And Through IoT Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='1145/3494322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='3494348 [7] Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Scented Dice: New interaction qualities for ideating connected devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In Workshop Smell, Taste, and Temperature Interfaces at Conference on Human Factors in Computing Systems (CHI ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Retrieved from https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/abs/2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='10484 [8] Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Explorative Co-Design-Werkzeuge zum Entwerfen von Smart Connected Things am Beispiel eines Workshops mit Blinden und Sehbehinderten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In Technische Unterstützungssysteme, die die Menschen wirklich wollen, 395–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Retrieved January 19, 2017 from http://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='com/janya26 [9] Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Loaded Dice: Exploring the Design Space of Connected Devices with Blind and Visually Impaired People.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In Proceedings of the 9th Nordic Conference on Human-Computer Interaction (NordiCHI ’16), 31:1-31:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='1145/2971485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='2971524 [10] Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Michael Storz, Albrecht Kurze, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' Loaded Dice: How to cheat your way to creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' In Proceedings of the 3rd Biennial Research Through Design Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='4746976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='v1 4 1st 2nd Goals Actors Spaces 3rd precise frendly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='4publie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='covered ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='apparent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='deterministic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='chaotic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='close ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='distant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fluent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='stepwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='prosaic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='poetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='papeas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='binary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='precise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='approximate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='metaphorical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='factual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spotial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='spatial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='proximity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='separation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='mediated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='familiar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='strange ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='targeted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='incidental ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='constant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='inconstant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='srabbing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='powerful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='gentle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='instant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='delayed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='harsh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='tender ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='diverging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='uniform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='slow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='angn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='friendlySENSORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Temperature Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Light Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Microphone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='MovementSensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Potentiometer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Distance Sensor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='ACTUATORDIE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Vibration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Heating Surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='LED-Bargraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Loudspeaker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='Power-LEDs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} +page_content='FanINPUT' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdFJT4oBgHgl3EQfDCyN/content/2301.11432v1.pdf'} diff --git a/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf b/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f1b27f6b411c446670ed0567352540b0ee941cd0 --- /dev/null +++ b/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:71908df53942bd57832e448ac01f8accc17a8df42d60ea185f8928763fcd4026 +size 1955411 diff --git a/H9AzT4oBgHgl3EQfU_x9/content/tmp_files/2301.01276v1.pdf.txt b/H9AzT4oBgHgl3EQfU_x9/content/tmp_files/2301.01276v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..572381ccbd195704c4a94a108cdf1aeaf95070a2 --- /dev/null +++ b/H9AzT4oBgHgl3EQfU_x9/content/tmp_files/2301.01276v1.pdf.txt @@ -0,0 +1,1008 @@ +arXiv:2301.01276v1 [cs.IT] 3 Jan 2023 +Age of Information of a Power Constrained +Scheduler in the Presence of a Power +Constrained Adversary +Subhankar Banerjee +Sennur Ulukus +Anthony Ephremides +Department of Electrical and Computer Engineering +University of Maryland, College Park, MD 20742 +sbanerje@umd.edu +ulukus@umd.edu +etony@umd.edu +Abstract—We consider a time slotted communication network +consisting of a base station (BS), an adversary, N users and +Ns communication channels. In the first part of the paper, we +consider the setting where Ns communication channels Ns are +heterogeneously divided among N users. The BS transmits an +update to the ith user on a subset of the communication channels +Ns,i where Ns,i ∩ Ns,j is not necessarily an empty set. At each +time slot, the BS transmits an update packet to a user through a +communication channel and the adversary aims to block the +update packet sent by the BS by blocking a communication +channel. The BS has n discrete transmission power levels to +communicate with the users and the adversary has m discrete +blocking power levels to block the communication channels. +The probability of successful transmission of an update packet +depends on these power levels. The BS and the adversary have a +transmission and blocking average power constraint, respectively. +We provide a universal lower bound for the average age of +information for this communication network. We prove that +the uniform user choosing policy, the uniform communication +channel choosing policy with any arbitrary feasible transmission +power choosing policy is 4 optimal; and the max-age user +choosing policy, the uniform communication channel choosing +policy with any arbitrary feasible transmission power choosing +policy is 2 optimal. In the second part of the paper, we consider +the setting where the BS chooses a transmission policy and the +adversary chooses a blocking policy from the set of randomized +stationary policies and Ns,i = Ns for all i, i.e., all users can +receive updates on all channels. We show that a Nash equilibrium +may or may not exist for this communication network, and +identify special cases where a Nash equilibrium always exists. +I. INTRODUCTION +We consider a wireless communication system consisting +of N users, one base station (BS), Ns communication chan- +nels and an adversary. A communication channel can have +different channel gains to different users, and thus, all the sub- +carriers may not be available to all the users for transmission +of an update packet. We consider the static setting. Thus, +the communication channels are divided into N potentially +overlapping sets, where each set corresponds to a user. We +denote the set of communication channels available to user +i as Ns,i. A sub-carrier can be an element of multiple sets, +and thus, the set Ns,i ∩ Ns,j is not necessarily empty. The +cardinality of Ns,i is Ns,i. The set of all available channels is +Ns = � +i Ns,i, and has cardinality Ns. There are n discrete +power levels available to the BS for transmission of an update +packet to the users and m discrete power levels available to +the adversary to block the transmission of an update packet. +We consider a slotted time model. At each time slot, the BS +chooses a transmission power to transmit an update packet to +a user via a communication channel and the adversary chooses +a communication channel and a blocking power to block any +update packet that is being sent on the chosen channel. +A large amount of work has been done on the analysis +of age of information for various applications and system +models, such as, scheduling policies for wireless networks, +gossip networks, caching systems, source coding problem, +remote estimation, energy harvesting systems and many more, +see e.g., [1]–[41]. These papers consider systems without +an adversary. The age of information in the presence of an +adversary in a wireless communication network has been +studied in the recent literature [42]–[50]. In particular, [49], +[50] consider an adversarial gossip network. In this paper, we +do not consider a gossip network, rather we consider that a +central node, i.e., the BS transmits the update packets to the +users. [42], [43] consider an adversary which decreases the +signal to noise ratio of a communication link through jamming, +due to which the rate of the communication decreases which +results in a higher age for the communication system. In +this paper, we consider that when the adversary blocks a +communication channel it completely eliminates the update +packet with a positive probability. [44] considers an adversary +which blocks the communication channel for a duration in time +which increases the average age of the system by disabling +communication in that interval. In this paper, we consider +that the adversary blocks the communication channel in a +time slotted manner. [45], [46] consider an adversary which +completely eliminates the update packet, however, they do not +consider any power constraint on the adversary. In this paper, +we consider a power constrained adversary. [47], [48] consider +a power constrained adversary which completely eliminates +the update packet. They have considered that on the time hori- +zon T , the adversary blocks αT time slots where 0 < α < 1. +On the contrary, in this paper, we consider that at each time +slot t, the adversary chooses one of the m blocking power +levels with a pmf d(t) and the expected power to be less than +or equal to a power constraint. Different than the adversary in +[47], [48], the adversary in this paper completely eliminates + +the update packet with a positive probability (strictly less than +1), and this probability depends on the blocking power chosen +by the adversary and the transmission power chosen by the BS. +In the first part of this paper, we propose algorithms to +minimize the average age of information for the described +wireless communication network. We show that the uniform +user choosing policy together with the uniform communication +channel choosing policy and any arbitrary feasible transmis- +sion power choosing policy is 4 optimal, and in a special case, +it is 2 optimal. We show that the maximum-age user choos- +ing policy together with the uniform communication channel +choosing policy and any arbitrary feasible transmission power +choosing policy is 2 optimal. +In the second part of this paper, we relax the system model +and consider that at each time slot the BS can choose any one +of the Ns sub-carriers for transmission of an update packet +to any one of the N users, i.e., Ns,i = Ns, for all i. We +also restrict the action space of the BS and the action space +of the adversary only to the stationary policies. If the power +level choosing algorithms are not fixed for the BS and for the +adversary and if those are included in the action space of the +BS and the action space of the adversary, then we show that +in the stationary policy regime a Nash equilibrium may not +exist. We give a counter example to prove this. We also show +a special case in which the Nash equilibrium exists. However, +when the power level choosing algorithms for the BS and for +the adversary are fixed, i.e., those are not included in the list +of the actions of the BS and the list of the actions of the +adversary, then the Nash equilibrium always exists. +II. SYSTEM MODEL AND PROBLEM FORMULATION +At each time slot, the BS schedules a user i out of N users, +N > 1, with a user choosing algorithm πu and chooses a +communication channel out of Ns,i communication channels, +Ns,i > 1, with a communication channel choosing algorithm +πs to transmit an update packet to the scheduled user i. In +this paper, we use sub-carrier and communication channel +interchangeably. We consider that n discrete transmission +powers, namely {p1, p2, · · · , pn} are available to the BS, and +at each time slot the BS chooses one of these n transmission +powers, following a power choosing algorithm πp. Thus, an +action of the BS is a triplet (πu, πs, πp) and we call a valid +triplet as a BS scheduling algorithm π. We call the set of all +causal scheduling algorithms as Π. Let us consider that πp is +such that at time slot t the BS chooses the ith transmission +power with probability ei(t). We consider the following power +constraint for the BS, +n +� +i=1 +ei(t)pi ≤ ¯p, +t ∈ {1, · · · , T } +(1) +We consider that an adversary is present in the system as +well. At each time slot, the adversary chooses a sub-carrier +out of Ns sub-carriers following an algorithm ψs to block any +update packet that is being transmitted by the BS in that sub- +carrier. We consider that m discrete blocking powers, namely +{p′ +1, p′ +2, · · · , p′ +m} are available to the adversary and at each +time slot the adversary chooses one of these powers, following +a blocking power choosing algorithm ψp, to block any update +packet on the sub-carrier chosen by ψs. Thus, an action of +the adversary is a pair (ψs, ψp) and we call a valid pair as an +adversarial action ψ. We call the set of all valid adversarial +actions as Ψ. Let us consider that ψp is such that at time +slot t, the adversary chooses the ith blocking power with +probability di(t). We consider the following power constraint +for the adversary, +m +� +i=1 +di(t)p′ +i ≤ ˜p, +t ∈ {1, · · · , T } +(2) +We create an n × m matrix F , whose (i, j)th element, +fi,j, represents the probability of successful transmission of +an update packet corresponding to the BS transmission power +pi and adversary blocking power p′ +j. Thus, at time slot t if +the BS schedules the user k, and chooses the sub-carrier l to +transmit an update packet with power pi and if the adversary +blocks the sub-carrier l with power p′ +j, then with probability +fi,j the age of the kth user at time slot (t+ 1) becomes 1 and +with probability 1 − fi,j the age of the kth user at time slot +t + 1 increases by one. +The age of user i at time slot t is defined as t−ti(t), where +ti(t) is the last time slot when the ith user has successfully +received an update packet. Note that the minimum value for +the age of user i is 1. We consider that at each time slot the BS +has a fresh update packet to transmit for every user present in +the system. Here by fresh update packet, we mean the update +packet for the ith user at time slot t is generated at time slot +t. As we are interested in freshness, we assume that if the ith +user does not receive the corresponding update packet at time +slot t, then that update packet gets dropped at the BS without +any cost. This is a valid assumption used in [45]–[48]. +The adversary has the knowledge of πu, πs and πp. How- +ever, as the BS uses a randomized algorithm at time slot t, the +adversary has no knowledge about which user will get sched- +uled, which sub-carrier will get chosen and which transmission +power will get used at time slot t′ when t ≤ t′ ≤ T . However, +at time slot t it has full knowledge about all these for time slot +t′ when 1 ≤ t′ < t, and the adversary can optimize its future +actions based on these available information. The adversary +has full knowledge about the elements of each set Ns,i. The +age of user i at time slot t corresponding to a BS scheduling +algorithm π and adversarial action ψ is denoted as v(π,ψ) +i +(t), +thus, v(π,ψ) +i +(t) = t − ti(t), and the expected age of user i +at time slot t, is denoted as ∆(π,ψ) +i +(t). Note that, if the BS +successfully transmits an update packet to user i at time slot t, +then v(π,ψ) +i +(t+1) = 1, otherwise v(π,ψ) +i +(t+1) = v(π,ψ) +i +(t)+1. +The average age of the overall system corresponding to the BS +scheduling algorithm π and adversarial action ψ is, +∆(π,ψ) = lim sup +T →∞ +1 +T +T +� +t=1 +1 +N +N +� +i=1 +∆(π,ψ) +i +(t) +(3) +For the simplicity of presentation, in the rest of the paper +we ignore the superscript (π, ψ), unless we specify otherwise. + +Now, as the BS has no control over the adversary, we consider +the following constrained optimization problem, +∆∗ = sup +ψ∈Ψ +inf +π∈Π +∆(π,ψ) +s.t. +(1), (2) +(4) +For the second part of the paper, we consider a relaxed +system model. We consider that at each time slot, all the +Ns sub-carriers are available to the BS to transmit an update +packet to any one of the N users, i.e., Ns,i = Ns for all +i. The BS chooses a scheduling algorithm and the adversary +chooses an adversarial action from the corresponding sets of +stationary randomized policies. In other words, πu is such +that at each time slot the BS chooses a user following a pmf +u = [u1, u2, · · · , uN], πs is such that at each time slot the BS +chooses a sub-carrier following a pmf s = [s1, s2, · · · , sNs] +and πp is such that at each time slot the the BS chooses a +power following a pmf e = [e1, e2, · · · , en]. Similarly, ψs is +such that at each time slot the adversary blocks a sub-carrier +following a pmf a = [a1, a2, · · · , aNs] and ψp is such that at +each time slot the adversary chooses a blocking power follow- +ing a pmf d = [d1, d2, · · · , dm]. Thus, the power constraints +for the adversary and the BS become �m +i=1 dip′(i) ≤ ˜p and +�n +i=1 eip(i) ≤ ¯p, respectively. When we restrict ourselves +only to the stationary randomized policies, instead of writing +∆π,ψ as in (3), we write the average age of the overall system +corresponding to pmfs u, s, e (these three pmfs are chosen +by the BS) and the pmfs a, d (these two pmfs are chosen by +the adversary) as ∆u,s,e,a,d. We denote the expected age of +user i at time slot t as ∆u,s,e,a,d +i +(t). Thus, the average age +for the ith user becomes +∆u,s,e,a,d +i += lim sup +T →∞ +1 +T +T +� +t=1 +∆u,s,e,a,d +i +(t) +(5) +Let us assume that the set of all valid user choosing pmfs, +the set of all valid sub-carrier choosing pmfs and the set of all +valid transmission power choosing pmfs are Fu, Fs and Fe, +respectively. Similarly, the set of all valid sub-carrier blocking +pmfs and the set for all valid blocking power choosing pmfs +are Fa and Fd, respectively. For a given adversarial action, +namely a sub-carrier blocking pmf a, and a blocking power +level choosing pmf d, the BS aims to minimize the average +age of the overall system by selecting a scheduling algorithm, +namely a user choosing pmf u, a sub-carrier choosing pmf +s and a transmission power choosing pmf e from the set +B(a, d), where B(a, d) is defined as follows, +B(a, d) = +arg min +(u∈Fu,s∈Fs,e∈Fe,�n +i=1 eipi≤¯p) +∆u,s,e,a,d +(6) +Similarly, for a given scheduling algorithm, i.e., a triplet of +pmfs (u, s, e), the adversary aims to maximize the average +age by choosing a pair of pmfs, namely (a, d) from the set +B(u, s, e), where B(u, s, e) is defined as +B(u, s, e) = +arg max +(a∈Fa,d∈Fd,�m +i=1 dip′(i)≤˜p) +∆u,s,e,a,d +(7) +We call a 5-tuple of pmfs, namely (u, s, e, a, d) as a Nash +equilibrium point if and only if (u, s, e) ∈ B(a, d) and +(a, d) ∈ B(u, s, e). +In the previous Nash equilibrium setting we consider that +the transmission power choosing pmf e and blocking power +choosing pmf d are components of the action space of the BS +and the action space of the adversary, respectively. However, +if e and d are fixed and not included in the action space of +the BS and the action space of the adversary, respectively, then +we define, +B(a) = +arg min +(u∈Fu,s∈Fs) +∆u,s,e,a,d +(8) +Similarly, we write, +B(u, s) = arg max +(a∈Fa) +∆u,s,e,a,d +(9) +We call a triplet of pmfs, namely (u, s, a) as a Nash equilib- +rium point if and only if (u, s) ∈ B(a, ) and a ∈ B(u, s). +III. ALGORITHM AND ANALYSIS OF AGE +We find a fundamental lower bound for the optimization +problem in (4). Let us define x = arg maxi∈{1,··· ,m} p′ +i ≤ ˜p. +Consider the following adversarial action: at each time slot +the adversary blocks any one of the Ns sub-carriers with a +uniform pmf and chooses the power level px. We denote this +adversarial action as ¯ψ = ( ¯ψs, ¯ψp). At each time slot, if the BS +schedules the user which has the maximum age and breaks the +tie with scheduling the lower indexed user, we call that user +choosing policy as the max-age policy. (In this paper, we will +present our results in a sequence of lemmas and theorems, +with some explanations. The proofs are skipped here due to +space limitations, and will be provided in the journal version.) +Lemma 1. For the adversarial action ¯ψ, an optimal user +choosing policy is the max-age policy; and if the ith user gets +chosen by the max-age policy, then an optimal sub-carrier +choosing policy is to choose a sub-carrier in Ns,i uniformly. +Let us define ¯y = arg mini∈{1,··· ,n} pi ≥ ¯p. +Theorem 1. The average age of the communication network +defined in (3) is lower bounded by +(N+1)Ns +2(Ns−1+f¯ +y,x). +Now, we consider that at each time slot the BS schedules +a user i with probability +1 +N and chooses one of the Ns,i sub- +carriers with probability +1 +Ns,i , to transmit an update packet to +the scheduled user with transmission power py with probability +β and with transmission power p¯y with probability (1 − β), +where β satisfies the following identity: +βpy + (1 − β)p¯y = ¯p +(10) +Let us denote this BS scheduling policy as ˆ˜π. Let us define +¯x = arg mini∈{1,··· ,m} p′ +i ≥ ˜p. +Theorem 2. The average age of the communication system +when the BS employs the scheduling algorithm ˆ˜π is upper +bounded by 2N; when Ns,i = Ns for all i, then the average +age is upper bounded by +NNs +Ns−1+βfy,¯x+(1−β)f¯ +y,¯x . + +Now, we consider that at each time slot the BS schedules the +max-age user, i, and chooses one of the Ns,i sub-carriers with +probability +1 +Ns,i . We also consider that the BS chooses power +py with probability β and power p¯y with probability 1 − β, +where β satisfies (10). Denote this BS scheduling policy as ˜˜π. +Theorem 3. The average age of the communication sys- +tem when the BS employs the scheduling algorithm ˜˜π is +upper bounded by +(N+1) ¯ +Ns +2( ¯ +Ns−1+βfy,¯x+(1−β)f¯ +y,¯x), where +¯Ns = +min {Ns,1, Ns,2, · · · , Ns,N}. +Next, we make some concluding remarks about the findings +of this section. From Theorem 1 and Theorem 2, we see that +in the general setting, ˆ˜π is 4N(Ns−1+f¯ +y,x) +(N+1)Ns +optimal, where +4N(Ns − 1 + f¯y,x) +(N + 1)Ns +≤ 4 +(11) +For the special case, when Ns,i = Ns, for all i, ˆ˜π is +2(N+1)(Ns−1+f¯ +y,x) +N(Ns−1+fy,¯x) +optimal, where +2(N + 1)(Ns − 1 + f¯y,x) +N(Ns − 1 + fy,¯x) +≤2(Ns − 1 + f¯y,x) +(Ns − 1 + fy,¯x) +(12) +≤ 2Ns +Ns − 1 +(13) +≤4 +(14) +If Ns is large, then the right side of (13) can be approximated +as 2. Thus, for the aforementioned special case and for large +Ns, ˆ˜π is 2 optimal. +From Theorem 1 and Theorem 3, we see that the scheduling +policy ˜˜π is +¯ +Ns +¯ +Ns−1 optimal and as Ns,i > 1, for all i, ˜˜π is 2 +optimal. Note that when ¯p exactly matches with one of the +powers from the sets {p1, p2, · · · , pn} and Ns,i = Ns, for all +i, then ˜˜π is the optimal scheduling policy. +IV. EQUILIBRIUM POINTS OF THE AVERAGE AGE FOR +RANDOMIZED STATIONARY ACTION SPACE +Let us assume that at each time slot the BS chooses a user +following a pmf u, chooses a sub-carrier following a pmf s, +chooses a transmission power with a pmf e and the adversary +chooses a sub-carrier with a pmf a and chooses a blocking +power following a pmf d. Recall that for this section we use +a relaxed system model, where we consider that Ns,i = Ns, +for all i. At some time slot t, user i successfully receives an +update packet transmitted by the BS and then after waiting for +Γi time slots it again receives another update packet from the +BS. Note that Γi is a random variable. The evolution of the +age for the ith user is a renewal process and Γi is a renewal +interval. Thus, from the renewal reward theorem, +∆u,s,e,a,d +i += E +� +Γ2 +i + Γi +� +2E [Γi] +(15) +Let the probability of successful transmission of the update +packet to user i be qi. Then, Γi is geometrically distributed +with success probability qi. Thus, (15) simplifies as, +∆u,s,e,a,d +i += 1 +qi +(16) +Theorem 4. The optimal sub-carrier choosing pmf s, for +a given adversarial action, namely, a pair of pmfs (a, d), +depends only on a and is independent of user choosing pmf u, +transmission power choosing pmf e and d. Moreover, if the +adversary blocks any l sub-carriers with lowest probability +then the optimal choice for the BS is to choose any subset of +these l sub-carriers with probability 1. Similarly, the optimal +user scheduling pmf u does not depend on a, s, d, e. The +optimal user scheduling pmf is the uniform pmf. +Theorem 5. The optimal sub-carrier blocking pmf, a, for +a given BS scheduling policy depends only on s and is +independent of u, e and d. Moreover, if the BS chooses any +l sub-carriers with the highest probability, then the optimal +choice for the adversary is to block any subset of these l sub- +carriers with probability 1. +Without loss of generality, let p1 ≤ p2 ≤ · · · ≤ pn and +p′ +1 ≤ p′ +2 ≤ · · · ≤ p′ +m. Thus, we have f1,j ≤ f2,j ≤ · · · ≤ fn,j +and fi,1 ≥ fi,2 ≥ · · · ≥ fi,m, i = 1, · · · , n, j = 1, · · · , m. +Algorithm 1 below provides an optimal transmission power +choosing pmf e for a given blocking power choosing pmf +d. The algorithm states that, if ¯p < p1, then there does +not exist a feasible e; if pn < ¯p, then the optimal e is to +choose the power pn with probability 1; If these two cases +do not occur, then we define x = arg maxi∈{1,··· ,n},pi<¯p i +and y = arg mini∈{1,··· ,n},pi>¯p i. Clearly, x < y. We define +a constant, gi = �m +j=1 djfi,j, i = 1, · · · , n. We call the +constant +� +gi + gx +py−pi +px−py − gy +px−pi +px−py +� +as the coefficient for +power pi, i ∈ {1, · · · , n}\{x, y}. Then, we traverse from +power py+1 to power pn, we call this procedure as the first +traversing procedure. During this traversing process, if we find +that +� +gj + gx +py−pj +px−py − gy +px−pj +px−py +� +, j > y, is a strictly positive +number, then we change the coefficient of the power pk as +� +gk + gx +pj−pk +px−pj − gj +px−pk +px−pj +� +, k ∈ {1, · · · , n}\{x, j}. We keep +on doing this procedure till we reach pn. Let us assume that +during this traversing procedure pi is the last power for which +we get a positive coefficient, then we define y = i. Then, +we start performing a second traversing procedure from the +power px−1 to the power p1. During this traversing process, +if we find that the coefficient of pl, l < x, is a strictly +positive number, then we change the coefficient of the power +pk as +� +gk + gl +py−pk +pl−py − gy +pl−pk +pl−py +� +, k ∈ {1, · · · , n}\{l, y}. We +keep on doing this procedure till we reach p1. Let us assume +that during this second traversing procedure pr is the last +power for which we get a positive coefficient, then we define +x = r. Now, if ¯p exactly matches one of the powers from +the set {p1, p2, · · · , pn}, without loss of generality assume +that pi += +¯p, then we compare the two vectors zi and +� +¯p−py +px−py zx + px−¯p +px−py zy +� +and select the one which minimizes +(15), otherwise we select +� +¯p−py +px−py zx + px−¯p +px−py zy +� +, where zi is +the ith basis vector of Rn. +We note that, Algorithm 1 finds an optimal solution in O(n) +time. Next, we state the optimality of Algorithm 1. + +Algorithm 1 For a given d finding an optimal e +Inputs: d, F , p, ¯p +Define: +g += +(g1, g2, · · · , gn), +where +gi += +�m +j=1 djfi,j, +x += +arg maxi∈{1,2,··· ,n},pi<¯p i +and +y += +arg mini∈{1,2,··· ,n},pi>¯p i, +zi +is +the +ith +basis +vector for Rn, x1 = x, y1 = y +if ¯p < p1 then +Return: Solution does not exist +else if pn < ¯p then +Return: zn +for i = y + 1 : n do +if +� +gi + gx +py−pi +px−py − gy +px−pi +px−py +� +> 0 then +y = i +for i = 1 : x − 1 do +if +� +gi + gx +py−pi +px−py − gy +px−pi +px−py +� +> 0 then +x = i +Define: e = +� +¯p−py +px−py zx + px−¯p +px−py zy +� +if x1 + 1 = y1 − 1 then +if �n +i=1 ei +�m +j=1 djfi,j ≤ �m +j=1 djfx1+1,j then +Return: zx+1 +else +Return: e +else +Return: e +Theorem 6. For a given blocking power pmf d, Algorithm 1 +gives an optimal transmission power pmf e. +Algorithm 2 provides an optimal blocking power choosing +pmf d for a given e. In Algorithm 2, we perform a similar +traversing procedure as Algorithm 1. The only difference is +while traversing in Algorithm 1, we change the coefficient +of a power level if the corresponding coefficient is strictly +positive, in Algorithm 2, we change the coefficient if it is +strictly negative. Next, we state the optimality of Algorithm 2. +Theorem 7. For a given transmission power choosing pmf e, +Algorithm 2 gives an optimal blocking power pmf d. +Next, we present a counter example which suggests that +when the transmission power choosing pmf and the blocking +power choosing pmf are not fixed and are part of the action +space of the BS and the action space of the adversary, +respectively, then a Nash equilibrium may not exist. Consider a +system where the BS has three power levels and the adversary +has also three power levels, i.e., n = m = 3. Both the power +constraint for the BS and the adversary is 3.5 watts. The +feasible powers for the BS and for the adversary are the same, +which is [1, 3, 5]. The matrix F is chosen as +F = + + +0.5 +0.35 +0.2 +0.6 +0.55 +0.4 +0.8 +0.7 +0.65 + + +(17) +We can show that for this example, for a given d, e cannot +be of the form [e1, e2, e3], where ei > 0, i ∈ {1, 2, 3} and +satisfy �3 +i=1 eipi ≤ ¯p. Now, from Algorithm 1, we know that +Algorithm 2 For a given e finding an optimal d +Inputs: e, F , p, ¯p +Define: +g += +(g1, g2, · · · , gm), +where +gi += +�n +j=1 ejfj,i, +x += +arg maxi∈{1,2,··· ,m},p′ +i<˜p i +and +y += +arg mini∈{1,2,··· ,m},p′ +i>˜p i, zi +is +the ith +basis +function for Rn, x1 = x, y1 = y +if ˜p < p′ +1 then +Return: Solution does not exist +else if p′ +n < ˜p then +Return: zn +for i = y + 1 : n do +if +� +gi + gx +p′ +y−p′ +i +p′ +x−p′ +y − gy +p′ +x−p′ +i +p′ +x−p′ +y +� +< 0 then +y = i +for i = 1 : x − 1 do +if +� +gi + gx +p′ +y−p′ +i +p′ +x−p′ +y − gy +p′ +x−p′ +i +p′ +x−p′ +y +� +< 0 then +x = i +Define: d = +� ˜p−p′ +y +p′x−p′y zx + p′ +x−˜p +p′x−p′y zy +� +if x1 + 1 = y1 − 1 then +if �m +j=1 dj +�n +i=1 eifi,j ≤ �n +i=1 eifi,x1+1 then +Return: d +else +Return: zx+1 +else +Return: d +if the adversary chooses powers 3 and 5, then the optimal +choice for the BS is to choose powers 3 and 5, similarly, if +the adversary chooses powers 1 and 5, then the optimal choice +for the BS is to choose powers 1 and 5. From Algorithm 2, +we know that if the BS chooses powers 1 and 5, then the +optimal choice for the adversary is to choose powers 3 and 5, +similarly, if the BS chooses powers 3 and 5, then the optimal +choice for the adversary is to choose powers 1 and 5. Thus, a +Nash equilibrium does not exist for this example. +In the next theorem, we consider the Nash equilibrium when +the transmission power choosing pmf and the blocking power +choosing pmf are not included in the action space of the BS +and in the action space of the adversary, respectively. +Theorem 8. The triplet of actions (ˆu, ˆs, ˆa) is the Nash +equilibrium point, where ˆa and ˆs are the uniform pmfs over +Ns sub-carriers and ˆu is the uniform pmf over N users. +Next, we present a special case in which the Nash equi- +librium exists even when the transmission power choosing +pmf and the blocking power choosing pmf are part of the +action space of the BS and the action space of the adversary, +respectively. Consider that the matrix F has the property, +fi,j − f1,j = li, +j ∈ {1, · · · , m}, i ∈ {1, · · · , n} +(18) +where li are non-negative constants. Consider a fixed blocking +power choosing pmf d. Then, gi in Algorithm 1 is +gi = +m +� +j=1 +djfi,j = +m +� +j=1 +djf1,j + li +(19) + +Thus, +gi + gx +py − pi +px − py +− gy +px − pi +px − py += + + +m +� +j=1 +djf1,j + + +� +1 + py − pi +px − py +− px − pi +px − py +� ++ lx +py − pi +px − py +− ly +px − pi +px − py ++ li +(20) +Thus, the sign of gi +gx +py−pi +px−py −gy +px−pi +px−py does not depend on +d, which implies that the optimal transmission power choosing +pmf is the same for all d. Similarly, the sign of gi+gx +p′ +y−p′ +i +p′x−p′y − +gy +p′ +x−p′ +i +p′x−p′y in Algorithm 2 does not depend on e, in other words +the optimal blocking power choosing pmf is independent of +e. Now, run Algorithm 1 for any arbitrary d and denote the +output as ˆe, similarly run Algorithm 2 for any arbitrary e and +denote the output as ˆd. Then, using Theorem 8, we have that +the 5-tuple (ˆb, ˆc, ˆe, ˆa, ˆd) is the unique Nash equilibrium. +REFERENCES +[1] A. Kosta, N. Pappas, and V. Angelakis. Age of information: A new +concept, metric, and tool. Foun. Trends Networ., 12(3):162–259, 2017. +[2] Y. Sun, I. Kadota, R. Talak, and E. Modiano. Age of information: A new +metric for information freshness. Synthesis Lectures on Communication +Networks, 12(2):1–224, December 2019. +[3] R. D. Yates, Y. Sun, R. Brown, S. K. Kaul, E. Modiano, and S. Ulukus. +Age of information: An introduction and survey. +IEEE Journal on +Selected Areas in Communications, 39(5):1183–1210, May 2021. +[4] I. Kadota, A. Sinha, E. Uysal-Biyikoglu, R. Singh, and E. Modiano. +Scheduling policies for minimizing age of information in broadcast +wireless networks. IEEE/ACM ToN, 26(6):2637–2650, December 2018. +[5] E. Najm, R. D. Yates, and E. Soljanin. Status updates through M/G/1/1 +queues with HARQ. In IEEE ISIT, June 2017. +[6] A. Soysal and S. Ulukus. Age of information in G/G/1/1 systems: Age +expressions, bounds, special cases, and optimization. IEEE Trans. on +Information Theory, 67(11):7477–7489, November 2021. +[7] R. D. Yates. The age of information in networks: Moments, distributions, +and sampling. IEEE Trans. Info. Theory, 66(9):5712–5728, Sep. 2020. +[8] S. Farazi, A. G. Klein, and D. R. Brown III. Average age of information +for status update systems with an energy harvesting server. In IEEE +Infocom, April 2018. +[9] X. Wu, J. Yang, and J. Wu. Optimal status update for age of information +minimization with an energy harvesting source. IEEE Trans. on Green +Communications and Networking, 2(1):193–204, March 2018. +[10] S. Leng and A. Yener. Age of information minimization for an energy +harvesting cognitive radio. IEEE TCCN, 5(2):427–43, June 2019. +[11] A. Arafa and S. Ulukus. +Timely updates in energy harvesting two- +hop networks: Offline and online policies. +IEEE Trans. on Wireless +Communications, 18(8):4017–4030, August 2019. +[12] Y. Gu, Q. Wang, H. Chen, Y. Li, and B. Vucetic. Optimizing information +freshness in two-hop status update systems under a resource constraint. +IEEE JSAC, 39(5):1380–1392, March 2021. +[13] A. Arafa, J. Yang, S. Ulukus, and H. V. Poor. Age-minimal transmission +for energy harvesting sensors with finite batteries: Online policies. IEEE +Trans. on Information Theory, 66(1):534–556, January 2020. +[14] Q. He, D. Yuan, and A. Ephremides. Optimal link scheduling for age +minimization in wireless systems. IEEE Trans. on Information Theory, +64(7):5381–5394, July 2018. +[15] B. Buyukates, A. Soysal, and S. Ulukus. Scaling laws for age of infor- +mation in wireless networks. IEEE Trans. on Wireless Communications, +20(4):2413–2427, April 2021. +[16] M. Abd-Elmagid, H. Dhillon, and N. Pappas. A reinforcement learning +framework for optimizing age of information in RF-powered communi- +cation systems. IEEE Trans. on Comm., 68(8):4747–4760, May 2020. +[17] B. Buyukates, A. Soysal, and S. Ulukus. Age of information in multihop +multicast networks. Jour. Comm. Netw., 21(3):256–267, July 2019. +[18] Y. Hsu. Age of information: Whittle index for scheduling stochastic +arrivals. In IEEE ISIT, June 2018. +[19] J. Liu, X. Wang, and H. Dai. Age-optimal trajectory planning for UAV- +assisted data collection. In IEEE Infocom, April 2018. +[20] E. Ceran, D. Gunduz, and A. Gyorgy. +A reinforcement learning +approach to age of information in multi-user networks. In IEEE PIMRC, +September 2018. +[21] B. Buyukates and S. Ulukus. +Timely distributed computation with +stragglers. IEEE Trans. on Comm., 68(9):5273–5282, September 2020. +[22] H. H. Yang, A. Arafa, T. Q. S. Quek, and H. V. Poor. +Age-based +scheduling policy for federated learning in mobile edge networks. In +IEEE ICASSP, May 2020. +[23] E. Ozfatura, B. Buyukates, D. Gunduz, and S. Ulukus. +Age-based +coded computation for bias reduction in distributed learning. In IEEE +Globecom, December 2020. +[24] B. Buyukates and S. Ulukus. +Timely communication in federated +learning. In IEEE Infocom, May 2021. +[25] M. Wang, W. Chen, and A. Ephremides. Reconstruction of counting +process in real-time: The freshness of information through queues. In +IEEE ICC, July 2019. +[26] M. Bastopcu and S. Ulukus. Who should Google Scholar update more +often? In IEEE Infocom, July 2020. +[27] Y. Sun, Y. Polyanskiy, and E. Uysal. Remote estimation of the Wiener +process over a channel with random delay. In IEEE ISIT, June 2017. +[28] J. Chakravorty and A. Mahajan. +Remote estimation over a packet- +drop channel with Markovian state. IEEE Trans. on Automatic Control, +65(5):2016–2031, July 2020. +[29] M. Bastopcu and S. Ulukus. Age of information for updates with dis- +tortion: Constant and age-dependent distortion constraints. IEEE/ACM +Trans. on Networking, 29(6):2425–2438, December 2021. +[30] N. Rajaraman, R. Vaze, and G. Reddy. Not just age but age and quality +of information. IEEE JSAC, 39(5):1325–1338, 2021. +[31] J. Zhong, R. D. Yates, and E. Soljanin. Timely lossless source coding +for randomly arriving symbols. In IEEE ITW, November 2018. +[32] P. Mayekar, P. Parag, and H. Tyagi. Optimal source codes for timely +updates. IEEE Trans. on Info. Theory, 66(6):3714–3731, March 2020. +[33] M. Bastopcu, B. Buyukates, and S. Ulukus. Selective encoding policies +for maximizing information freshness. IEEE Trans. on Communications, +69(9):5714–5726, September 2021. +[34] R. D. Yates, P. Ciblat, A. Yener, and M. Wigger. Age-optimal constrained +cache updating. In IEEE ISIT, June 2017. +[35] M. Bastopcu and S. Ulukus. Information freshness in cache updating +systems. IEEE Trans. Wireless Comm., 20(3):1861–1874, March 2021. +[36] B. Abolhassani, J. Tadrous, A. Eryilmaz, and E. Yeh. Fresh caching for +dynamic content. In IEEE Infocom, May 2021. +[37] P. Kaswan, M. Bastopcu, and S. Ulukus. Freshness based cache updating +in parallel relay networks. In IEEE ISIT, July 2021. +[38] R. D. Yates. The age of gossip in networks. In IEEE ISIT, July 2021. +[39] B. Buyukates, M. Bastopcu, and S. Ulukus. Age of gossip in networks +with community structure. In IEEE SPAWC, September 2021. +[40] M. Bastopcu, B. Buyukates, and S. Ulukus. +Gossiping with binary +freshness metric. In IEEE Globecom, December 2021. +[41] C. Kam, S. Kompella, and A. Ephremides. Age of incorrect information +for remote estimation of a binary Markov source. In Infocom, July 2020. +[42] A. Garnaev, W. Zhang, J. Zhong, and R. D. Yates. +Maintaining +information freshness under jamming. In IEEE Infocom, May 2019. +[43] G. +D. +Nguyen, +S. +Kompella, +C. +Kam, +J. E. +Wieselthier, +and +A. Ephremides. Impact of hostile interference on information freshness: +A game approach. In IEEE WiOpt, May 2017. +[44] Y. Xiao and Y. Sun. +A dynamic jamming game for real-time status +updates. In IEEE Infocom, April 2018. +[45] S. Banerjee, R. Bhattacharjee, and A. Sinha. Fundamental limits of age- +of-information in stationary and non-stationary environments. In IEEE +ISIT, June 2020. +[46] R. Bhattacharjee and A. Sinha. Competitive algorithms for minimizing +the maximum age-of-information. In ACM Perf. Eval., September 2020. +[47] S. Banerjee and S. Ulukus. Age of information in the presence of an +adversary. In IEEE Infocom, May 2022. +[48] S. Banerjee and S. Ulukus. Game theoretic analysis of an adversarial +status updating system. In IEEE ISIT, June 2022. +[49] P. Kaswan and S. Ulukus. Age of gossip in ring networks in the presence +of jamming attacks. In Asilomar Conference, October 2022. +[50] P. Kaswan and S. Ulukus. Susceptibility of age of gossip to timestomp- +ing. In IEEE ITW, November 2022. + diff --git a/H9AzT4oBgHgl3EQfU_x9/content/tmp_files/load_file.txt b/H9AzT4oBgHgl3EQfU_x9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ad4c2b6092c3e12711d6907a6d0fceb4b3b7a01 --- /dev/null +++ b/H9AzT4oBgHgl3EQfU_x9/content/tmp_files/load_file.txt @@ -0,0 +1,603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf,len=602 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='01276v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='IT] 3 Jan 2023 Age of Information of a Power Constrained Scheduler in the Presence of a Power Constrained Adversary Subhankar Banerjee Sennur Ulukus Anthony Ephremides Department of Electrical and Computer Engineering University of Maryland, College Park, MD 20742 sbanerje@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='edu ulukus@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='edu etony@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='edu Abstract—We consider a time slotted communication network consisting of a base station (BS), an adversary, N users and Ns communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In the first part of the paper, we consider the setting where Ns communication channels Ns are heterogeneously divided among N users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The BS transmits an update to the ith user on a subset of the communication channels Ns,i where Ns,i ∩ Ns,j is not necessarily an empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' At each time slot, the BS transmits an update packet to a user through a communication channel and the adversary aims to block the update packet sent by the BS by blocking a communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The BS has n discrete transmission power levels to communicate with the users and the adversary has m discrete blocking power levels to block the communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The probability of successful transmission of an update packet depends on these power levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The BS and the adversary have a transmission and blocking average power constraint, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We provide a universal lower bound for the average age of information for this communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We prove that the uniform user choosing policy, the uniform communication channel choosing policy with any arbitrary feasible transmission power choosing policy is 4 optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' and the max-age user choosing policy, the uniform communication channel choosing policy with any arbitrary feasible transmission power choosing policy is 2 optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In the second part of the paper, we consider the setting where the BS chooses a transmission policy and the adversary chooses a blocking policy from the set of randomized stationary policies and Ns,i = Ns for all i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', all users can receive updates on all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We show that a Nash equilibrium may or may not exist for this communication network, and identify special cases where a Nash equilibrium always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' INTRODUCTION We consider a wireless communication system consisting of N users, one base station (BS), Ns communication chan- nels and an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' A communication channel can have different channel gains to different users, and thus, all the sub- carriers may not be available to all the users for transmission of an update packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider the static setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, the communication channels are divided into N potentially overlapping sets, where each set corresponds to a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We denote the set of communication channels available to user i as Ns,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' A sub-carrier can be an element of multiple sets, and thus, the set Ns,i ∩ Ns,j is not necessarily empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The cardinality of Ns,i is Ns,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The set of all available channels is Ns = � i Ns,i, and has cardinality Ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' There are n discrete power levels available to the BS for transmission of an update packet to the users and m discrete power levels available to the adversary to block the transmission of an update packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider a slotted time model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' At each time slot, the BS chooses a transmission power to transmit an update packet to a user via a communication channel and the adversary chooses a communication channel and a blocking power to block any update packet that is being sent on the chosen channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' A large amount of work has been done on the analysis of age of information for various applications and system models, such as, scheduling policies for wireless networks, gossip networks, caching systems, source coding problem, remote estimation, energy harvesting systems and many more, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', [1]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' These papers consider systems without an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The age of information in the presence of an adversary in a wireless communication network has been studied in the recent literature [42]–[50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In particular, [49], [50] consider an adversarial gossip network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In this paper, we do not consider a gossip network, rather we consider that a central node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', the BS transmits the update packets to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [42], [43] consider an adversary which decreases the signal to noise ratio of a communication link through jamming, due to which the rate of the communication decreases which results in a higher age for the communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In this paper, we consider that when the adversary blocks a communication channel it completely eliminates the update packet with a positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [44] considers an adversary which blocks the communication channel for a duration in time which increases the average age of the system by disabling communication in that interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In this paper, we consider that the adversary blocks the communication channel in a time slotted manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [45], [46] consider an adversary which completely eliminates the update packet, however, they do not consider any power constraint on the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In this paper, we consider a power constrained adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [47], [48] consider a power constrained adversary which completely eliminates the update packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' They have considered that on the time hori- zon T , the adversary blocks αT time slots where 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' On the contrary, in this paper, we consider that at each time slot t, the adversary chooses one of the m blocking power levels with a pmf d(t) and the expected power to be less than or equal to a power constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Different than the adversary in [47], [48], the adversary in this paper completely eliminates the update packet with a positive probability (strictly less than 1), and this probability depends on the blocking power chosen by the adversary and the transmission power chosen by the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In the first part of this paper, we propose algorithms to minimize the average age of information for the described wireless communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We show that the uniform user choosing policy together with the uniform communication channel choosing policy and any arbitrary feasible transmis- sion power choosing policy is 4 optimal, and in a special case, it is 2 optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We show that the maximum-age user choos- ing policy together with the uniform communication channel choosing policy and any arbitrary feasible transmission power choosing policy is 2 optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In the second part of this paper, we relax the system model and consider that at each time slot the BS can choose any one of the Ns sub-carriers for transmission of an update packet to any one of the N users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', Ns,i = Ns, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We also restrict the action space of the BS and the action space of the adversary only to the stationary policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' If the power level choosing algorithms are not fixed for the BS and for the adversary and if those are included in the action space of the BS and the action space of the adversary, then we show that in the stationary policy regime a Nash equilibrium may not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We give a counter example to prove this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We also show a special case in which the Nash equilibrium exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' However, when the power level choosing algorithms for the BS and for the adversary are fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', those are not included in the list of the actions of the BS and the list of the actions of the adversary, then the Nash equilibrium always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' SYSTEM MODEL AND PROBLEM FORMULATION At each time slot, the BS schedules a user i out of N users, N > 1, with a user choosing algorithm πu and chooses a communication channel out of Ns,i communication channels, Ns,i > 1, with a communication channel choosing algorithm πs to transmit an update packet to the scheduled user i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In this paper, we use sub-carrier and communication channel interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider that n discrete transmission powers, namely {p1, p2, · · · , pn} are available to the BS, and at each time slot the BS chooses one of these n transmission powers, following a power choosing algorithm πp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, an action of the BS is a triplet (πu, πs, πp) and we call a valid triplet as a BS scheduling algorithm π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We call the set of all causal scheduling algorithms as Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Let us consider that πp is such that at time slot t the BS chooses the ith transmission power with probability ei(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider the following power constraint for the BS, n � i=1 ei(t)pi ≤ ¯p, t ∈ {1, · · · , T } (1) We consider that an adversary is present in the system as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' At each time slot, the adversary chooses a sub-carrier out of Ns sub-carriers following an algorithm ψs to block any update packet that is being transmitted by the BS in that sub- carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider that m discrete blocking powers, namely {p′ 1, p′ 2, · · · , p′ m} are available to the adversary and at each time slot the adversary chooses one of these powers, following a blocking power choosing algorithm ψp, to block any update packet on the sub-carrier chosen by ψs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, an action of the adversary is a pair (ψs, ψp) and we call a valid pair as an adversarial action ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We call the set of all valid adversarial actions as Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Let us consider that ψp is such that at time slot t, the adversary chooses the ith blocking power with probability di(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider the following power constraint for the adversary, m � i=1 di(t)p′ i ≤ ˜p, t ∈ {1, · · · , T } (2) We create an n × m matrix F , whose (i, j)th element, fi,j, represents the probability of successful transmission of an update packet corresponding to the BS transmission power pi and adversary blocking power p′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, at time slot t if the BS schedules the user k, and chooses the sub-carrier l to transmit an update packet with power pi and if the adversary blocks the sub-carrier l with power p′ j, then with probability fi,j the age of the kth user at time slot (t+ 1) becomes 1 and with probability 1 − fi,j the age of the kth user at time slot t + 1 increases by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The age of user i at time slot t is defined as t−ti(t), where ti(t) is the last time slot when the ith user has successfully received an update packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Note that the minimum value for the age of user i is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider that at each time slot the BS has a fresh update packet to transmit for every user present in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Here by fresh update packet, we mean the update packet for the ith user at time slot t is generated at time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' As we are interested in freshness, we assume that if the ith user does not receive the corresponding update packet at time slot t, then that update packet gets dropped at the BS without any cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' This is a valid assumption used in [45]–[48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The adversary has the knowledge of πu, πs and πp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' How- ever, as the BS uses a randomized algorithm at time slot t, the adversary has no knowledge about which user will get sched- uled, which sub-carrier will get chosen and which transmission power will get used at time slot t′ when t ≤ t′ ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' However, at time slot t it has full knowledge about all these for time slot t′ when 1 ≤ t′ < t, and the adversary can optimize its future actions based on these available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The adversary has full knowledge about the elements of each set Ns,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The age of user i at time slot t corresponding to a BS scheduling algorithm π and adversarial action ψ is denoted as v(π,ψ) i (t), thus, v(π,ψ) i (t) = t − ti(t), and the expected age of user i at time slot t, is denoted as ∆(π,ψ) i (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Note that, if the BS successfully transmits an update packet to user i at time slot t, then v(π,ψ) i (t+1) = 1, otherwise v(π,ψ) i (t+1) = v(π,ψ) i (t)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The average age of the overall system corresponding to the BS scheduling algorithm π and adversarial action ψ is, ∆(π,ψ) = lim sup T →∞ 1 T T � t=1 1 N N � i=1 ∆(π,ψ) i (t) (3) For the simplicity of presentation, in the rest of the paper we ignore the superscript (π, ψ), unless we specify otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Now, as the BS has no control over the adversary, we consider the following constrained optimization problem, ∆∗ = sup ψ∈Ψ inf π∈Π ∆(π,ψ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' (1), (2) (4) For the second part of the paper, we consider a relaxed system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We consider that at each time slot, all the Ns sub-carriers are available to the BS to transmit an update packet to any one of the N users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', Ns,i = Ns for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The BS chooses a scheduling algorithm and the adversary chooses an adversarial action from the corresponding sets of stationary randomized policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In other words, πu is such that at each time slot the BS chooses a user following a pmf u = [u1, u2, · · · , uN], πs is such that at each time slot the BS chooses a sub-carrier following a pmf s = [s1, s2, · · · , sNs] and πp is such that at each time slot the the BS chooses a power following a pmf e = [e1, e2, · · · , en].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Similarly, ψs is such that at each time slot the adversary blocks a sub-carrier following a pmf a = [a1, a2, · · · , aNs] and ψp is such that at each time slot the adversary chooses a blocking power follow- ing a pmf d = [d1, d2, · · · , dm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, the power constraints for the adversary and the BS become �m i=1 dip′(i) ≤ ˜p and �n i=1 eip(i) ≤ ¯p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' When we restrict ourselves only to the stationary randomized policies, instead of writing ∆π,ψ as in (3), we write the average age of the overall system corresponding to pmfs u, s, e (these three pmfs are chosen by the BS) and the pmfs a, d (these two pmfs are chosen by the adversary) as ∆u,s,e,a,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We denote the expected age of user i at time slot t as ∆u,s,e,a,d i (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, the average age for the ith user becomes ∆u,s,e,a,d i = lim sup T →∞ 1 T T � t=1 ∆u,s,e,a,d i (t) (5) Let us assume that the set of all valid user choosing pmfs, the set of all valid sub-carrier choosing pmfs and the set of all valid transmission power choosing pmfs are Fu, Fs and Fe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Similarly, the set of all valid sub-carrier blocking pmfs and the set for all valid blocking power choosing pmfs are Fa and Fd, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' For a given adversarial action, namely a sub-carrier blocking pmf a, and a blocking power level choosing pmf d, the BS aims to minimize the average age of the overall system by selecting a scheduling algorithm, namely a user choosing pmf u, a sub-carrier choosing pmf s and a transmission power choosing pmf e from the set B(a, d), where B(a, d) is defined as follows, B(a, d) = arg min (u∈Fu,s∈Fs,e∈Fe,�n i=1 eipi≤¯p) ∆u,s,e,a,d (6) Similarly, for a given scheduling algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', a triplet of pmfs (u, s, e), the adversary aims to maximize the average age by choosing a pair of pmfs, namely (a, d) from the set B(u, s, e), where B(u, s, e) is defined as B(u, s, e) = arg max (a∈Fa,d∈Fd,�m i=1 dip′(i)≤˜p) ∆u,s,e,a,d (7) We call a 5-tuple of pmfs, namely (u, s, e, a, d) as a Nash equilibrium point if and only if (u, s, e) ∈ B(a, d) and (a, d) ∈ B(u, s, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In the previous Nash equilibrium setting we consider that the transmission power choosing pmf e and blocking power choosing pmf d are components of the action space of the BS and the action space of the adversary, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' However, if e and d are fixed and not included in the action space of the BS and the action space of the adversary, respectively, then we define, B(a) = arg min (u∈Fu,s∈Fs) ∆u,s,e,a,d (8) Similarly, we write, B(u, s) = arg max (a∈Fa) ∆u,s,e,a,d (9) We call a triplet of pmfs, namely (u, s, a) as a Nash equilib- rium point if and only if (u, s) ∈ B(a, ) and a ∈ B(u, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' ALGORITHM AND ANALYSIS OF AGE We find a fundamental lower bound for the optimization problem in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Let us define x = arg maxi∈{1,··· ,m} p′ i ≤ ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Consider the following adversarial action: at each time slot the adversary blocks any one of the Ns sub-carriers with a uniform pmf and chooses the power level px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We denote this adversarial action as ¯ψ = ( ¯ψs, ¯ψp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' At each time slot, if the BS schedules the user which has the maximum age and breaks the tie with scheduling the lower indexed user, we call that user choosing policy as the max-age policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' (In this paper, we will present our results in a sequence of lemmas and theorems, with some explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The proofs are skipped here due to space limitations, and will be provided in the journal version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=') Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' For the adversarial action ¯ψ, an optimal user choosing policy is the max-age policy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' and if the ith user gets chosen by the max-age policy, then an optimal sub-carrier choosing policy is to choose a sub-carrier in Ns,i uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Let us define ¯y = arg mini∈{1,··· ,n} pi ≥ ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The average age of the communication network defined in (3) is lower bounded by (N+1)Ns 2(Ns−1+f¯ y,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Now, we consider that at each time slot the BS schedules a user i with probability 1 N and chooses one of the Ns,i sub- carriers with probability 1 Ns,i , to transmit an update packet to the scheduled user with transmission power py with probability β and with transmission power p¯y with probability (1 − β), where β satisfies the following identity: βpy + (1 − β)p¯y = ¯p (10) Let us denote this BS scheduling policy as ˆ˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Let us define ¯x = arg mini∈{1,··· ,m} p′ i ≥ ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The average age of the communication system when the BS employs the scheduling algorithm ˆ˜π is upper bounded by 2N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' when Ns,i = Ns for all i, then the average age is upper bounded by NNs Ns−1+βfy,¯x+(1−β)f¯ y,¯x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Now, we consider that at each time slot the BS schedules the max-age user, i, and chooses one of the Ns,i sub-carriers with probability 1 Ns,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We also consider that the BS chooses power py with probability β and power p¯y with probability 1 − β, where β satisfies (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Denote this BS scheduling policy as ˜˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The average age of the communication sys- tem when the BS employs the scheduling algorithm ˜˜π is upper bounded by (N+1) ¯ Ns 2( ¯ Ns−1+βfy,¯x+(1−β)f¯ y,¯x), where ¯Ns = min {Ns,1, Ns,2, · · · , Ns,N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Next, we make some concluding remarks about the findings of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' From Theorem 1 and Theorem 2, we see that in the general setting, ˆ˜π is 4N(Ns−1+f¯ y,x) (N+1)Ns optimal, where 4N(Ns − 1 + f¯y,x) (N + 1)Ns ≤ 4 (11) For the special case, when Ns,i = Ns, for all i, ˆ˜π is 2(N+1)(Ns−1+f¯ y,x) N(Ns−1+fy,¯x) optimal, where 2(N + 1)(Ns − 1 + f¯y,x) N(Ns − 1 + fy,¯x) ≤2(Ns − 1 + f¯y,x) (Ns − 1 + fy,¯x) (12) ≤ 2Ns Ns − 1 (13) ≤4 (14) If Ns is large, then the right side of (13) can be approximated as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, for the aforementioned special case and for large Ns, ˆ˜π is 2 optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' From Theorem 1 and Theorem 3, we see that the scheduling policy ˜˜π is ¯ Ns ¯ Ns−1 optimal and as Ns,i > 1, for all i, ˜˜π is 2 optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Note that when ¯p exactly matches with one of the powers from the sets {p1, p2, · · · , pn} and Ns,i = Ns, for all i, then ˜˜π is the optimal scheduling policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' EQUILIBRIUM POINTS OF THE AVERAGE AGE FOR RANDOMIZED STATIONARY ACTION SPACE Let us assume that at each time slot the BS chooses a user following a pmf u, chooses a sub-carrier following a pmf s, chooses a transmission power with a pmf e and the adversary chooses a sub-carrier with a pmf a and chooses a blocking power following a pmf d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Recall that for this section we use a relaxed system model, where we consider that Ns,i = Ns, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' At some time slot t, user i successfully receives an update packet transmitted by the BS and then after waiting for Γi time slots it again receives another update packet from the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Note that Γi is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The evolution of the age for the ith user is a renewal process and Γi is a renewal interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, from the renewal reward theorem, ∆u,s,e,a,d i = E � Γ2 i + Γi � 2E [Γi] (15) Let the probability of successful transmission of the update packet to user i be qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Then, Γi is geometrically distributed with success probability qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, (15) simplifies as, ∆u,s,e,a,d i = 1 qi (16) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The optimal sub-carrier choosing pmf s, for a given adversarial action, namely, a pair of pmfs (a, d), depends only on a and is independent of user choosing pmf u, transmission power choosing pmf e and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Moreover, if the adversary blocks any l sub-carriers with lowest probability then the optimal choice for the BS is to choose any subset of these l sub-carriers with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Similarly, the optimal user scheduling pmf u does not depend on a, s, d, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The optimal user scheduling pmf is the uniform pmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The optimal sub-carrier blocking pmf, a, for a given BS scheduling policy depends only on s and is independent of u, e and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Moreover, if the BS chooses any l sub-carriers with the highest probability, then the optimal choice for the adversary is to block any subset of these l sub- carriers with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Without loss of generality, let p1 ≤ p2 ≤ · · · ≤ pn and p′ 1 ≤ p′ 2 ≤ · · · ≤ p′ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, we have f1,j ≤ f2,j ≤ · · · ≤ fn,j and fi,1 ≥ fi,2 ≥ · · · ≥ fi,m, i = 1, · · · , n, j = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Algorithm 1 below provides an optimal transmission power choosing pmf e for a given blocking power choosing pmf d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The algorithm states that, if ¯p < p1, then there does not exist a feasible e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' if pn < ¯p, then the optimal e is to choose the power pn with probability 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' If these two cases do not occur, then we define x = arg maxi∈{1,··· ,n},pi<¯p i and y = arg mini∈{1,··· ,n},pi>¯p i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Clearly, x < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We define a constant, gi = �m j=1 djfi,j, i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We call the constant � gi + gx py−pi px−py − gy px−pi px−py � as the coefficient for power pi, i ∈ {1, · · · , n}\\{x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Then, we traverse from power py+1 to power pn, we call this procedure as the first traversing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' During this traversing process, if we find that � gj + gx py−pj px−py − gy px−pj px−py � , j > y, is a strictly positive number, then we change the coefficient of the power pk as � gk + gx pj−pk px−pj − gj px−pk px−pj � , k ∈ {1, · · · , n}\\{x, j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We keep on doing this procedure till we reach pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Let us assume that during this traversing procedure pi is the last power for which we get a positive coefficient, then we define y = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Then, we start performing a second traversing procedure from the power px−1 to the power p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' During this traversing process, if we find that the coefficient of pl, l < x, is a strictly positive number, then we change the coefficient of the power pk as � gk + gl py−pk pl−py − gy pl−pk pl−py � , k ∈ {1, · · · , n}\\{l, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We keep on doing this procedure till we reach p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Let us assume that during this second traversing procedure pr is the last power for which we get a positive coefficient, then we define x = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Now, if ¯p exactly matches one of the powers from the set {p1, p2, · · · , pn}, without loss of generality assume that pi = ¯p, then we compare the two vectors zi and � ¯p−py px−py zx + px−¯p px−py zy � and select the one which minimizes (15), otherwise we select � ¯p−py px−py zx + px−¯p px−py zy � , where zi is the ith basis vector of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' We note that, Algorithm 1 finds an optimal solution in O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Next, we state the optimality of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Algorithm 1 For a given d finding an optimal e Inputs: d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' ¯p Define: g = (g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' gn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' where gi = �m j=1 djfi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' x = arg maxi∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='n},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='pi<¯p i and y = arg mini∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='n},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='pi>¯p i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' zi is the ith basis vector for Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' x1 = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' y1 = y if ¯p < p1 then Return: Solution does not exist else if pn < ¯p then Return: zn for i = y + 1 : n do if � gi + gx py−pi px−py − gy px−pi px−py � > 0 then y = i for i = 1 : x − 1 do if � gi + gx py−pi px−py − gy px−pi px−py � > 0 then x = i Define: e = � ¯p−py px−py zx + px−¯p px−py zy � if x1 + 1 = y1 − 1 then if �n i=1 ei �m j=1 djfi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='j ≤ �m j=1 djfx1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='j then Return: zx+1 else Return: e else Return: e Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' For a given blocking power pmf d, Algorithm 1 gives an optimal transmission power pmf e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Algorithm 2 provides an optimal blocking power choosing pmf d for a given e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In Algorithm 2, we perform a similar traversing procedure as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The only difference is while traversing in Algorithm 1, we change the coefficient of a power level if the corresponding coefficient is strictly positive, in Algorithm 2, we change the coefficient if it is strictly negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Next, we state the optimality of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' For a given transmission power choosing pmf e, Algorithm 2 gives an optimal blocking power pmf d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Next, we present a counter example which suggests that when the transmission power choosing pmf and the blocking power choosing pmf are not fixed and are part of the action space of the BS and the action space of the adversary, respectively, then a Nash equilibrium may not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Consider a system where the BS has three power levels and the adversary has also three power levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', n = m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Both the power constraint for the BS and the adversary is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='5 watts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The feasible powers for the BS and for the adversary are the same, which is [1, 3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The matrix F is chosen as F = \uf8ee \uf8f0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='65 \uf8f9 \uf8fb (17) We can show that for this example, for a given d, e cannot be of the form [e1, e2, e3], where ei > 0, i ∈ {1, 2, 3} and satisfy �3 i=1 eipi ≤ ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' from Algorithm 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' we know that Algorithm 2 For a given e finding an optimal d Inputs: e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' ¯p Define: g = (g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' gm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' where gi = �n j=1 ejfj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' x = arg maxi∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='m},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='p′ i<˜p i and y = arg mini∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='m},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='p′ i>˜p i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' zi is the ith basis function for Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' x1 = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' y1 = y if ˜p < p′ 1 then Return: Solution does not exist else if p′ n < ˜p then Return: zn for i = y + 1 : n do if � gi + gx p′ y−p′ i p′ x−p′ y − gy p′ x−p′ i p′ x−p′ y � < 0 then y = i for i = 1 : x − 1 do if � gi + gx p′ y−p′ i p′ x−p′ y − gy p′ x−p′ i p′ x−p′ y � < 0 then x = i Define: d = � ˜p−p′ y p′x−p′y zx + p′ x−˜p p′x−p′y zy � if x1 + 1 = y1 − 1 then if �m j=1 dj �n i=1 eifi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='j ≤ �n i=1 eifi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content='x1+1 then Return: d else Return: zx+1 else Return: d if the adversary chooses powers 3 and 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' then the optimal choice for the BS is to choose powers 3 and 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' if the adversary chooses powers 1 and 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' then the optimal choice for the BS is to choose powers 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' From Algorithm 2, we know that if the BS chooses powers 1 and 5, then the optimal choice for the adversary is to choose powers 3 and 5, similarly, if the BS chooses powers 3 and 5, then the optimal choice for the adversary is to choose powers 1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Thus, a Nash equilibrium does not exist for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In the next theorem, we consider the Nash equilibrium when the transmission power choosing pmf and the blocking power choosing pmf are not included in the action space of the BS and in the action space of the adversary, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The triplet of actions (ˆu, ˆs, ˆa) is the Nash equilibrium point, where ˆa and ˆs are the uniform pmfs over Ns sub-carriers and ˆu is the uniform pmf over N users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Next, we present a special case in which the Nash equi- librium exists even when the transmission power choosing pmf and the blocking power choosing pmf are part of the action space of the BS and the action space of the adversary, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Consider that the matrix F has the property, fi,j − f1,j = li, j ∈ {1, · · · , m}, i ∈ {1, · · · , n} (18) where li are non-negative constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Consider a fixed blocking power choosing pmf d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Then, gi in Algorithm 1 is gi = m � j=1 djfi,j = m � j=1 djf1,j + li (19) Thus, gi + gx py − pi px − py − gy px − pi px − py = \uf8eb \uf8ed m � j=1 djf1,j \uf8f6 \uf8f8 � 1 + py − pi px − py − px − pi px − py � + lx py − pi px − py − ly px − pi px − py + li (20) Thus, the sign of gi +gx py−pi px−py −gy px−pi px−py does not depend on d, which implies that the optimal transmission power choosing pmf is the same for all d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Similarly, the sign of gi+gx p′ y−p′ i p′x−p′y − gy p′ x−p′ i p′x−p′y in Algorithm 2 does not depend on e, in other words the optimal blocking power choosing pmf is independent of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Now, run Algorithm 1 for any arbitrary d and denote the output as ˆe, similarly run Algorithm 2 for any arbitrary e and denote the output as ˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Then, using Theorem 8, we have that the 5-tuple (ˆb, ˆc, ˆe, ˆa, ˆd) is the unique Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kosta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Pappas, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Angelakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information: A new concept, metric, and tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Foun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Trends Networ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', 12(3):162–259, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Sun, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kadota, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Talak, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Modiano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information: A new metric for information freshness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Synthesis Lectures on Communication Networks, 12(2):1–224, December 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yates, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Sun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Brown, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kaul, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Modiano, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information: An introduction and survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Journal on Selected Areas in Communications, 39(5):1183–1210, May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [4] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kadota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Sinha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Uysal-Biyikoglu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Singh, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Modiano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Scheduling policies for minimizing age of information in broadcast wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE/ACM ToN, 26(6):2637–2650, December 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Najm, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yates, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Soljanin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Status updates through M/G/1/1 queues with HARQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Soysal and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information in G/G/1/1 systems: Age expressions, bounds, special cases, and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Information Theory, 67(11):7477–7489, November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The age of information in networks: Moments, distributions, and sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theory, 66(9):5712–5728, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Farazi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Klein, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Brown III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Average age of information for status update systems with an energy harvesting server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, April 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [9] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Optimal status update for age of information minimization with an energy harvesting source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Green Communications and Networking, 2(1):193–204, March 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Leng and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information minimization for an energy harvesting cognitive radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE TCCN, 5(2):427–43, June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Arafa and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Timely updates in energy harvesting two- hop networks: Offline and online policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Wireless Communications, 18(8):4017–4030, August 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Gu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Li, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Vucetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Optimizing information freshness in two-hop status update systems under a resource constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE JSAC, 39(5):1380–1392, March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Arafa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age-minimal transmission for energy harvesting sensors with finite batteries: Online policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Information Theory, 66(1):534–556, January 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [14] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' He, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yuan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ephremides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Optimal link scheduling for age minimization in wireless systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Information Theory, 64(7):5381–5394, July 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Soysal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Scaling laws for age of infor- mation in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Wireless Communications, 20(4):2413–2427, April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Abd-Elmagid, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Dhillon, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Pappas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' A reinforcement learning framework for optimizing age of information in RF-powered communi- cation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', 68(8):4747–4760, May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [17] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Soysal, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information in multihop multicast networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Jour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', 21(3):256–267, July 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Hsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information: Whittle index for scheduling stochastic arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, June 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age-optimal trajectory planning for UAV- assisted data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, April 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ceran, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Gunduz, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Gyorgy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' A reinforcement learning approach to age of information in multi-user networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE PIMRC, September 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Timely distributed computation with stragglers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', 68(9):5273–5282, September 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Arafa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Quek, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age-based scheduling policy for federated learning in mobile edge networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ICASSP, May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ozfatura, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Gunduz, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age-based coded computation for bias reduction in distributed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Globecom, December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [24] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Timely communication in federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Chen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ephremides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Reconstruction of counting process in real-time: The freshness of information through queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ICC, July 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bastopcu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Who should Google Scholar update more often?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [27] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Sun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Polyanskiy, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Uysal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Remote estimation of the Wiener process over a channel with random delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Chakravorty and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Mahajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Remote estimation over a packet- drop channel with Markovian state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Automatic Control, 65(5):2016–2031, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bastopcu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information for updates with dis- tortion: Constant and age-dependent distortion constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE/ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Networking, 29(6):2425–2438, December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [30] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Rajaraman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Vaze, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Reddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Not just age but age and quality of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE JSAC, 39(5):1325–1338, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Zhong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yates, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Soljanin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Timely lossless source coding for randomly arriving symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ITW, November 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Mayekar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Parag, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Tyagi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Optimal source codes for timely updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Theory, 66(6):3714–3731, March 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bastopcu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Selective encoding policies for maximizing information freshness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' on Communications, 69(9):5714–5726, September 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yates, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ciblat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yener, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age-optimal constrained cache updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bastopcu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Information freshness in cache updating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wireless Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', 20(3):1861–1874, March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [36] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Abolhassani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Tadrous, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Eryilmaz, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Fresh caching for dynamic content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, May 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kaswan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bastopcu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Freshness based cache updating in parallel relay networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [38] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' The age of gossip in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [39] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bastopcu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of gossip in networks with community structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE SPAWC, September 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bastopcu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Buyukates, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Gossiping with binary freshness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Globecom, December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kompella, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ephremides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of incorrect information for remote estimation of a binary Markov source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In Infocom, July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Garnaev, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Zhong, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Yates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Maintaining information freshness under jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, May 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [43] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Nguyen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kompella, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kam, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Wieselthier, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ephremides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Impact of hostile interference on information freshness: A game approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE WiOpt, May 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [44] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Xiao and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' A dynamic jamming game for real-time status updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, April 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Banerjee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bhattacharjee, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Sinha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Fundamental limits of age- of-information in stationary and non-stationary environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, June 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [46] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Bhattacharjee and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Sinha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Competitive algorithms for minimizing the maximum age-of-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In ACM Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=', September 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Banerjee and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of information in the presence of an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE Infocom, May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Banerjee and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Game theoretic analysis of an adversarial status updating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ISIT, June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [49] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kaswan and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Age of gossip in ring networks in the presence of jamming attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In Asilomar Conference, October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' [50] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Kaswan and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' Susceptibility of age of gossip to timestomp- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} +page_content=' In IEEE ITW, November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf'} diff --git a/HNAzT4oBgHgl3EQfjP3H/content/tmp_files/2301.01514v1.pdf.txt b/HNAzT4oBgHgl3EQfjP3H/content/tmp_files/2301.01514v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..589e845a1e4cb757ecb378a17931faa3f1b49692 --- /dev/null +++ b/HNAzT4oBgHgl3EQfjP3H/content/tmp_files/2301.01514v1.pdf.txt @@ -0,0 +1,814 @@ +1 +PENDANTSS: PEnalized Norm-ratios Disentangling +Additive Noise, Trend and Sparse Spikes +Paul Zheng, Student Member, IEEE, Emilie Chouzenoux, Senior Member, IEEE, Laurent Duval, Senior Member, +IEEE +Abstract—Denoising, detrending, deconvolution: usual restora- +tion tasks, traditionally decoupled. Coupled formulations entail +complex ill-posed inverse problems. We propose PENDANTSS +for joint trend removal and blind deconvolution of sparse peak- +like signals. It blends a parsimonious prior with the hypothesis +that smooth trend and noise can somewhat be separated by +low-pass filtering. We combine the generalized quasi-norm ratio +SOOT/SPOQ sparse penalties ℓp/ℓq with the BEADS ternary +assisted source separation algorithm. This results in a both +convergent and efficient tool, with a novel Trust-Region block +alternating variable metric forward-backward approach. It out- +performs comparable methods, when applied to typically peaked +analytical chemistry signals. Reproducible code is provided. +Index Terms—Blind deconvolution, sparse signal, trend es- +timation, non-convex optimization, forward-backward splitting, +alternating minimization, source separation +I. INTRODUCTION AND BACKGROUND +Restoration recovers information from observations with +amplitude distortion, level displacement or random distur- +bance. A discrete additive-convolutive degradation model is: +y = s ∗ π + t + n . +(1) +Among N sample values, series of spikes (or impulses, events, +“diracs”, spectral lines) prototype the first component sought +sparse signal s ∈ RN. Its convolution with an unknown +short-support kernel π ∈ RL — typically peak-shaped — +yields the peak-signal x = s ∗ π ∈ RN. Second component +t ∈ RN displaces the reference level, harming quantitative +estimations. It can be called baseline, background, continuum, +drift, wander. We opt here for trend, a reference above which +peaks are detected, evaluated, measured. This “trend” notion +goes from mere offsets to slowly-varying amplitude shifts +(seasonality, calibration distortion, sensor decline), making its +automated removal challenging. Third component n ∈ RN +(noise) gathers stochastic residuals. Given (1), one goal is +to perform jointly denoising, detrending and deconvolution. +Namely, given y, to retrieve an estimation of the spiky signal +and the trend. Figure 1 is reminiscent of standard spectral +subtraction [1], and motivated here by peak-signal retrieval in +separative analytical chemistry (AC): chromatography, spec- +trometry, spectroscopy [2], where peak localization, amplitude, +width or area provide useful chemical quantitative information. +This work was supported by the European Research Council Starting Grant +MAJORIS ERC-2019-STG-850925. +P. Zheng is currently with Chair of Information Theory and Data Analyt- +ics, RWTH Aachen University, Germany (paul.zheng@inda.rwth-aachen.de); +work conducted while at Univ. Paris-Saclay, CentraleSup´elec, CVN, Inria, +Gif-sur-Yvette, France. +E. Chouzenoux is with Univ. Paris-Saclay, CentraleSup´elec, CVN, Inria, +Gif-sur-Yvette, France (emilie.chouzenoux@centralesupelec.fr). +L. Duval is with IFP Energies nouvelles, France (laurent.duval@ifpen.fr). +Whether acquired in its natural domain [3] or after spar- +sification [4], noise/trend/spike models (1) cover many mul- +tidimensional issues: signal (1D), image (2D), video, volume +(3D+). We focus here on 1D data common to distinct domains: +Fourier spectral analysis, econometrics, stocks, biomedical +measurements (ECG, EEG, EMG), environment, astronomy. .. +On the one hand, joint denoising and detrending is a long- +standing preprocessing question from time series analysis to +imaging. Background issues are commonly solved using a host +of filling, fitting and filtering methods. We refer to overviews +in [5], [6], and for AC to background corrections backcor [7] +and BEADS [8]. +On the other hand, joint denoising and deconvolution +matters from channel estimation in communications [9] to +image deblurring [10]. We refer to [11], [12], and especially +emphasize on sparsity-promoting methods like SOOT [13] and +SPOQ [14], using smoothed ”scale-invariant” norm ratios. +PENDANTSS contributions are a fully coupled and solvable +non-convex formulation for (1) (Section II) and an efficient +joint disentangling algorithm (forward-backward-based [15], +[16]) with proved convergence (Section III), validated by its +comparative performance (Section IV). +II. PROPOSED PROBLEM FORMULATION +A. BEADS peak/trend/noise separation paradigm +We seek estimates (�s, �t, �π) of (s, t, π) to penalized problem +minimize s,t∈RN +π∈RL +1 +2∥y − π ∗ s − t∥2 + R(s, t, π). +(2) +The squared loss is supplemented with regularization R, +incorporating prior knowledge. Disentangling trend and signal +is tedious [17]. As in BEADS [8], we assume that the trend +can be recovered from a peakless observation through a low- +pass filter L: +�t = L(y − �π ∗ �s), +(3) +This motivates the rewriting of the data fidelity term as: +(∀s ∈ RN)(∀π ∈ RL) ρ(s, π) = 1 +2∥y − Ly − H(π ∗ s)∥2 += 1 +2∥H(y − π ∗ s)∥2, +(4) +where H = IdN − L is a high-pass filter, and IdN the +identity operator of RN. We introduce a regularization term Ψ, +promoting signal sparsity. We add two extra terms to constrain +estimates �s and �π. The indicator function ιA of the non-empty +convex set A: zero when the value evaluated belongs to A, ++∞ otherwise. Sets C1 ⊂ RN and C2 ⊂ RL limiting the +arXiv:2301.01514v1 [eess.SP] 4 Jan 2023 + +2 +search space for the signal and the kernel are assumed closed, +non-empty and convex. Optimization problem (2) becomes: +minimize +s∈RN, π∈RL +1 +2||H(y−π∗s)||2+ιC1(s)+ιC2(π)+λΨ(s). (5) +The estimated trend can be obtained from (3) with �π and �s +obtained by (5). +B. SPOQ/SOOT (quasi-)norm ratio penalties +Being scale-invariant, ratios of norms are promising proxies +for sparsity characterization [18]. We promote sparse solutions +s by the Smoothed p-Over-q (SPOQ) family of penalties, +introduced in [14], a generalization to the Smoothed One- +Over-Two norm (SOOT) ratio [13], for sparse spectroscopic +signals. Let p ∈]0, 2[ and q ∈ [2, +∞[. We first define +two smoothed approximations to the ℓp quasi-norm and ℓq +norm, parameterized by constants (α, η) ∈]0, +∞[2 . For +s = (sn)1≤n≤N ∈ RN: +ℓp,α(s) = +� N +� +n=1 +� +(s2 +n + α2)p/2 − αp��1/p +, +(6) +and +ℓq,η(s) = +� +ηq + +N +� +n=1 +|sn|q +�1/q +. +(7) +The non-convex SPOQ penalty is given, for β ∈]0, +∞[, as: +(∀s ∈ RN) +Ψ(s) = log +� +(ℓp +p,α(s) + βp)1/p +ℓq,η(s) +� +. +(8) +Ψ is Lipschitz differentiable on RN [14, Prop. 2] and admits +0N as a local minimizer when [14, Prop. 1]: +q > 2, +or +q = 2 +and +η2αp−2 > βp. +(9) +Condition (9) is assumed throughout this paper. +III. PROPOSED OPTIMIZATION ALGORITHM +A. Problem structure +The objective function in (5) is the sum of a differentiable +function (least squares + SPOQ) and terms acting separably +on s or π (i.e., indicator terms). In the differentiable part +(∀s ∈ RN)(∀π ∈ RL) +f(s, π) = ρ(s, π) + λΨ(s),. (10) +function ρ from (4) is quadratic in s and π. In particular, +for every π ∈ RL (resp. ∀s ∈ RN), the gradient ∇ρ1(·, π) +(resp. ∇ρ2(s, ·)) of ρ with respect to its first (resp. sec- +ond) variable is Lipschitz continuous with constant Λ1(π) +(resp. Λ2(s)). As aforementioned, ∇Ψ is Lipschitz continuous +too. The second part of the objective function reads as: +(∀s ∈ RN)(∀π ∈ RL) +g(s, π) = ιC1(s) + ιC2(π). +(11) +In a nutshell, Problem (5) amounts to minimizing: +(∀s ∈ RN)(∀π ∈ RL) +Ω(s, π) = f(s, π) + g(s, π). (12) +B. Proposed Trust-Region PENDANTSS algorithm +The structure of (12) suggests using a block alternating +approach where signal s and kernel π are updated sequentially. +We hereby generalize the BC-VMFB algorithm [16], also used +in [13] for blind deconvolution. +Algorithm 1: TR-BC-VMFB for solving (5) +Settings: Kmax > 0, ε > 0, I > 0, θ ∈]0, 1[, +(γs,k)k∈N ∈ [γ, 2 − γ] and (γπ,k)k∈N ∈ [γ, 2 − γ] for +some (γ, γ) ∈]0, +∞[2, (p, q) ∈]0, 2[×[2, +∞[ +satisfying (9), convex sets (C1, C2) ⊂ RN × RL. +Initialize: s0 ∈ C1, π0 ∈ C2 +for k = 0, 1, . . . do +Update of the signal +for i = 1, . . . , I do +Set TR radius ρk,i using (17) with parameter θ; +Construct MM metric using (15): +A1,ρk,i(sk, πk) = Λ1(πk)Id + λAq,ρk,i(sk) +Find sk,i ∈ C1 such that (18) holds. +if sk,i ∈ ¯Bq,ρk,i then +Stop loop +end +end +sk+1 = sk,i; +Update of the kernel +Find πk+1 ∈ C2 such that (20) holds. +Stopping criterion +if ∥sk − sk+1|| ≤ ε or k ≥ Kmax then +Stop loop +end +end +(�s, �π) = (sk+1, πk+1) and �t given by (3); +Result: �s, �π, �t +1) Signal update: Let k ∈ N and (sk, πk) ∈ C1 ×C2. The +computation of sk+1 follows one Majoration-Minimization +(MM) iteration [19]. First, we build a majorization for Ω(·, πk) +around sk. Second, sk+1 is defined as a minimizer to the +majorant. In practice, both steps can be approximated for +speedup and robustness to numerical errors. As emphasized +in [14], [20], we need the majorization to be valid only within +a neighborhood of the current iterate. For ρ ∈ [0, +∞[, the ℓq- +ball complement set is: +¯Bq,ρ = {s = (sn)1≤n≤N ∈ RN| +N +� +n=1 +|sn|q ≥ ρq}. +(13) +From [14, Prop. 2], we can show that +(∀s ∈ ¯Bq,ρ ∩ C1) +Ω(s, πk) ≤ f(sk, πk) ++ (s − sk)⊤∇1f(sk, πk) + 1 +2∥s − sk∥2 +A1,ρ(sk,πk), +(14) +where we define the so-called MM metric as: +A1,ρ(sk, πk) = (Λ1(πk) + λχq,ρ)IdN+ +λ +ℓp +p,α(sk) + βp Diag((s2 +n,k + α2)p/2−1)1≤n≤N, +(15) +with the constant +χq,ρ = +q − 1 +(ηq + ρq)2/q . +(16) +In (14), ∥.∥A denotes the weighted Euclidean norm related +to a symmetric definite positive (SDP) matrix A ∈ RN×N, + +3 +i.e., ∀z ∈ RN, ∥z∥A = (z⊤Az)1/2. Since inequality (14) +only holds on a limited region, we introduce a Trust-Region- +based (TR) loop [21] to make sure that the minimizer of the +majorant is indeed in the validity domain of (14). Namely, we +set I > 0, a maximum number of trials of TR approach. For +i ∈ {1, . . . , I}, we define the TR radius as: +ρk,i = +� +� +� +� +� +�N +n=1 |sn,k|q +if i = 1 , +θρk,i−1 +if 2 ≤ i ≤ I − 1 , +0 +if i = I . +(17) +We compute the associated MM metric A1,ρk,i(sk, πk) and +define sk,i as a minimizer of the right term in (14). The loop +stops whenever sk,i belongs to ¯Bq,ρk,i, which is ensured to +arise in a finite number of steps according to [14]. There re- +mains to explain how we practically compute sk,i. Depending +on the choice for C1, the right term in (14) might not have a +closed-form minimizer. Actually, as we will show, it appears +sufficient for convergence purpose to search for sk,i ∈ C1 +satisfying the first order optimality conditions: +� +(sk,i−sk)⊤∇1f(sk, πk)+γ−1 +s,k||sk,i−sk||2 +A1,ρk,i(sk,πk) ≤0, +||∇1f(sk, πk)+r(1) +k,i|| ≤ κ1||sk,i−sk||A1,ρk,i(sk,πk) +(18) +for some r(1) +k,i +∈ NC1(sk,i) (i.e., the normal cone of C1 at +sk,i [22]), and some κ1 > 0. The existence of such an sk,i +can be shown from [23, Rem. 3.3]. In particular, a minimizer +over C1 of the right term in (14) satisfies (18). +2) Kernel update: It follows a similar approach. The main +difference is that we do not use the TR loop in that case, +as the function to minimize here is simpler. Let k ∈ N, +and (sk+1, πk) ∈ C1 × C2. Using the descent lemma, it is +straightforward to show that: +(∀π ∈ C2) +Ω(sk+1, π) ≤ f(sk+1, πk) ++ (π − πk)⊤∇2f(sk+1, πk) + Λ2(sk+1) +2 +∥π − πk∥2. +(19) +The new iterate πk+1 is then defined as a minimizer of the +right term of (19). Hereagain, we can solve this problem in +an inexact manner, that is to search for some πk+1 ∈ C2 +satisfying +� +� +� +� +� +(πk+1 − πk)⊤∇2f(sk+1, πk) ++γ−1 +π,kΛ2(sk+1)∥πk+1 − πk∥2 ≤ 0, +∥∇2f(sk+1, πk) + r(2) +k ∥ ≤ κ2 +� +Λ2(sk+1)∥πk+1 − πk∥, +(20) +for some r(2) +k +∈ NC2(πk+1) and κ2 > 0. The existence of +πk+1 can be shown from [23, Rem. 3.3]. In particular, a +minimizer over C2 of the right term in (19) satisfies (20). +C. Convergence Result +We establish the following convergence theorem for Algo- +rithm 1. Its proof is provided in the supplementary material. +Theorem 1. Let (sk)k∈N and (πk)k∈N be sequences gener- +ated by Algorithm 1. If C1 and C2 are semi-algebraic sets then +the sequence (sk, πk)k∈N converges to a critical point (�s, �π) +of Problem (5). +The above result extends [14, Theo.1] to the block alternat- +ing case using proof ingredients reminiscent from [16], [24]. +IV. NUMERICAL RESULTS +A. Datasets +Two datasets A and B were considered. The original sparse +signal s and the observed signal y are shown in Fig. 1, both +of size N = 200. The observed signal y is obtained from (1) +where π is a normalized Gaussian kernel with standard devi- +ation 0.15 and size L = 21. The noise n is zero-mean white +Gaussian with variance σ2 either equals 0.5 % or 1.0 % of +xmax defined as the maximum amplitude of x = π ∗s. Signal +and kernel convolution is implemented with zero padding. +B. Algorithmic settings +We choose C1 = [0, +∞[N and C2 the simplex unit set, +i.e. C2 =S ={π =(πℓ)1≤ℓ≤L ∈ [0, +∞[L +s.t. +�L +ℓ=1 πℓ = +1}. For such choices, and giving the fact that the metric (15) is +diagonal, the resolution of (18) and (20) is straightforward, by +[22, Prop. 24.11] and [25, Cor. 9]. Namely, for every k ∈ N, +and i ∈ {1, . . . , I}, +� +sk,i =ProjC1 +� +sk−γs,kA1,ρk,i(sk, πk)−1∇1f(sk, πk) +� +, +πk+1 = ProjC2 +� +πk − γπ,kΛ2(sk+1)−1∇2f(sk+1, πk) +� +. +Hereabove, ProjC1 is the projection over the positive orthant, +that has a simple closed form expression, while ProjC2 is the +projection over the simplex unit set, that can be computed +using the fast procedure from [26]. +For simplicity, we set constant stepsizes γs,k ≡ 1.9 and +γπ,k ≡ 1.9, thus satisfying the required range assumption. +Moreover, we take θ = 0.5 in the TR update, and a maximum +of I = 50 of TR trials. We use the same initialization strategy +for all methods as in [13], namely s0 ∈ C1 is a constant +positive valued signal and π0 ∈ C2 is a centered Gaussian +filter with standard deviation of 1. The stopping criterion +parameters are set as ε = +√ +N × 10−6 and Kmax = 2000. +C. Numerical results +PENDANTSS jointly performs blind deconvolution and +trend removal, using SPOQ penalty. Let us recall that SOOT +penalty from [13] is retrieved by setting (p, q) = (1, 2) in +SPOQ. Another setting will be analyzed, namely (p, q) = +(0.75, 2), which was shown to be competitive in the problem +considered in [14]. In the spirit of an ablation study, we com- +pare: (i) applying the state-of-the-art background estimation +method backcor [7] to estimate and remove the trend and then +the blind deconvolution method [13] to estimate the signal �s +and the kernel �π, (ii) applying our pipeline when using either +SPOQ (p, q) = (0.75, 2), SPOQ (p, q) = (1, 2) (i.e., SOOT). +We use signal-to-noise ratios to evaluate our estimations, +respectively for the for signal (SNRs), kernel (SNRπ) and +trend (SNRt). For instance, SNRs = 20 log10(∥s∥2/∥s−�s∥2). +Moreover, TSNR evaluates the SNR only on the support of +the original sparse signal. While their support are not known +in general, it reveals how peak-derived quantities (height, + +4 +width, area), important for downstream quantitative chemical +analysis, would be impacted by detrending and deconvolution. +Hyperparameters, e.g. regularization parameters of back- +cor [7] and SPOQ/SOOT parameters (λ, α, β, η), are adjusted +to maximize a weighted sum of SNRs for one completely +known reference realization, i.e. 2SNRs + SNRπ + SNRt. +The cutoff frequency of the low-pass filter in (3) is chosen +as the best performing point over the first ten peak points of +the modulus of the signal frequency spectrum. To assure the +kernel is centered, a spatial shift on the estimated kernel and +the sparse signal is applied as a post-processing step because +spatially shifted kernels and sparse signals result in the same +observed signal. A grid search determines the number of inner +loops to maximize the SNRs of the sparse signal. +Table I summarizes the results of mean SNR values, and +standard deviations after the “±” sign, calculated over two +hundred noise realizations. The highest among the four com- +pared methods are followed by two asterisks (**); the second +best are denoted by only one (*). We notice that the best values +and the second best values are almost always achieved by +the proposed PENDANTSS approach with (p, q) = (0.75, 2) +or (1, 2). The difference with the baseline methods is also +significant for all cases in terms of TSNRs and SNRt. One +exception lies on SNRπ with dataset B with the noise level +of 1.0 % of xmax, where the second best is achieved by +the combination backcor+SPOQ. We stress out that in such +problems, correct estimations of sparse signal and baseline +are usually more important than kernel estimation. The per- +formance of PENDANTSS for the two penalty parameters +(p, q) = (0.75, 2), (1, 2) is dependent on the datasets and +the noise level. +In terms of sparse signal recovery SNRs +and TSNRs, PENDANTSS with (p, q) = (0.75, 2) achieves +slightly higher performance than PENDANTSS with (p, q) = +(1, 2) for dataset A. However, its outcomes are notably lower +for dataset B, a less sparse signal, while remaining the second +best method. For dataset A, both PENDANTSS methods have +similar baseline estimation accuracy, while for dataset B, +PENDANTSS (p, q) = (0.75, 2) performs better with lower +noise level and PENDANTSS (p, q) = (1, 2) better with +greater noise level, with a difference of SNR of about 2 +dB. As for the estimation of SNRπ, PENDANTSS with +(p, q) = (1, 2) performs the best for all four cases with +little difference for dataset A but a larger difference for more +challenging cases with dataset B and higher noise levels. +Considering various SPOQ parameters is indeed beneficial. +According to the presented simulation results, PENDANTSS +with (p, q) = (0.75, 2) is better for datasets with sparser, well- +separable peaks whereas PENDANTSS with (p, q) = (1, 2) for +more challenging datasets. Graphical details on the quality of +estimated peaks are provided as supplementary material. +V. CONCLUSION AND PERSPECTIVES +We propose to solve a complicated joint sparse signal blind +deconvolution and additive trend problem. Our method handles +smooth trend removal by exploiting their low-pass property +and simplifies the problem into a blind deconvolution prob- +lem. The blind deconvolution problem uses the recent SPOQ +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +0 +1 +2 +3 +4 +5 +6 +7 +(a) Dataset A. +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +0 +5 +10 +15 +20 +25 +(b) Sparse spike signal for dataset A. +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +0 +2 +4 +6 +8 +10 +12 +(c) Dataset B. +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +0 +5 +10 +15 +20 +25 +30 +(d) Sparse spike signal for dataset B. +Fig. 1. Unknown sparse signal s (b) and (d); in (a) and (c) observation y +(blue) and baseline t (black) (bottom) for datasets A and B. Signal A has 10 +spikes (5.0 % of sparsity) while signal B has 20 spikes (10.0 % of sparsity). +TABLE I +NUMERICAL RESULTS ON DATASETS A AND B. SNR QUANTITIES IN DB. +BEST PERFORMING METHOD FOLLOWED BY **, SECOND BY *. +Dataset A +Dataset B +Noise level σ (% of xmax) +0.5 % +1.0 % +0.5 % +1.0 % +SNRs +backcor+SOOT +29.2±0.7 +28.5±1.9 +14.9±4.0 +11.5±4.7 +backcor+SPOQ +29.2±0.7 +29.3±1.3 +12.9±3.5 +11.3±4.4 +PENDANTS (1, 2) +32.9±1.5* +30.9±2.2* +22.3±8.2** +17.5±8.4** +PENDANTS (0.75, 2) +33.2±2.3** +31.0±4.2** +15.9±4.5* +12.9±4.6* +TSNRs +backcor+SOOT +29.2±0.7 +29.3±1.3 +16.6±3.5 +13.4±4.3 +backcor+SPOQ +29.2±0.7 +29.3±1.3 +15.1±3.0 +13.7±3.7 +PENDANTS (1, 2) +34.1±1.4* +32.2±2.1* +24.9±8.0** +19.2±7.7** +PENDANTS (0.75, 2) +35.4±1.7** +32.6±3.8** +17.7±4.0* +14.5±4.1* +SNRt +backcor+SOOT +20.5±0.2 +20.3±0.4 +15.5±0.5 +14.8±0.8 +backcor+SPOQ +20.5±0.2 +20.3±0.4 +15.5±0.5 +14.8±0.8 +PENDANTS (1, 2) +26.9±0.5** +26.0±0.8** +22.0±0.4* +21.6±1.0** +PENDANTS (0.75, 2) +26.9±0.6** +26.0±1.0** +24.6±0.6** +19.6±3.9* +SNRπ +backcor+SOOT +36.3±1.3 +33.9±1.7 +30.3±1.3 +28.5±1.8 +backcor+SPOQ +36.3±1.3 +34.0±1.7 +33.1±1.9 +31.2±2.1* +PENDANTS (1, 2) +41.3±2.0** +34.4±2.4** +38.3±1.9** +33.6±2.2** +PENDANTS (0.75, 2) +41.3±2.0** +34.2±2.5* +35.7±1.5* +25.4±5.5 +sparse penalty. Simulation results confirm that PENDANTSS +outperforms comparable methods on typical sparse analytical +signals. Further works include its validation on a variety of +other sparse spike signals. The appropriate parameters for the +sparsity-promoting norm ratio penalty ought to be investigated, +for instance with respect to the alleged signal sparsity or +peak separability. PENDANTSS Matlab code is available at +https://github.com/paulzhengfr/PENDANTSS. + +5 +REFERENCES +[1] S. Boll, “Suppression of acoustic noise in speech using spectral subtrac- +tion,” IEEE Trans. Acoust. Speech Signal Process., vol. 27, no. 2, pp. +113–120, Apr. 1979. +[2] J. Lynch, Ed., Physico-Chemical Analysis of Industrial Catalysts. A +Practical Guide to Characterisation. +´Edition Technip, Sep. 2003. +[3] A. Marmin, M. Castella, J.-C. Pesquet, and L. Duval, “Sparse signal +reconstruction for nonlinear models via piecewise rational optimization,” +Signal Process., vol. 179, no. 107835, Feb. 2021. +[4] J. Gauthier, L. Duval, and J.-C. Pesquet, “Optimization of synthesis +oversampled complex filter banks,” IEEE Trans. Signal Process., vol. 57, +no. 10, pp. 3827–3843, Oct. 2009. +[5] L. Duval, L. T. Duarte, and C. Jutten, “An overview of signal processing +issues in chemical sensing,” in Proc. Int. Conf. Acoust. Speech Signal +Process., Vancouver, BC, Canada, May 26-31, 2013, pp. 8742–8746. +[6] Z. Zhao, S. Wang, D. Wong, C. Sun, R. Yan, and X. Chen, “Robust +enhanced trend filtering with unknown noise,” Signal Process., vol. 180, +p. 107889, Mar. 2021. +[7] V. Mazet, C. Carteret, D. Brie, J. Idier, and B. Humbert, “Background +removal from spectra by designing and minimising a non-quadratic cost +function,” Chemometr. Intell. Lab. Syst., vol. 76, no. 2, pp. 121–133, +2005. +[8] X. Ning, I. W. Selesnick, and L. Duval, “Chromatogram baseline +estimation and denoising using sparsity (BEADS),” Chemometr. Intell. +Lab. Syst., vol. 139, pp. 156–167, Dec. 2014. +[9] S. Amari, S. C. Douglas, A. Cichocki, and H. H. Yang, “Multichannel +blind deconvolution and equalization using the natural gradient,” in Proc. +IEEE Workshop Signal Process. Adv. Wireless Comm., 1997, pp. 101– +104. +[10] D. Krishnan, T. Tay, and R. Fergus, “Blind deconvolution using a +normalized sparsity measure,” in Proc. IEEE Conf. Comput. Vis. Pattern +Recogn., Jun. 21-25, 2011, pp. 233–240. +[11] S. Chaudhuri, R. Velmurugan, and R. Rameshan, “Blind deconvolution +methods: A review,” in Blind Image Deconvolution. Methods and +Convergence. +Springer, 2014, pp. 37–60. +[12] Q. +Sun +and +D. +Donoho, +“Convex +sparse +blind +deconvolution,” +PREPRINT, Jun. 2021. +[13] A. Repetti, M. Q. Pham, L. Duval, E. Chouzenoux, and J.-C. Pesquet, +“Euclid in a taxicab: Sparse blind deconvolution with smoothed ℓ1/ℓ2 +regularization,” IEEE Signal Process. Lett., vol. 22, no. 5, pp. 539–543, +May 2015. +[14] A. Cherni, E. Chouzenoux, L. Duval, and J.-C. Pesquet, “SPOQ ℓp- +over-ℓq regularization for sparse signal recovery applied to mass spec- +trometry,” IEEE Trans. Signal Process., vol. 68, pp. 6070–6084, 2020. +[15] J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized +minimization fon nonconvex and nonsmooth problems,” Math. Pro- +gramm., vol. 146, no. 1-2, pp. 459–494, Aug. 2014. +[16] E. Chouzenoux, J.-C. Pesquet, and A. Repetti, “A block coordinate +variable metric forward-backward algorithm,” J. Global Optim., vol. 66, +no. 3, pp. 457–485, Feb. 2016. +[17] I. W. Selesnick, H. L. Graber, D. S. Pfeil, and R. L. Barbour, “Simul- +taneous low-pass filtering and total variation denoising,” IEEE Trans. +Signal Process., vol. 62, no. 5, pp. 1109–1124, Mar. 2014. +[18] N. Hurley and S. Rickard, “Comparing measures of sparsity,” IEEE +Trans. Inform. Theory, vol. 55, no. 10, pp. 4723–4741, Oct. 2009. +[19] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algo- +rithms in signal processing, communications, and machine learning,” +IEEE Trans. Signal Process., vol. 65, no. 3, pp. 794–816, Feb. 2017. +[20] E. Chouzenoux, S. Martin, and J.-C. Pesquet, “A local MM subspace +method for solving constrained variational problems in image recovery,” +J. Math. Imaging Vision, 2022. +[21] A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region Methods, ser. +MOS-SIAM Series on Optimization. +Society for Industrial Mathemat- +ics, 2000. +[22] H. H. Bauschke and P. L. Combettes, Convex analysis and monotone +operator theory in Hilbert spaces, 2nd ed., ser. CMS books in mathe- +matics. +Springer, 2011. +[23] E. Chouzenoux, J.-C. Pesquet, and A. Repetti, “Variable metric forward- +backward algorithm for minimizing the sum of a differentiable function +and a convex function,” J. Optim. Theory Appl., vol. 162, no. 1, pp. +107–132, Jul. 2014. +[24] L. T. K. Hien, N. Gillis, and P. Patrinos, “Inertial block proximal +methods for non-convex non-smooth optimization,” in Proc. Int. Conf. +Mach. Learn., vol. 119, Jul. 13–18, 2020, pp. 5671–5681. +[25] S. Becker and M. J. Fadili, “A quasi-Newton proximal splitting method,” +in Proc. Ann. Conf. Neur. Inform. Proc. Syst., vol. 2, Dec. 3-6, 2012, +pp. 2618–2626. +[26] L. Condat, “Fast projection onto the simplex and the l1 ball,” Math. +Programm., vol. 158, no. 1-2, pp. 575–585, 2016. + +1 +PENDANTSS: Supplementary Material +I. PROOF OF THEOREM 1 FOR ALGORITHM 1 +We first provide a useful majorant metric matrix property. +Lemma 1 There exists (λ, λ) ∈]0, +∞[2 such that for every +k ∈ N, and for every i ∈ {1, . . . , I}, +� +λIdN ⪯ A1,ρk,i(sk, πk) ⪯ λIdN, +λ ≤ Λ2(sk) ≤ λ. +(A1) +Proof. Direct consequence of [14, Prop. 2] and [13, Prop. 1]. +We then show that Algorithm 1 satisfies two essential descent +properties, that are key for the convergence analysis. +Lemma 2 There exists (µ1, µ2) ∈]0, +∞[ such that, for every +k ∈ N, the following descent properties hold: +Ω(sk+1, πk) ≤ Ω(sk, πk) − µ1 +2 ||sk+1 − sk||2, +(A2) +Ω(sk+1, πk+1)≤Ω(sk+1, πk) − µ2 +2 ||πk+1 − πk||2. +(A3) +Proof. Let k ∈ N. We remind that the objective function Ω +is defined in (12), with g specified in (11). By construction, +sk+1 ∈ ¯Bq,ρ ∩ C1 for some i ∈ {1, . . . , I}. Summing the +majoration (14) and the first inequality in (18) yields: +Ω(sk+1, πk) ≤ f(sk, πk)−(γ−1 +s,k − 1 +2)∥sk −sk+1∥2 +A1,ρ(sk,πk). +We notice that f(sk, πk) = Ω(sk, πk) since sk +∈ C1 +and πk ∈ C2. Using Lemma 1 and the range assumption on +γs,k allows to show (A2) for µ1 = λγ/(2 − γ). Again by +construction, πk+1 ∈ C2. Summing inequalities (19) and (20) +leads to: +Ω(sk+1, πk+1) ≤ f(sk+1, πk)− +(γ−1 +π,k − 1 +2)Λ2(sk+1)∥πk+1 − πk∥2. +Here again, we use f(sk+1, πk) = Ω(sk+1, πk) as sk+1 ∈ +C1 and πk ∈ C2. The descent property (A3) is obtained by +using Lemma 1, the range constraint on γπ,k, and setting µ2 = +λ¯γ(2 − ¯γ). +The rest of the proof of Theorem 1 is obtained by following +the same lines than the one of [16, Theorem 3.1]. +II. ADDITIONAL RESULTS +Figures 2 and 3 provide additional insights into PEN- +DANTSS restoration. Dataset A in Figure 2-(a) presents +sparse and well-isolated peaks. Accurate peak restoration is +secured. Peak shapes are well recovered (left-hand zoom), +and the estimated trend matches well the actual baseline. As +a consequence, peak features that are computed with respect +to the trend (height, area) are likely to be well-estimated with +PENDANTSS. The less sparse Dataset B in Figure 2-(b) shows +that the separation and the height of close peaks are accurately +matched. Some overshoot in trend estimation can be noticed. It +is however not likely to drastically affect relative peak height +or area computations. +Retrieved spikes are exposed in Figure 3. For Dataset A, +well-separated spikes are accurately recovered using PEN- +DANTSS. Estimated amplitudes and locations are almost +indistinguishable from the original ones. This is exemplified +for the less sparse Dataset B in Figure 3-(b). Isolated peaks +are well-estimated. However, some spikes (for instance around +index 175) for Dataset B in Figure 3-(b) remain unelucidated. +Three contiguous spikes are estimated, instead of two. Such an +ambiguous solution is typical to source separation problems. +(a) Dataset A reconstruction and trend. +(b) Dataset B reconstruction and trend. +Fig. 2: Ground truth (thick black line) and proposed estimation results (thin +blue line), and the baseline t (dashed dot) and the signal s ∗ p (continuous). +(a) Dataset A sparse spike signal. +(b) Dataset B sparse spike signal. +Fig. 3: Ground truth (black line with circle marker) and proposed estimation +results (blue line with cross marker). +arXiv:2301.01514v1 [eess.SP] 4 Jan 2023 + diff --git a/HNAzT4oBgHgl3EQfjP3H/content/tmp_files/load_file.txt b/HNAzT4oBgHgl3EQfjP3H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd02fe0ee0daa486e4770d7dfd915202bb65f56c --- /dev/null +++ b/HNAzT4oBgHgl3EQfjP3H/content/tmp_files/load_file.txt @@ -0,0 +1,709 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf,len=708 +page_content='1 PENDANTSS: PEnalized Norm-ratios Disentangling Additive Noise, Trend and Sparse Spikes Paul Zheng, Student Member, IEEE, Emilie Chouzenoux, Senior Member, IEEE, Laurent Duval, Senior Member, IEEE Abstract—Denoising, detrending, deconvolution: usual restora- tion tasks, traditionally decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Coupled formulations entail complex ill-posed inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We propose PENDANTSS for joint trend removal and blind deconvolution of sparse peak- like signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' It blends a parsimonious prior with the hypothesis that smooth trend and noise can somewhat be separated by low-pass filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We combine the generalized quasi-norm ratio SOOT/SPOQ sparse penalties ℓp/ℓq with the BEADS ternary assisted source separation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' This results in a both convergent and efficient tool, with a novel Trust-Region block alternating variable metric forward-backward approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' It out- performs comparable methods, when applied to typically peaked analytical chemistry signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Reproducible code is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Index Terms—Blind deconvolution, sparse signal, trend es- timation, non-convex optimization, forward-backward splitting, alternating minimization, source separation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' INTRODUCTION AND BACKGROUND Restoration recovers information from observations with amplitude distortion, level displacement or random distur- bance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' A discrete additive-convolutive degradation model is: y = s ∗ π + t + n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (1) Among N sample values, series of spikes (or impulses, events, “diracs”, spectral lines) prototype the first component sought sparse signal s ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Its convolution with an unknown short-support kernel π ∈ RL — typically peak-shaped — yields the peak-signal x = s ∗ π ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Second component t ∈ RN displaces the reference level, harming quantitative estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' It can be called baseline, background, continuum, drift, wander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We opt here for trend, a reference above which peaks are detected, evaluated, measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' This “trend” notion goes from mere offsets to slowly-varying amplitude shifts (seasonality, calibration distortion, sensor decline), making its automated removal challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Third component n ∈ RN (noise) gathers stochastic residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Given (1), one goal is to perform jointly denoising, detrending and deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Namely, given y, to retrieve an estimation of the spiky signal and the trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Figure 1 is reminiscent of standard spectral subtraction [1], and motivated here by peak-signal retrieval in separative analytical chemistry (AC): chromatography, spec- trometry, spectroscopy [2], where peak localization, amplitude, width or area provide useful chemical quantitative information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' This work was supported by the European Research Council Starting Grant MAJORIS ERC-2019-STG-850925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Zheng is currently with Chair of Information Theory and Data Analyt- ics, RWTH Aachen University, Germany (paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='zheng@inda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='de);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' work conducted while at Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Paris-Saclay, CentraleSup´elec, CVN, Inria, Gif-sur-Yvette, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chouzenoux is with Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Paris-Saclay, CentraleSup´elec, CVN, Inria, Gif-sur-Yvette, France (emilie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='chouzenoux@centralesupelec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duval is with IFP Energies nouvelles, France (laurent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='duval@ifpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Whether acquired in its natural domain [3] or after spar- sification [4], noise/trend/spike models (1) cover many mul- tidimensional issues: signal (1D), image (2D), video, volume (3D+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We focus here on 1D data common to distinct domains: Fourier spectral analysis, econometrics, stocks, biomedical measurements (ECG, EEG, EMG), environment, astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='. On the one hand, joint denoising and detrending is a long- standing preprocessing question from time series analysis to imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Background issues are commonly solved using a host of filling, fitting and filtering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We refer to overviews in [5], [6], and for AC to background corrections backcor [7] and BEADS [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' On the other hand, joint denoising and deconvolution matters from channel estimation in communications [9] to image deblurring [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We refer to [11], [12], and especially emphasize on sparsity-promoting methods like SOOT [13] and SPOQ [14], using smoothed ”scale-invariant” norm ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' PENDANTSS contributions are a fully coupled and solvable non-convex formulation for (1) (Section II) and an efficient joint disentangling algorithm (forward-backward-based [15], [16]) with proved convergence (Section III), validated by its comparative performance (Section IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' PROPOSED PROBLEM FORMULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' BEADS peak/trend/noise separation paradigm We seek estimates (�s, �t, �π) of (s, t, π) to penalized problem minimize s,t∈RN π∈RL 1 2∥y − π ∗ s − t∥2 + R(s, t, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (2) The squared loss is supplemented with regularization R, incorporating prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Disentangling trend and signal is tedious [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' As in BEADS [8], we assume that the trend can be recovered from a peakless observation through a low- pass filter L: �t = L(y − �π ∗ �s), (3) This motivates the rewriting of the data fidelity term as: (∀s ∈ RN)(∀π ∈ RL) ρ(s, π) = 1 2∥y − Ly − H(π ∗ s)∥2 = 1 2∥H(y − π ∗ s)∥2, (4) where H = IdN − L is a high-pass filter, and IdN the identity operator of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We introduce a regularization term Ψ, promoting signal sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We add two extra terms to constrain estimates �s and �π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The indicator function ιA of the non-empty convex set A: zero when the value evaluated belongs to A, +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Sets C1 ⊂ RN and C2 ⊂ RL limiting the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='01514v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='SP] 4 Jan 2023 2 search space for the signal and the kernel are assumed closed, non-empty and convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Optimization problem (2) becomes: minimize s∈RN, π∈RL 1 2||H(y−π∗s)||2+ιC1(s)+ιC2(π)+λΨ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (5) The estimated trend can be obtained from (3) with �π and �s obtained by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' SPOQ/SOOT (quasi-)norm ratio penalties Being scale-invariant, ratios of norms are promising proxies for sparsity characterization [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We promote sparse solutions s by the Smoothed p-Over-q (SPOQ) family of penalties, introduced in [14], a generalization to the Smoothed One- Over-Two norm (SOOT) ratio [13], for sparse spectroscopic signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Let p ∈]0, 2[ and q ∈ [2, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We first define two smoothed approximations to the ℓp quasi-norm and ℓq norm, parameterized by constants (α, η) ∈]0, +∞[2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For s = (sn)1≤n≤N ∈ RN: ℓp,α(s) = � N � n=1 � (s2 n + α2)p/2 − αp��1/p , (6) and ℓq,η(s) = � ηq + N � n=1 |sn|q �1/q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (7) The non-convex SPOQ penalty is given, for β ∈]0, +∞[, as: (∀s ∈ RN) Ψ(s) = log � (ℓp p,α(s) + βp)1/p ℓq,η(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (8) Ψ is Lipschitz differentiable on RN [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2] and admits 0N as a local minimizer when [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1]: q > 2, or q = 2 and η2αp−2 > βp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (9) Condition (9) is assumed throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' PROPOSED OPTIMIZATION ALGORITHM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Problem structure The objective function in (5) is the sum of a differentiable function (least squares + SPOQ) and terms acting separably on s or π (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', indicator terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' In the differentiable part (∀s ∈ RN)(∀π ∈ RL) f(s, π) = ρ(s, π) + λΨ(s),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (10) function ρ from (4) is quadratic in s and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' In particular, for every π ∈ RL (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' ∀s ∈ RN), the gradient ∇ρ1(·, π) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' ∇ρ2(s, ·)) of ρ with respect to its first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' sec- ond) variable is Lipschitz continuous with constant Λ1(π) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Λ2(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' As aforementioned, ∇Ψ is Lipschitz continuous too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The second part of the objective function reads as: (∀s ∈ RN)(∀π ∈ RL) g(s, π) = ιC1(s) + ιC2(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (11) In a nutshell, Problem (5) amounts to minimizing: (∀s ∈ RN)(∀π ∈ RL) Ω(s, π) = f(s, π) + g(s, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (12) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Proposed Trust-Region PENDANTSS algorithm The structure of (12) suggests using a block alternating approach where signal s and kernel π are updated sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We hereby generalize the BC-VMFB algorithm [16], also used in [13] for blind deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Algorithm 1: TR-BC-VMFB for solving (5) Settings: Kmax > 0, ε > 0, I > 0, θ ∈]0, 1[, (γs,k)k∈N ∈ [γ, 2 − γ] and (γπ,k)k∈N ∈ [γ, 2 − γ] for some (γ, γ) ∈]0, +∞[2, (p, q) ∈]0, 2[×[2, +∞[ satisfying (9), convex sets (C1, C2) ⊂ RN × RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Initialize: s0 ∈ C1, π0 ∈ C2 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' do Update of the signal for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' , I do Set TR radius ρk,i using (17) with parameter θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Construct MM metric using (15): A1,ρk,i(sk, πk) = Λ1(πk)Id + λAq,ρk,i(sk) Find sk,i ∈ C1 such that (18) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' if sk,i ∈ ¯Bq,ρk,i then Stop loop end end sk+1 = sk,i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Update of the kernel Find πk+1 ∈ C2 such that (20) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Stopping criterion if ∥sk − sk+1|| ≤ ε or k ≥ Kmax then Stop loop end end (�s, �π) = (sk+1, πk+1) and �t given by (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Result: �s, �π, �t 1) Signal update: Let k ∈ N and (sk, πk) ∈ C1 ×C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The computation of sk+1 follows one Majoration-Minimization (MM) iteration [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' First, we build a majorization for Ω(·, πk) around sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Second, sk+1 is defined as a minimizer to the majorant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' In practice, both steps can be approximated for speedup and robustness to numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' As emphasized in [14], [20], we need the majorization to be valid only within a neighborhood of the current iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For ρ ∈ [0, +∞[, the ℓq- ball complement set is: ¯Bq,ρ = {s = (sn)1≤n≤N ∈ RN| N � n=1 |sn|q ≥ ρq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (13) From [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2], we can show that (∀s ∈ ¯Bq,ρ ∩ C1) Ω(s, πk) ≤ f(sk, πk) + (s − sk)⊤∇1f(sk, πk) + 1 2∥s − sk∥2 A1,ρ(sk,πk), (14) where we define the so-called MM metric as: A1,ρ(sk, πk) = (Λ1(πk) + λχq,ρ)IdN+ λ ℓp p,α(sk) + βp Diag((s2 n,k + α2)p/2−1)1≤n≤N, (15) with the constant χq,ρ = q − 1 (ηq + ρq)2/q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (16) In (14), ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='∥A denotes the weighted Euclidean norm related to a symmetric definite positive (SDP) matrix A ∈ RN×N, 3 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', ∀z ∈ RN, ∥z∥A = (z⊤Az)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Since inequality (14) only holds on a limited region, we introduce a Trust-Region- based (TR) loop [21] to make sure that the minimizer of the majorant is indeed in the validity domain of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Namely, we set I > 0, a maximum number of trials of TR approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' , I}, we define the TR radius as: ρk,i = � � � � � �N n=1 |sn,k|q if i = 1 , θρk,i−1 if 2 ≤ i ≤ I − 1 , 0 if i = I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (17) We compute the associated MM metric A1,ρk,i(sk, πk) and define sk,i as a minimizer of the right term in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The loop stops whenever sk,i belongs to ¯Bq,ρk,i, which is ensured to arise in a finite number of steps according to [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' There re- mains to explain how we practically compute sk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Depending on the choice for C1, the right term in (14) might not have a closed-form minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Actually, as we will show, it appears sufficient for convergence purpose to search for sk,i ∈ C1 satisfying the first order optimality conditions: � (sk,i−sk)⊤∇1f(sk, πk)+γ−1 s,k||sk,i−sk||2 A1,ρk,i(sk,πk) ≤0, ||∇1f(sk, πk)+r(1) k,i|| ≤ κ1||sk,i−sk||A1,ρk,i(sk,πk) (18) for some r(1) k,i ∈ NC1(sk,i) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', the normal cone of C1 at sk,i [22]), and some κ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The existence of such an sk,i can be shown from [23, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' In particular, a minimizer over C1 of the right term in (14) satisfies (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2) Kernel update: It follows a similar approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The main difference is that we do not use the TR loop in that case, as the function to minimize here is simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Let k ∈ N, and (sk+1, πk) ∈ C1 × C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Using the descent lemma, it is straightforward to show that: (∀π ∈ C2) Ω(sk+1, π) ≤ f(sk+1, πk) + (π − πk)⊤∇2f(sk+1, πk) + Λ2(sk+1) 2 ∥π − πk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (19) The new iterate πk+1 is then defined as a minimizer of the right term of (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Hereagain, we can solve this problem in an inexact manner, that is to search for some πk+1 ∈ C2 satisfying � � � � � (πk+1 − πk)⊤∇2f(sk+1, πk) +γ−1 π,kΛ2(sk+1)∥πk+1 − πk∥2 ≤ 0, ∥∇2f(sk+1, πk) + r(2) k ∥ ≤ κ2 � Λ2(sk+1)∥πk+1 − πk∥, (20) for some r(2) k ∈ NC2(πk+1) and κ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The existence of πk+1 can be shown from [23, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' In particular, a minimizer over C2 of the right term in (19) satisfies (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Convergence Result We establish the following convergence theorem for Algo- rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Its proof is provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Let (sk)k∈N and (πk)k∈N be sequences gener- ated by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' If C1 and C2 are semi-algebraic sets then the sequence (sk, πk)k∈N converges to a critical point (�s, �π) of Problem (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The above result extends [14, Theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1] to the block alternat- ing case using proof ingredients reminiscent from [16], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' NUMERICAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Datasets Two datasets A and B were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The original sparse signal s and the observed signal y are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1, both of size N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The observed signal y is obtained from (1) where π is a normalized Gaussian kernel with standard devi- ation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='15 and size L = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The noise n is zero-mean white Gaussian with variance σ2 either equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 % or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 % of xmax defined as the maximum amplitude of x = π ∗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Signal and kernel convolution is implemented with zero padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Algorithmic settings We choose C1 = [0, +∞[N and C2 the simplex unit set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' C2 =S ={π =(πℓ)1≤ℓ≤L ∈ [0, +∞[L s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' �L ℓ=1 πℓ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For such choices, and giving the fact that the metric (15) is diagonal, the resolution of (18) and (20) is straightforward, by [22, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='11] and [25, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Namely, for every k ∈ N, and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' , I}, � sk,i =ProjC1 � sk−γs,kA1,ρk,i(sk, πk)−1∇1f(sk, πk) � , πk+1 = ProjC2 � πk − γπ,kΛ2(sk+1)−1∇2f(sk+1, πk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Hereabove, ProjC1 is the projection over the positive orthant, that has a simple closed form expression, while ProjC2 is the projection over the simplex unit set, that can be computed using the fast procedure from [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For simplicity, we set constant stepsizes γs,k ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9 and γπ,k ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9, thus satisfying the required range assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Moreover, we take θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 in the TR update, and a maximum of I = 50 of TR trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We use the same initialization strategy for all methods as in [13], namely s0 ∈ C1 is a constant positive valued signal and π0 ∈ C2 is a centered Gaussian filter with standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The stopping criterion parameters are set as ε = √ N × 10−6 and Kmax = 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Numerical results PENDANTSS jointly performs blind deconvolution and trend removal, using SPOQ penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Let us recall that SOOT penalty from [13] is retrieved by setting (p, q) = (1, 2) in SPOQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Another setting will be analyzed, namely (p, q) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2), which was shown to be competitive in the problem considered in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' In the spirit of an ablation study, we com- pare: (i) applying the state-of-the-art background estimation method backcor [7] to estimate and remove the trend and then the blind deconvolution method [13] to estimate the signal �s and the kernel �π, (ii) applying our pipeline when using either SPOQ (p, q) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2), SPOQ (p, q) = (1, 2) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', SOOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We use signal-to-noise ratios to evaluate our estimations, respectively for the for signal (SNRs), kernel (SNRπ) and trend (SNRt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For instance, SNRs = 20 log10(∥s∥2/∥s−�s∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Moreover, TSNR evaluates the SNR only on the support of the original sparse signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' While their support are not known in general, it reveals how peak-derived quantities (height, 4 width, area), important for downstream quantitative chemical analysis, would be impacted by detrending and deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Hyperparameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' regularization parameters of back- cor [7] and SPOQ/SOOT parameters (λ, α, β, η), are adjusted to maximize a weighted sum of SNRs for one completely known reference realization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2SNRs + SNRπ + SNRt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The cutoff frequency of the low-pass filter in (3) is chosen as the best performing point over the first ten peak points of the modulus of the signal frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' To assure the kernel is centered, a spatial shift on the estimated kernel and the sparse signal is applied as a post-processing step because spatially shifted kernels and sparse signals result in the same observed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' A grid search determines the number of inner loops to maximize the SNRs of the sparse signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Table I summarizes the results of mean SNR values, and standard deviations after the “±” sign, calculated over two hundred noise realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The highest among the four com- pared methods are followed by two asterisks (**);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' the second best are denoted by only one (*).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We notice that the best values and the second best values are almost always achieved by the proposed PENDANTSS approach with (p, q) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) or (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The difference with the baseline methods is also significant for all cases in terms of TSNRs and SNRt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' One exception lies on SNRπ with dataset B with the noise level of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 % of xmax, where the second best is achieved by the combination backcor+SPOQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We stress out that in such problems, correct estimations of sparse signal and baseline are usually more important than kernel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The per- formance of PENDANTSS for the two penalty parameters (p, q) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2), (1, 2) is dependent on the datasets and the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' In terms of sparse signal recovery SNRs and TSNRs, PENDANTSS with (p, q) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) achieves slightly higher performance than PENDANTSS with (p, q) = (1, 2) for dataset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' However, its outcomes are notably lower for dataset B, a less sparse signal, while remaining the second best method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For dataset A, both PENDANTSS methods have similar baseline estimation accuracy, while for dataset B, PENDANTSS (p, q) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) performs better with lower noise level and PENDANTSS (p, q) = (1, 2) better with greater noise level, with a difference of SNR of about 2 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' As for the estimation of SNRπ, PENDANTSS with (p, q) = (1, 2) performs the best for all four cases with little difference for dataset A but a larger difference for more challenging cases with dataset B and higher noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Considering various SPOQ parameters is indeed beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' According to the presented simulation results, PENDANTSS with (p, q) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) is better for datasets with sparser, well- separable peaks whereas PENDANTSS with (p, q) = (1, 2) for more challenging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Graphical details on the quality of estimated peaks are provided as supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' CONCLUSION AND PERSPECTIVES We propose to solve a complicated joint sparse signal blind deconvolution and additive trend problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Our method handles smooth trend removal by exploiting their low-pass property and simplifies the problem into a blind deconvolution prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The blind deconvolution problem uses the recent SPOQ 0 20 40 60 80 100 120 140 160 180 200 220 0 1 2 3 4 5 6 7 (a) Dataset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 180 200 0 5 10 15 20 25 (b) Sparse spike signal for dataset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 180 200 220 0 2 4 6 8 10 12 (c) Dataset B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 180 200 0 5 10 15 20 25 30 (d) Sparse spike signal for dataset B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Unknown sparse signal s (b) and (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' in (a) and (c) observation y (blue) and baseline t (black) (bottom) for datasets A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Signal A has 10 spikes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 % of sparsity) while signal B has 20 spikes (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 % of sparsity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' TABLE I NUMERICAL RESULTS ON DATASETS A AND B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' SNR QUANTITIES IN DB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' BEST PERFORMING METHOD FOLLOWED BY **, SECOND BY *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Dataset A Dataset B Noise level σ (% of xmax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 % SNRs backcor+SOOT 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 backcor+SPOQ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4 PENDANTS (1, 2) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5* 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2* 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2** 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4** PENDANTS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3** 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2** 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5* 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6* TSNRs backcor+SOOT 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3 backcor+SPOQ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 PENDANTS (1, 2) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4* 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1* 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0** 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7** PENDANTS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7** 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='8** 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0* 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1* SNRt backcor+SOOT 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='8 backcor+SPOQ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='8 PENDANTS (1, 2) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5** 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='8** 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4* 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0** PENDANTS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6** 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0** 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6** 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9* SNRπ backcor+SOOT 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='8 backcor+SPOQ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1* PENDANTS (1, 2) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0** 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4** 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='9** 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2** PENDANTS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='75, 2) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='0** 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5* 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5* 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='4±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='5 sparse penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Simulation results confirm that PENDANTSS outperforms comparable methods on typical sparse analytical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Further works include its validation on a variety of other sparse spike signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The appropriate parameters for the sparsity-promoting norm ratio penalty ought to be investigated, for instance with respect to the alleged signal sparsity or peak separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' PENDANTSS Matlab code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='com/paulzhengfr/PENDANTSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 5 REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Boll, “Suppression of acoustic noise in speech using spectral subtrac- tion,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Acoust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Speech Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 113–120, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Lynch, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', Physico-Chemical Analysis of Industrial Catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' A Practical Guide to Characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' ´Edition Technip, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Marmin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Castella, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pesquet, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duval, “Sparse signal reconstruction for nonlinear models via piecewise rational optimization,” Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 179, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 107835, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Gauthier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duval, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pesquet, “Optimization of synthesis oversampled complex filter banks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 3827–3843, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duval, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duarte, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Jutten, “An overview of signal processing issues in chemical sensing,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Acoust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Speech Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', Vancouver, BC, Canada, May 26-31, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 8742–8746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [6] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Wong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Sun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Yan, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chen, “Robust enhanced trend filtering with unknown noise,” Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 180, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 107889, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [7] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Mazet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Carteret, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Brie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Idier, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Humbert, “Background removal from spectra by designing and minimising a non-quadratic cost function,” Chemometr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 76, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 121–133, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [8] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Ning, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Selesnick, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duval, “Chromatogram baseline estimation and denoising using sparsity (BEADS),” Chemometr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 139, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 156–167, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Amari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Douglas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Cichocki, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Yang, “Multichannel blind deconvolution and equalization using the natural gradient,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' IEEE Workshop Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Wireless Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', 1997, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 101– 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Krishnan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Tay, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Fergus, “Blind deconvolution using a normalized sparsity measure,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pattern Recogn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 21-25, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 233–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chaudhuri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Velmurugan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Rameshan, “Blind deconvolution methods: A review,” in Blind Image Deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Methods and Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Springer, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 37–60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [12] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Sun and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Donoho, “Convex sparse blind deconvolution,” PREPRINT, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Repetti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pham, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duval, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chouzenoux, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pesquet, “Euclid in a taxicab: Sparse blind deconvolution with smoothed ℓ1/ℓ2 regularization,” IEEE Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 539–543, May 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Cherni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chouzenoux, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Duval, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pesquet, “SPOQ ℓp- over-ℓq regularization for sparse signal recovery applied to mass spec- trometry,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 68, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 6070–6084, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Bolte, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Sabach, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Teboulle, “Proximal alternating linearized minimization fon nonconvex and nonsmooth problems,” Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pro- gramm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 146, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 459–494, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chouzenoux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pesquet, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Repetti, “A block coordinate variable metric forward-backward algorithm,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Global Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 457–485, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Selesnick, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Graber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pfeil, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Barbour, “Simul- taneous low-pass filtering and total variation denoising,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 62, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1109–1124, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Hurley and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Rickard, “Comparing measures of sparsity,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 4723–4741, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Sun, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Babu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Palomar, “Majorization-minimization algo- rithms in signal processing, communications, and machine learning,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 65, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 794–816, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chouzenoux, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Martin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pesquet, “A local MM subspace method for solving constrained variational problems in image recovery,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Imaging Vision, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Conn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Gould, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Toint, Trust-Region Methods, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' MOS-SIAM Series on Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Society for Industrial Mathemat- ics, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Bauschke and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Combettes, Convex analysis and monotone operator theory in Hilbert spaces, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' CMS books in mathe- matics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Springer, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Chouzenoux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Pesquet, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Repetti, “Variable metric forward- backward algorithm for minimizing the sum of a differentiable function and a convex function,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 162, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 107–132, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [24] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Hien, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Gillis, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Patrinos, “Inertial block proximal methods for non-convex non-smooth optimization,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 119, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 13–18, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 5671–5681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Becker and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Fadili, “A quasi-Newton proximal splitting method,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Neur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 3-6, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2618–2626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Condat, “Fast projection onto the simplex and the l1 ball,” Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Programm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 158, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1-2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 575–585, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1 PENDANTSS: Supplementary Material I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' PROOF OF THEOREM 1 FOR ALGORITHM 1 We first provide a useful majorant metric matrix property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Lemma 1 There exists (λ, λ) ∈]0, +∞[2 such that for every k ∈ N, and for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' , I}, � λIdN ⪯ A1,ρk,i(sk, πk) ⪯ λIdN, λ ≤ Λ2(sk) ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (A1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Direct consequence of [14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2] and [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We then show that Algorithm 1 satisfies two essential descent properties, that are key for the convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Lemma 2 There exists (µ1, µ2) ∈]0, +∞[ such that, for every k ∈ N, the following descent properties hold: Ω(sk+1, πk) ≤ Ω(sk, πk) − µ1 2 ||sk+1 − sk||2, (A2) Ω(sk+1, πk+1)≤Ω(sk+1, πk) − µ2 2 ||πk+1 − πk||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (A3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We remind that the objective function Ω is defined in (12), with g specified in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' By construction, sk+1 ∈ ¯Bq,ρ ∩ C1 for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' , I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Summing the majoration (14) and the first inequality in (18) yields: Ω(sk+1, πk) ≤ f(sk, πk)−(γ−1 s,k − 1 2)∥sk −sk+1∥2 A1,ρ(sk,πk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' We notice that f(sk, πk) = Ω(sk, πk) since sk ∈ C1 and πk ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Using Lemma 1 and the range assumption on γs,k allows to show (A2) for µ1 = λγ/(2 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Again by construction, πk+1 ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Summing inequalities (19) and (20) leads to: Ω(sk+1, πk+1) ≤ f(sk+1, πk)− (γ−1 π,k − 1 2)Λ2(sk+1)∥πk+1 − πk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Here again, we use f(sk+1, πk) = Ω(sk+1, πk) as sk+1 ∈ C1 and πk ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The descent property (A3) is obtained by using Lemma 1, the range constraint on γπ,k, and setting µ2 = λ¯γ(2 − ¯γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The rest of the proof of Theorem 1 is obtained by following the same lines than the one of [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' ADDITIONAL RESULTS Figures 2 and 3 provide additional insights into PEN- DANTSS restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Dataset A in Figure 2-(a) presents sparse and well-isolated peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Accurate peak restoration is secured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Peak shapes are well recovered (left-hand zoom), and the estimated trend matches well the actual baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' As a consequence, peak features that are computed with respect to the trend (height, area) are likely to be well-estimated with PENDANTSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' The less sparse Dataset B in Figure 2-(b) shows that the separation and the height of close peaks are accurately matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Some overshoot in trend estimation can be noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' It is however not likely to drastically affect relative peak height or area computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Retrieved spikes are exposed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' For Dataset A, well-separated spikes are accurately recovered using PEN- DANTSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Estimated amplitudes and locations are almost indistinguishable from the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' This is exemplified for the less sparse Dataset B in Figure 3-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Isolated peaks are well-estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' However, some spikes (for instance around index 175) for Dataset B in Figure 3-(b) remain unelucidated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Three contiguous spikes are estimated, instead of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Such an ambiguous solution is typical to source separation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (a) Dataset A reconstruction and trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (b) Dataset B reconstruction and trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 2: Ground truth (thick black line) and proposed estimation results (thin blue line), and the baseline t (dashed dot) and the signal s ∗ p (continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (a) Dataset A sparse spike signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' (b) Dataset B sparse spike signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' 3: Ground truth (black line with circle marker) and proposed estimation results (blue line with cross marker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='01514v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} +page_content='SP] 4 Jan 2023' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfjP3H/content/2301.01514v1.pdf'} diff --git a/I9E2T4oBgHgl3EQfUQfh/content/tmp_files/2301.03812v1.pdf.txt b/I9E2T4oBgHgl3EQfUQfh/content/tmp_files/2301.03812v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e36b24147234ccacf4be8e630323102ef80c7f7 --- /dev/null +++ b/I9E2T4oBgHgl3EQfUQfh/content/tmp_files/2301.03812v1.pdf.txt @@ -0,0 +1,2053 @@ +Draft version January 11, 2023 +Preprint typeset using LATEX style emulateapj v. 12/16/11 +DETECTING ISOLATED STELLAR-MASS BLACK HOLES BY THE Roman TELESCOPE +Sedighe Sajadian1 +Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran +Kailash C. Sahu2 +Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +and +Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA +Draft version January 11, 2023 +Abstract +Isolated Stellar-Mass BlackHoles (ISMBHs) are potentially discernible through microlensing obser- +vations because they are expected to be long-duration microlensing events. In this work, we study +detecting and characterizing ISMBHs with the Roman observations. We simulate a big ensemble of +these events as seen by Roman and estimate the errors in the physical parameters of the lens objects, +including their masses, distances, and proper motions through calculating Fisher and Covariance +matrices. Since the ∼2.3-year time gap between Roman’s first three observing seasons and the others +may lower the efficiency of realizing microlensing events and characterizing ISMBHs, we additionally +consider a scenario where we add a small amount of additional observations –one hour of observations +every 10 days when the Bulge is observable during the large time gap– which is equivalent to a total +of about one additional day of observations with the Roman telescope. +These extra observations +increase Roman’s efficiency for characterizing ISMBHs by ∼ 1-2% and, more importantly, improve +the robustness of the results by avoiding possible degenerate solutions. By considering uniform, and +power-law mass functions (dN/dM ∝ M −α, α = 2, 1, 0.5) for ISMBHs in the range of [2, 50]M⊙, we +conclude that the Roman telescope will determine the physical parameters of the lenses within < 5% +uncertainty, with efficiencies of 21%, and 16-18%, respectively. By considering these mass functions, +we expect that the Roman telescope during its mission will detect and characterize 3-4, 15-17 and +22-24 ISMBHs through astrometric microlensing, with the relative errors for all physical parameters +less than 1, 5, 10%, respectively. Microlensing events owing to ISMBHs with a mass ≃ 10-25M⊙ +and located close to the observer with Dl ≲ 0.5Ds while the source is inside the Galactic disk can be +characterized with least errors. +Subject headings: (cosmology:) gravitational lensing; astrometry; techniques: photometric; methods: +numerical +1. INTRODUCTION +A black hole (BH) refers to a massive object where +the escape velocity from it exceeds the speed of light. +Therefore, a BH can not reflect any light. However, it +radiates what is called the Hawking radiation (Hawking +1974), which is generally faint (Malyshev et al. 2022; +Auffinger 2022). +Their formation mechanisms are as follows: (a) BHs +can be formed by the death of massive stars with an ini- +tial mass higher than 20M⊙ (Bailyn et al. 1998; Fryer +& Kalogera 2001; Bambi 2018). (b) The interstellar gas +at the centre of massive galaxies can directly collapse to +form massive BHs (Volonteri 2010; Haiman 2013; Wise +et al. 2019). (c) Initial spatial fluctuations in the early +universe (during the first second after the Big Bang) +could potentially lead to the formation of primordial BHs +as proposed by S. Hawking (Hawking 1971). +BHs are generally classified based on their mass +into three categories: +(i) Super-massive BHs, +(ii) +Intermediate-Mass BHs (IMBHs), and (iii) Stellar-Mass +BHs. +1 Email: s.sajadian@iut.ac.ir +2 Email: ksahu@stsci.edu +The first class—the super-massive BHs—have masses +M ≥ 105M⊙. These objects can be found at the centers +of massive galaxies (such as the Milky Way Galaxy, and +M87), bright quasars, and Active Galactic Nuclei (AGN). +These massive objects can be detected and characterized +by tracking stars near massive galaxies’ centre (Volonteri +et al. 2021). +The second class—the IMBHs—have masses in the +range of 100-105 M⊙ and are thought to reside at cen- +tres of globular clusters (Koliopanos 2017; Greene et al. +2020). One method to indirectly detect these objects is +through gravitational waves caused by the merging of +these massive objects (Abbott et al. 2016, 2017). +At- +tempts have also been made to detect IMBHs through +astrometric microlensing of background stars caused by +the IMBHs (Kains et al. 2016, 2018). +The third class—the stellar-mass BHs—form after the +gravitational collapse of massive stars. +These objects +have masses as high as a few tens of solar mass. The num- +ber of such BHs in our galaxy has been predicted to be +more than 10 million (Shapiro & Teukolsky 1983; Lam- +berts et al. 2018). +The lowest-mass confirmed stellar- +mass BHs have a mass in the range of 3-4.5 M⊙ (Thomp- +son et al. 2019; Jayasinghe et al. 2021), whereas the most +arXiv:2301.03812v1 [astro-ph.GA] 10 Jan 2023 + +2 +Sajadian and Sahu +massive neutron stars (NSs) confirmed up to now have +masses of ≲ 2M⊙ (Fonseca et al. 2021), so there is a mass +gap between confirmed NSs and stellar-mass BHs (see, +e.g., Gao et al. 2022). +Stellar-mass BHs in binary systems can be detected +either through transient X-rays emitted by the accretion +of matter (from companions or close objects) onto the +BHs’ surface, or through observations of Doppler shifts +in the spectra of stellar companions orbiting the BHs, +or through both of them (Webster & Murdin 1972). +In these systems, the Doppler shifts provide radial +velocity measurements which are used to determine the +dynamic masses of BHs. +Up to now, more than 65 +stellar-mass BHs have been discovered in binary systems +and through X-ray transient observations, mostly in +our galaxy 3 (Corral-Santana et al. 2016). This method +is restricted only to cases where the stellar-mass BHs +are in binary systems with luminous companion objects, +thus ISMBHs cannot be detected by this method. +A unique and powerful method for discovering ISMBHs +is gravitational microlensing, which refers to a temporary +enhancement in the brightness of a background star while +passing behind a massive foreground object (the so-called +gravitational lens) (Einstein 1936; Liebes 1964; Refsdal +1964). In this phenomenon, the lens could be completely +dark. Hence, microlensing observations can reveal the +existence of dark (or faint) and massive compact objects, +e.g., stellar-mass BHs, even ones located outside of our +galaxy (Paczynski 1986; Sajadian & Rahvar 2012; Sahu +et al. 2017). +The important observing issue is that the photometric +light curve alone is not sufficient to calculate the physi- +cal parameters of the lens, such as its mass, distance and +proper motion. However, by additionally measuring the +parallax effect and astrometric shift in the source star +position which is proportional to the angular Einstein +radius, θE, a length-scale in the lensing formalism (see, +e.g., Walker 1995; Hog et al. 1995; Miyamoto & Yoshii +1995; Dominik & Sahu 2000)), the lensing degeneracy can +be resolved. Instead of measuring the astrometric mo- +tion of the source star, the interferometry observations +by even ground-based telescopes can resolve the lensing +images. This leads to a direct measurement of θE, which +also resolves the lensing degeneracy (Dong et al. 2019; +Zang et al. 2020). Measuring finite source effects in tran- +sit, caustic-crossing and high-magnification microlensing +events is another method to estimate θE and resolve the +lensing degeneracy (An et al. 2002). +The first unambiguous detection of an ISMBH in the +Galactic disk has been reported recently based on the +combined observations by the Hubble Space Telescope +(HST) and ground-based telescopes in the microlensing +event OGLE-2011-BLG-0462 (Sahu et al. 2022). There +were some claims that this long-duration microlensing +event could also be due to lower-mass objects (Lam +et al. 2022), but recently Mroz et al. (2022) have shown +that the lower mass estimates come from systematic er- +rors and the lens mass should be ≃ 7M⊙. There were +other reports of microlensing events due to ISMBHs, but +their lensing parameters or the nature of the lens objects +were not determined uniquely (Mao et al. 2002; Bennett +3 https://www.astro.puc.cl/BlackCAT/ +et al. 2002; Agol et al. 2002; Poindexter et al. 2005; Lu +et al. 2016).The Optical Gravitational Lensing Experi- +ment group (OGLE) (Udalski et al. 2015; Udalski 2003) +has also found 13 long-duration microlensing events from +observations in the years 2001-2009 which were due to +white dwarfs, neutron stars, or black holes (Wyrzykowski +et al. 2016). +In this work, we aim to study the possible detection +and characterization of ISMBHs by the Roman mission. +The Nancy Grace Roman Telescope will observe the +Galactic-bulge field during six 62-day seasons in its +5-year mission (Penny et al. 2019). +Even though its +observing strategy is aimed at detecting free-floating +planets and exoplanets beyond the snow line, we expect +that the Roman telescope will also detect microlensing +events due to other lens objects (Sajadian 2021a,b). +Additionally, +because of high photometric accuracy +during microlensing observations, it can resolve some +second-order perturbations (Bagheri et al. 2019; Sa- +jadian & Salehi 2020). +Roman is also expected to +detect ISMBHs through observations of long-duration +microlensing events. +The relatively long lifespan of +the Roman mission is very appropriate for detecting +long-duration microlensing events and measuring both +annual parallax effects and astrometric trajectories of +source stars. +The scheme of the paper is as follows. In Section 2, +we explain all the details for simulating astrometric mi- +crolensing events as seen by the Roman telescope. In Sec- +tion 3, we first explain how to calculate Fisher and Co- +variance matrices for photometry and astrometry mea- +surements by Roman from microlensing events due to +ISMBHs. Then, we illustrate the results of our simula- +tions and statistical calculations. Finally, in Section 4, +we briefly review our results and conclusions. +2. FORMALISM +Here we review the known formalism for astrometric +microlensing. We start with ignoring the parallax effect +but add this at a later stage. The temporary enhance- +ment in the stellar brightness due to the gravitational +lensing of a point-like and massive object which is called +the magnification factor versus time, t, is given by (see, +e.g., Gaudi 2012; Tsapras 2018): +A(t) = +u2 + 2 +u +√ +u2 + 4 +, +u = +� +u2 +0 + +�t − t0 +tE +�2, +(1) +where, u is the lens-source distance projected on the sky +plane and normalized to the Einstein radius (i.e., RE the +radius of the image ring at the complete alignment), u0 +is the lens impact parameter (the smallest lens-source +distance), and t0 is the time of the closest approach. +The Einstein crossing time, tE, represents the lensing +timescale which is given by: +tE = +θE +µrel,⊙ += +1 +µrel,⊙ +� +Ml πrel κ, +(2) +Here, Ml is the lens mass, κ = 8.14 mas.M−1 +⊙ +is a con- +stant, and πrel = au +� +1/Dl −1/Ds +� +is the relative parallax +amplitude, and Dl, Ds are the lens and source distances + +Detecting stellar-mass black holes by Roman +3 +Fig. 1.— Two examples of simulated magnification curves. The left panels show the magnification curves with (dashed curves) and +without (dotted curves) the parallax effect. The right panels show the corresponding astrometric motions of the source stars (blue curves), +lens objects (magenta curves), and their relative motions (dark red curves) projected on the sky plane. The synthetic data are taken with +the Roman telescope. The observable parameters used to make them are mentioned at the top of their lightcurves and astrometric plots. +from the observer. We note that θE = RE +� +Dl is an an- +gular length-scale in the lensing formalism. +µrel,⊙ is the size of the relative lens-source angular veloc- +ity. If we ignore the observer’s motion around the Sun, +the relative velocity vector (with respect to the Sun) is +given by: +µrel,⊙ = µs − µl = vs − v⊙ +Ds +− vl − v⊙ +Dl +, +(3) +where, vs, vl, and v⊙ are the source, lens and the Sun +velocity vectors projected on the sky plane. In Appendix +A, we explain how to convert the stellar velocities from +the Galactic coordinate frame to the observer frame. +Parallax effect: We know that the observer (here, +the Roman telescope) rotates around the Sun, so the real +relative lens-source angular velocity will be a function of +time and is given by: +µrel(t) = µrel,⊙ + πrel +au vo(t), +(4) +vo being the velocity vector of the observer with respect +to the Sun projected on the sky plane as explained in +Appendix A 4. Hence, the observer’s rotation around the +Sun, which is a function of time, causes the relative lens- +source angular velocity to be a function of time, and as +a result, it makes a periodic perturbation in the magnifi- +cation curve, the so-called parallax effect (Gould 1994). +By considering this effect in the lensing formalism, the +normalized source-lens angular displacement (which de- +termines the magnification factor) versus time is given +by: +u = u0 +� +− sin ξ +cos ξ +� ++ t − t0 +tE +� +cos ξ +sin ξ +� ++ πE +au +� t +t0 +dt +� +vo,n1 +vo,n2 +� +(5) +where, πE = πrel/θE which is a dimensionless parameter, +and ξ is the angle between the relative source-lens +trajectory and the direction of increasing Galactic +longitude, i.e. n1 (as defined in Appendix A) which is +given by tan ξ = µrel,⊙,n2/µrel,⊙,n1. +4 For projection of the observer orbit on the sky plane, first +we should project the observer orbit on the Galactic plane by a +rotation 60◦ around the intersection line of the orbital plane and +the Galactic plane. + +te(days) =134.7, Q(mas) =2.77, TE =0.007 +Magnification +19.75 +Magnification + parallax +19.80 +119.95 +149 +W1 +20.00 +20.05 +20.10 +0 +2 +3 +1 +4 +5 +time(yrs)Uo =0.75, mbase(mag) =20.11, to(years) =2.6 +-6 +4 +position(mas) +2 +0 +2 +source(undeflected) + parallax + source(deflected) + parallax +4 +-lens - source(undeflected) + parallax + Lens ＋ parallax + Deflection +6 +-20 +-10 +0 +10 +20 +30 +x position(mas)te(days) =113.9, Qe(mas) =2.64, TE =0.013 +Magnification +17.8 +Magnification + parallax +18.0 +18.2 +18.4 +18.6 +18.8 +19.0 +19.2 +0 +1 +2 +3 +4 +5 +time(yrs)Uo =0.16, mbase(mag) =19.25, to(years) =1.0 +source(undeflected) + parallax +source(deflected) + parallax +-20 + lens - source(undeflected) + parallax + Lens + parallax +-15 +-Deflection +position(mas) +10 +-5 +y +0 +5 +10 +-40 +-30 +-20 +-10 +0 +10 +20 +x position(mas)4 +Sajadian and Sahu +Fig. 2.— Same as Figure 1, but by considering extra observations, one-hour observations of the Galactic bulge every 10 days when the +Bulge is observable during the ∼2.3-year time gap, with the Roman telescope. These extra data points are depicted in green color. + +Uo =0.64, mbase(mag) =15.9, to(years) =2.8 +source(undeflected) + parallax +source(deflected) + parallax +-10 + lens - source(undeflected) + parallax +Lens + parallax + Deflection +5 +position(mas) +0 +y +5 +10 +-7.5 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +x position(mas)te(days) =249.4, Q(mas) =5.19, TE =0.013 +17.4 +Magnification +Magnification + parallax +17.6 + magnitude +17.8 +18.0 +18.2 +149 +M +18.4 +18.6 +18.8 +0 +1 +2 +3 +4 +5 +time(yrs)Uo =0.17, mbase(mag) =18.85, to(years) =3.7 +-30 +source(undeflected) + parallax +source(deflected) + parallax +lens - source(undeflected) + parallax +20 +Lens + parallax +Deflection +position(mas) +10 +0 +y +10 +20 +0 +5 +10 +15 +x position(mas)te(days) =125.2, Qe(mas) =3.03, Te =0.024 +Magnification +19.5 +Magnification + parallax + magnitude +19.6 +19.7 +W149 +19.8 +19.9 +0 +2 +3 +4 +5 +time(yrs)Uo =0.41, mbase(mag) =19.87, to(years) =2.1 +source(undeflected) + parallax +6 +source(deflected) + parallax + lens - source(undeflected) + parallax +.4 +: Lens + parallax +Deflection +position(mas) +2 +0 +2 +4 +6 +-20 +-10 +0 +10 +x position(mas)te(days) =275.5, Qe(mas) =1.99, TE =0.009 +15.3 +Magnification +Magnification + parallax +15.4 +15.5 +15.6 +15.7 +15.8 +15.9 +0 +1 +2 +3 +4 +5 +time(yrs)Detecting stellar-mass black holes by Roman +5 +According to the literature, we could define πE as a vec- +tor which is parallel with the relative lens-source proper +motion, i.e., +πE = +� +πn1, πn2 +� += πE +� +cos ξ, sin ξ +� +. +(6) +The initial parameters that can be derived from the +simple form of microlensing lightcurves (Eq. 1) are t0, u0, +and tE . In observations toward the Galactic bulge, most +of the source stars are located in the Galactic bulge, at a +distance Ds = 8 kpc from us. Measuring tE gives us only +a relation between the lens mass, the lens distance, and +the relative lens-source angular velocity, even by fixing +the source distance. +However, discerning the parallax +effect in the lightcurve allows us to measure the vector +of the parallax amplitude, πE, which is still not enough +to resolve the lensing degeneracy completely. +Astrometric microlensing: One way to resolve this +degeneracy and determine these parameters specially for +long-duration microlensing events due to ISMBHs is re- +solving the source angular trajectory projected on the +sky plane: +θs(t) = θs,0(t) + +u +u2 + 2θE, +(7) +where, the last term is the astrometric shift in the ap- +parent brightness center of the source star which is an- +other result of the lensing effect. In the lensing formal- +ism where a background star is lensed by a point-like +and massive lens object, two distorted images are formed +whose brightness center does not coincide with the source +center. +We note that this astrometric shift is propor- +tional to the Einstein angular radius which is a function +of the lens mass and its distance (see, e.g., Miyamoto & +Yoshii 1995; Dominik & Sahu 2000). +In Equation 7, θs,0(t), is the position vector of the +source star projected on the sky plane as a function of +time as seen by the observer, which is: +θs,0(t) = θs,0(t0) + µs(t − t0) − 1 +Ds +� t +t0 +vo(t)dt, +(8) +where, the first term, θs,0(t0) = u0 θE +� +− sin ξ, cos ξ +� +, +is the source position on the sky plane at the time of +the closest approach with respect to the lens position +(i.e., the coordinate center). The second term specifies a +straight line over the sky plane. The last term, which is +related to the effect of the observer’s motion around the +Sun on the source position, is mostly very small because +of the large source distance from the observer. This can +be clearly seen by comparing the blue dotted lines (which +do not take the parallax effect into account) and the blue +dashed lines (which take the parallax effect into account) +in the right panels of Figures 1 and 2. This term makes a +periodic perturbation on the source trajectory projected +on the sky plane. +The lens also has a similar angular trajectory projected +on the sky plane, given by +θl(t) = µl(t − t0) − 1 +Dl +� t +t0 +vo(t)dt. +(9) +Here, we have set the lens location at the coordinate +center at the time of the closest approach. However, in +most of the gravitational microlensing events the lens +objects are dark and their angular trajectories cannot be +determined. We note that +u(t) = θs(t) − θl(t) +θE +Let’s come back to Equation 7, which describes the +source angular trajectory projected on the sky plane ver- +sus time. In the case of astrometric observations where +we discern this source trajectory, the observables that +we can measure are: (a) θE, which is the angular size +of the Einstein ring radius, (b) µs, the angular source +velocity projected on the sky plane with respect to the +observer, and (c) the sign of the lens impact parameter +(e.g., Sajadian & Rahvar 2015). +However, for discerning the second one, observations +are necessary either long after or long before the lensing +event. Additionally, the astrometric shift due to lensing +effect has longer timescale than tE. It tends to zero as +u−1, while the magnification factor is proportional to +∝ u−4 for u ≫ 1 (see, e.g., +Dominik & Sahu 2000). +Its long timescale helps to resolve the time dependent +perturbations, such as the orbital-motion effect in binary +lensing (Sajadian 2014). +By measuring both astrometric shift due to microlens- +ing and the parallax effect in the magnification curve, +we determine tE, θE, πE, ξ, and µs, which allows us to +completely resolve the lensing degeneracy and determine +Dl, Ml, µrel,⊙, and µl. +We note that u0, and t0 are +measurable from magnification curve and are necessary +while modeling the astrometric motion of the source +star, but they are not directly involved in extracting the +physical parameters. +One class of microlensing events that are specially +interesting are the long-duration events caused by +ISMBHs. In these events, the astrometric shift in the +source angular position is considerable, because of the +large angular Einstein radius. +Additionally the paral- +lax effect potentially could be measured, because of long +duration of such events. We note that in most of the mi- +crolensing events due to ISMBHs, the finite source effect +is negligible, unless the lens passes over the source sur- +face. This is is rare since the impact parameter has to be +less than the normalized angular source radius, u0 < ρs, +ρs = θs/θE, where θs is the angular source radius, and +the large value of θE decreases ρs. +Using the introduced formalism, we simulate the astro- +metric microlensing events due to ISMBHs toward the +Galactic bulge. We also make the synthetic data points +according to the Roman observing strategy. In this re- +gard, the observing cadence is fixed at 15.16 min. The +observations include six 62-day seasons, three of them at +the first part of the Roman mission with a time interval +110-day between seasons, and three other seasons at the +end. +The photometric observations are mostly done in +the W149 filter. +This filter roughly corresponds to +W149 = (K + H + J) +� +3 (Montet et al. 2017). +Its +photometric precision, σm, is a function of the apparent +magnitude (Penny et al. 2019; Johnson et al. 2020). The +astrometric precision of the Roman observations also + +6 +Sajadian and Sahu +Fig. 3.— The normalized (fractional) distributions of tE, mbase, t0, and u0 for all the detected microlensing events by Roman are depicted +in green. Also, the normalized distributions of the events for which the physical parameters of the lenses are measurable with ≤ 5% relative +errors (after considering the extra observations during ∼2.3-year time gap) are shown as black stepped curves. The average values of these +parameters calculated from related distributions are mentioned in the legends. +strongly depends on the apparent stellar brightness. S. +Calchi Novati (private communication) has modelled +the Roman astrometric precisions for stars of different +magnitudes through Jitter +simulations and in this +work we use his simulations to determine the Roman +astrometric precision. He has used the Roman observing +strategy described by Penny et al. (2019), and calculated +the astrometry precisions through simulations (see, e.g., +Monet et al. 2010). +Two examples of simulated astrometric microlensing +events are shown in Figure 1. +The left panels show +the magnification curves with (dashed curves) and with- +out (dotted curves) the parallax effect and their cor- +responding right panels show the related astrometric +motions of the source stars (blue curves), lens objects +(magenta curves), and their relative motions (dark red +curves). The observable parameters which characterize +these events are specified at the top of the light curve +and astrometric motion plots. +There is a large time gap of ∼2.3 years between the first +three and the last three observing seasons of Roman5, +5 +https://roman.gsfc.nasa.gov/galactic_bulge_time_ +which lowers the detection efficiency of ISMBHs. If the +peak of the light curve happens during this large time +gap (which lasts ∼ 2.3 years), discerning such events will +have large uncertainties, and several degenerate models +will fit the data well. For instance, the peak of the first +lightcurve in the top panel of Figure 1 was not covered by +Roman data which would have been useful in correctly +determining the microlensing parameters, including the +parallax. +Hence, for a robust determination of the microlensing pa- +rameters, we additionally consider a case where the Ro- +man telescope observes the seven Galactic-bulge fields +for a total of one hour every 10 days when the Galac- +tic bulge is observable during the ∼2.3-year time gap. +Although these observations are sparse and use a total +of ∼1-day of Roman time, they are very helpful in dis- +cerning the source trajectories during the Roman mission +(see the first astrometry microlensing event in Figure 1), +and fully characterizing the microlensing lightcurves with +high confidence. In Figure 2, we show three more simu- +lated astrometric microlensing events due to ISMBHs as +detected by Roman, by assuming additional sparse obser- +domain_survey.html + +0.20 +303.01 +0.17 +556.93 +0.11 +0.08 +0.06 +0.03 +0.00 +1.5 +2.0 +3.0 +3.5 +log1o[te(days)]0.11 +20.06 +0.10 +19.31 +0.08 +Distribution +0.07 +0.06 +0.04 +0.03 +0.01 +0.00 +16 +17 +18 +19 +20 +21 +22 +23 +24 +mbase(mag)0.05 +2.47 +0.04 +2.45 +0.03 +Distribution +0.03 +0.02 +Normalized [ +0.02 +0.01 +0.01 +0.00 +0 +3 +2 +4 +5 +to(years)0.05 +0.49 +0.04 +0.36 +0.03 +Normalized Distribution +0.03 +0.02 +0.02 +0.01 +0.01 +0.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +UoDetecting stellar-mass black holes by Roman +7 +vations as discussed above. In these plots the extra data +points are depicted in green. We note that the astrom- +etry data points during the time gap (green points) can +jump to the observing seasons (shown by the red points) +because of the added noise in the simulated data. +In the next section, we evaluate the expected errors +in the physical parameters of ISMBHs detected through +astrometric microlensing by the Roman telescope. +3. OBSERVATIONS OF ASTROMETRIC MICROLENSING +To study detection and characterization of the ISMBHs +by microlensing observations during the Roman mission, +we extend our simulation and make a big ensemble of +detectable astrometric microlensing events. +Since the mass function for ISMBHs are not well de- +termined, so we consider several different mass functions. +A simple form for ISMBHs’ mass function is a uniform +function versus mass in the range of Ml ∈ [2, 50]M⊙. +Through modeling of black holes, Sicilia et al. (2022) +have found that the mass function of ISMBHs is almost +flat up to 50M⊙. Additionally, we examine three more +mass functions, which are log-uniform (dN/dM ∝ 1/M) +and power-law (dN/dM ∝ M −0.5, and dN/dM ∝ M −2) +ones. +Other parameters are determined according to their +distribution functions, as explained in the previous pa- +pers (see, e.g., +Sajadian & Poleski 2019; Moniez et al. +2017). For each mass function, we perform the simula- +tions two times, i.e., with and without considering sparse +observations during the ∼2.3-year time gap. +We choose the discernible events. Our criteria for de- +tectability are (i) ∆χ2(= +��χ2 +base − χ2 +real +��) > 800 for pho- +tometry data points, and (ii) at least three photome- +try data points above the baseline by 4σm, where σm +is the photometric accuracy. In Figure 3, we show the +normalized (fractional) distributions for four observing +parameters including tE, mbase, t0, u0 of detectable mi- +crolensing events due to ISMBHs (by considering a uni- +form mass function and sparse observations during the +large time gap) in green color. In order to study for which +kind of these microlensing events the physical parame- +ters of their lens objects are measurable with reasonable +accuracy, we also plot the corresponding normalized dis- +tributions of events with the relative errors in the lens +mass, distance, and proper motion ≤ 5% (black stepped +curves). +Accordingly, detectable microlensing events due to +ISMBHs have the average timescale of ⟨tE⟩ = 303 days +and their average source magnitude at the baseline is +⟨mbase⟩ = 20.1 mag. Discerning these microlensing light +curves (by adding extra observations during the large +time gap) does not highly depend on the time of the +closest approach and the lens impact parameter. +The +events with measurable physical parameters of their lens +objects have on average smaller lens impact parameters +(by 0.13), and mostly happen during either three first or +three last observing seasons of the Roman telescope. +For each discernible event, we determine the errors in +the physical parameters of microlenses through calculat- +ing Fisher and Covariance matrices (see, e.g., Boutreux +& Gould 1996; Gould & Salim 1999; Sajadian 2015). In +this regard, we make several simple assumptions which +are listed here. +(i) We separate the photometry and +astrometry measurements completely and calculate two +Fisher matrices corresponding to these measurements, +A, and B for each event. (ii) We assume that the lens- +ing parameters such as t0, u0, tE, and ξ are determined +through photometry observations well and their real val- +ues are used for astrometric modeling. In fact, the photo- +metric accuracy is better than the astrometric accuracy. +(iii) We ignore the parallax effect on the source trajec- +tories, which are too small to be measured (compare the +dotted and dashed blue lines in right panels in Figures +1, and 2). +(iv) We ignore the finite source effects on +both microlensing lightcurves and astrometric shifts in +the source position. (v) We assume that the source dis- +tances from the observer, Ds, are determined by other +observations, and we do not need to tune them through +microlensing observations. For instance, the Gaia obser- +vations provide stellar parallax distances for some source +stars. +Photometry and astrometry Fisher matrices are: +Aij = +N +� +k=1 +1 +σ2m(tk) +∂2ms(tk) +∂pi∂pj +, +Bij = +N +� +k=1 +1 +σ2a(tk) +�∂2θs,n1(tk) +∂qi ∂qj ++ ∂2θs,n2(tk) +∂qi ∂qj +� +, (10) +where, ms(tk) = mbase − 2.5 log10 +� +fbA(tk) + 1 − fb +� +is +the apparent source magnitude at the given time tk. fb +is the blending factor in W149 filter, mbase is the base- +line magnitude without lensing effect in that filter (its +distribution for detectable events is shown in the second +panel of Figure 3). pis, and qis are observable parameters +that affect on photometry and astrometry measurements +(ms, θs), respectively. +Observable parameters: A microlensing light curve +by considering the parallax effect can be modeled with 7 +parameters which are: pi ∈ t0, u0, tE, ξ, fb, mbase, πE. +The finite source effect can be ignored in long-duration +microlensing events due to ISMBHs, so we put aside this +effect while calculating A. The source apparent trajec- +tory on the sky plane can be modeled with 3 parameters: +qi ∈ θE, µs,n1, µs,n2. +We calculate Fisher matrices numerically. +Their in- +verses (i.e., covariance matrices, A−1 and B−1) are de- +rived using the Python module Numpy 6. +The square +roots of diagonal elements are the errors in the observ- +able parameters, e.g., σpi = +� +A−1 +ii and σqi = +� +B−1 +ii , +and non-diagonal elements are the correlation coefficients +between errors in the parameters. +Taking these errors into account, we determine the errors +in the physical parameters of ISMBHs, which is explained +in the next subsection. +3.1. Errors in the physical parameters +According to Equation 2, the lens mass and its error +as a function of observable parameters are: +6 https://numpy.org/ + +8 +Sajadian and Sahu +Fig. 4.— The fractional distributions of the relative errors in the normalized parallax amplitude, the lens mass, the lens distance, and the +lens proper motion for a big samples of microlensing events due to ISMBHs detectable by the Roman telescope with (green distributions) +and without (black step ones) considering sparse observations when the Galactic bulge is observable during the large time gap. +The +vertical (solid, dashed and dotted) lines show the thresholds of the relative errors 10%, 5%, and 1%, respectively. The samples due to both +distributions have the same entrances. +Ml = θE +κ πE +, +σMl =Ml +��σθE +θE +�2 ++ +�σπE +πE +�2 +, +(11) +where σMl, σθE, and σπE are the error in the lens mass, +error in the angular Einstein radius, and the error in +normalized parallax amplitude, respectively. +We note +that there is no correlation between σπE and σθE, because +these two parameters are determined from photometry +and astrometry Fisher matrices independently. The next +parameter is the lens distance which is given by: +1 +Dl += 1 +Ds ++ πE θE +au +, +σDl =Dl +Ds − Dl +Ds +σMl +Ml +, +(12) +Here, we assume that the error in source distance is very +small and can be ignored. The last parameter is the lens +angular velocity components which are: +µl,n1 =µs,n1 − θE +tE +cos ξ, +µl,n2 =µs,n2 − θE +tE +sin ξ, +(13) +Accordingly, the errors in the lens angular velocity com- +ponents are given by: +σ2 +l,n1 = σ2 +s,n1 +µ2 +rel,⊙ cos2 ξ +��σθ +θE +�2 + +�σt +tE +�2 ++ +� σξ +cot ξ +�2 − 2σt +tE +σξ +cot ξ +ˆ +A−1 +ij +� +, +σ2 +l,n2 = σ2 +s,n2 +µ2 +rel,⊙ sin2 ξ +��σθ +θE +�2 + +�σt +tE +�2 ++ +� σξ +tan ξ +�2 − 2σt +tE +σξ +tan ξ +ˆ +A−1 +ij +� +. +(14) +where, σl,i, σs,i are the errors in ith component of the lens +and source angular velocity projected on the sky plane, +and ˆ +A−1 +ij = A−1 +ij / +� +A−1 +ii A−1 +jj is the correlation coefficient + +0.11 +0.09 +Distribution +0.06. +.. +.......... + Normalized +0.04 +0.02 +0.00 +0 +2 +3 +5 +0g10l0E +/ TE(%))0.12 +0.11 +0.09 +0.08 +istribu +20.07 +30.05 +0.04 +0.03 +................. +0.01 +0.00 +0 +3 +5 +[(%)W / W0]0160l0.13 +0.12 +≤0.10 +ibutior +0.08- +Distril +D +0.07 +lormalized +0.05 +Z0.03 +0.02 - +0.00 +2 +log10[g D// Di(%)0.15 +0.13 +0.12 +ution +0.10 +istribu +- +20.08 +D +8 +30.07 +Normalize +0.05 +0.03 +0.02 +0.00 +0 +3 +5 +l0g10[0μ/μ(%)]Detecting stellar-mass black holes by Roman +9 +TABLE 1 +Statistical information about simulated microlensing events due to ISMBHs detectable with the Roman telescope +by assuming different ISMBHs mass functions. +σtE +� +tE +σπE +� +πE +σθE +� +θE +σMl +� +Ml +σDl +� +Dl +σµs +� +µs +σµl +� +µl +ϵm(%) +Ne,BHs +dN/dM = const +No observations during the time gap +≤ 1% +23.60 +7.50 +85.56 +6.11 +21.15 +99.67 +5.16 +4.21 +2 +≤ 5% +53.26 +24.35 +99.32 +24.08 +50.59 +99.98 +22.32 +19.37 +11 +≤ 10% +65.91 +34.86 +99.88 +34.77 +64.11 +100.00 +33.00 +29.29 +17 +Sparse observations during the time gap +≤ 1% +30.81 +8.32 +83.15 +6.93 +22.99 +99.66 +6.10 +5.15 +4 +≤ 5% +63.72 +25.66 +98.85 +25.40 +52.37 +99.98 +24.27 +21.48 +17 +≤ 10% +76.00 +36.14 +99.75 +36.05 +65.26 +99.99 +34.98 +31.54 +24 +dN/dM ∝ M−0.5 +No observations during the time gap +≤ 1% +22.20 +7.52 +75.03 +5.34 +19.43 +99.68 +4.38 +3.64 +2 +≤ 5% +49.88 +22.52 +98.29 +21.98 +45.97 +99.99 +20.34 +17.57 +12 +≤ 10% +62.02 +31.84 +99.65 +31.66 +59.07 +99.99 +29.94 +26.30 +18 +Sparse observations during the time gap +≤ 1% +25.77 +7.70 +71.64 +5.65 +19.49 +99.66 +4.94 +4.22 +3 +≤ 5% +56.57 +22.29 +97.40 +21.82 +45.21 +99.98 +20.81 +18.25 +15 +≤ 10% +69.18 +31.33 +99.32 +31.15 +57.54 +99.99 +30.05 +26.75 +22 +dN/dM ∝ M−1 +No observations during the time gap +≤ 1% +21.89 +7.52 +71.23 +5.11 +18.85 +99.67 +4.19 +3.51 +3 +≤ 5% +48.83 +22.00 +97.82 +21.34 +44.75 +99.98 +19.79 +17.00 +14 +≤ 10% +61.02 +31.20 +99.56 +30.97 +57.68 +99.99 +29.15 +25.56 +21 +Sparse observations during the time gap +≤ 1% +24.48 +7.55 +67.89 +5.30 +18.56 +99.67 +4.56 +3.92 +3 +≤ 5% +54.23 +21.38 +96.75 +20.81 +43.22 +99.99 +19.85 +17.33 +15 +≤ 10% +66.95 +30.00 +99.17 +29.79 +55.42 +100.00 +28.79 +25.61 +22 +dN/dM ∝ M−2 +No observations during the time gap +≤ 1% +21.75 +7.15 +59.45 +4.51 +16.60 +99.69 +3.83 +3.34 +3 +≤ 5% +49.50 +19.65 +95.20 +18.83 +39.16 +99.99 +17.93 +15.53 +12 +≤ 10% +62.21 +27.56 +98.69 +27.24 +50.89 +100.00 +26.30 +23.07 +18 +Sparse observations during the time gap +≤ 1% +21.00 +7.57 +62.54 +4.46 +17.91 +99.67 +3.71 +3.31 +3 +≤ 5% +46.86 +21.33 +96.58 +20.35 +42.25 +99.98 +18.81 +16.08 +15 +≤ 10% +58.57 +29.98 +99.28 +29.61 +54.68 +100.00 +27.93 +24.39 +23 +Note. — Each entry represents the persentage of simulated events with the desired relativel error (specified in its row) be less +than the given threshold (determined in its column). ϵm is the Roman efficiency for measuing the lens mass, distance, and its proper +motion with the relative errors less than the given threshold. The last column reports the estimated number of ISMBHs that can +be detected in the Roman observations by considering different mass functions, as explained in Subsection 3.4. +between errors in tE, and ξ. The errors in the lens and +source proper motion can be determined using the errors +in their components. +3.2. Results +The normalized distributions for four relevant param- +eters (i.e., tE, mbase, t0, and u0) for simulated events +whose relative errors in the lens mass, distance and +proper motion are ≤ 5%, are shown in Figure 3 with +black step lines. Accordingly, longer microlensing events +from brighter source stars, whose times of the closest ap- +proach happen during either the first three or the last +three observing seasons are more favourable for the mea- +surement of the physical parameters of the lens objects +with reasonable accuracy. +In Figure 4, we show the normalized distributions +of the relative errors in the physical parameters of +ISMBHs (as microlenses), resulting from Monte Carlo +simulations, by considering a uniform mass function +for ISMBHs. Green and black distributions are related +to detectable events by the Roman telescope with and +without considering sparse data points during the time +gap, respectively. +These parameters are the normalized +parallax amplitude, the lens mass, the lens distance and +the lens proper motion. The threshold amounts of the +relative errors in the given parameters of 10%, 5%, and +1% are depicted with solid, dashed, and dotted lines. +Accordingly, adding extra observations during the time +gap (one hour of observations every 10 days when the +Galactic bulge is observable) improves the relative errors +in all physical parameters, especially the lens distance +from the observer. +For numerical evaluation, in Table 1 we give the per- + +10 +Sajadian and Sahu +centages of simulated detectable events with the rela- +tive errors (specified in the first row) less than the given +thresholds (i.e., 1, 5, 10% as mentioned in the first col- +umn) are reported. Hence, sparse observations during +the time gap improve the Roman efficiencies by ∼ 1%, +∼ 2%, and ∼ 2% for measuring the physical parameters +by the relative errors less than 1, 5, 10%, respectively. +In 20-25% detectable events, the lens mass can be de- +termined with the relative error less than 5%. +These +events have smaller relative errors in the lens distance, +because the factor (Ds − Dl)/Ds is less than one. +The source proper motion can be determined by +monitoring the source positions during 6 observing +seasons (with a 15 min cadence) of the Roman mission +even without taking sparse data points during the +∼2.3-year time gap very well. +Nevertheless, the lens +proper motion can be determined with the relative error +less than 5% in 19-24% of these events. +Even though ISMBHs produce long-duration mi- +crolensing events, +which are suitable for discerning +the annual parallax effects, the normalized parallax +amplitude, πE, decreases with increasing the lens mass. +Hence, the parallax effect can be discerned in these +long-duration microlensing events with the relative +errors less than 5% only in 21-26% of all detectable +events. +In order to determine which kinds of ISMBHs might +be well characterized through astrometric microlensing +observations with the Roman telescope, we show the de- +pendence of the relative errors in the lens mass, the lens +distance, its proper motion, and the parallax amplitude +to Ml, xls, Ds, and mbase in Figure 5, in different panels, +respectively. For these plots, we only use the events with +the relative errors less than 5%. There are several factors +which determine their dependencies. +According to the first panel, the relative error in the +lens mass minimize when Ml ≃ 10-25M⊙. +Increasing +the lens mass has two against effects: (i) The lens mass +enhances the Einstein crossing time and decreases the +average photometry errors. +Because more data points +are taken while the source is being lensed, and less data +points are recorded over the baseline. (ii) Enhancing the +lens mass decreases the normalized parallax amplitude +πE significantly, and makes hard measure it (see the dot- +ted red step line in the top panel). This point was also +expressed by Karolinski & Zhu (2020) and while model- +ing OGLE-2006-BLG-044 microlensing event. For that +reason, the optimum value for the lens mass with least +errors is neither the least (2-3 solar mass), nor the most +(40-50 solar mass). The relative error in the lens distance +decreases with the lens mass. In fact, by increasing the +lens mass xls enhances to keep the Einstein crossing times +close to reasonable values for detection. +The relative error in the lens proper motion weakly de- +pends on the lens mass. In fact, σtE/tE is an increasing +function versus the lens mass. By fixing the observing +time and cadence (considering a determined observing +platform) and increasing tE, its error increases. In to- +tal, the relative errors in the lens physical parameters +enhance with the lens mass slowly. +The second panel of Figure 5 shows the relative errors +in the lens mass, lens distance, its proper motion, and +the parallax amplitude versus xls = Dl/Ds. The smaller +xls make larger πE and θE, with smaller observing errors. +That increases the relative error in the lens mass versus +xls. However, this enhancement is slower in the relative +error in the lens distance, because of the factor (Ds − +Dl)/Ds in Equation 12. +In the next panel of Figure 5, we show the depen- +dence of the relative errors with the source distance from +the observer. The source distance decreases πE, and θE, +which increases the relative errors in the lens mass and +its distance. We note that decreasing the parallax am- +plitude increases both errors in the parallax amplitude, +and ξ. Comparing these panels, we find that the effect +of the source distance and the lens relative position (xls) +on the errors is higher than the effect of the lens mass. +In the last panel, the relative errors versus the apparent +magnitude of the source star at the baseline are depicted. +As shown here, they enhance with the source magnitude. +Both Roman photometric and astrometric errors increase +with the apparent magnitude of source stars. Worse ac- +curacies cause higher relative errors in the lens physical +parameters. +Therefore, long-duration microlensing events due to +ISMBHs with the mass Ml ≃ 10-25M⊙, close to the ob- +server (xls ≲ 0.5) while the source is inside the Galactic +disk (Ds ≲ 6kpc) can be characterized with the least +errors. +3.3. Different mass function for ISMBHs +We know that there is no accurate mass function +for ISMBHs based on observations yet, so we perform +the simulation by considering several mass functions for +ISMBHs, which are given in the following: +dN +dM =const., +dN +dM ∝1 +�√ +M, +dN +dM ∝M −1, +dN +dM ∝M −2. +(15) +The results from simulations based on each of these mass +functions are reported in Table 1. Accordingly, by chang- +ing ISMBHs mass function, the Roman efficiency to mea- +sure the lens physical parameters can change up to 2-7%. +Also, the first mass function makes more ISMBHs with +mass Ml ∈ [10, 25]M⊙ than other mass functions. So it +has larger efficiencies to measure the physical parameters +of lens objects than others. +In the next subsection, we do some statistical estima- +tions about detecting and characterizing such events dur- +ing the Roman mission. +3.4. Statistical estimations +The number of microlensing events that the Ro- +man telescope will detect is Ne,tot = 27000, which were +estimated in Penny et al. (2019); Johnson et al. (2020). +Here, we want to evaluate what fraction of this total +number of microlensing events detectable by the Ro- +man telescope are due to ISMBHs. In this regard, there +are two factors: (i) the optical depth, and (ii) the av- +erage microlensing duration which are discussed in the + +Detecting stellar-mass black holes by Roman +11 +Fig. 5.— The dependence of the average relative errors in the lens mass (solid green lines), the lens distance (dashed blue lines), its +proper motion (dot-dashed magenta lines), and the normalized parallax amplitude (dotted red lines) versus the lens mass, the ratio of the +lens distance to the source distance from the observer (xls), the source distance, and the source apparent magnitude at the baseline. +following. +(i) The number of detectable microlensing events is pro- +portional to the optical depth. The microlensing optical +depth at a given line of sight (l, b) and one specified dis- +tance from the observer, (D), is proportional to the lens +mass Ml, because it is given by: +dτ(l, b, D) +dD += π θ2 +E n(l, b, D) D2, +(16) +where, (l, b) are the Galactic longitude and latitude, +respectively. n(l, b, D) is the number density of stars in +our galaxy which is the Galactic mass density divided by +the average stellar mass. +Accordingly, the ratio of the optical depth (and as a re- +sult the number of microlensing events) due to ISMBHs +to the overall optical depth due to all potential lens ob- +jects can be estimated by: +F1 = +� ∞ +20M⊙ +Ml η(Ml) dMl +� � ∞ +13MJ +Ml η(Ml) dMl,(17) +where,MJ is the Jupiter mass, η(Ml) is the initial mass +function in the Galactic disk. +In fact, F1 determines +the contribution of the ISMBHs in producing the effec- +tive lensing surface in comparison with the total lens- +ing surfaces covered by all possible Einstein rings. +In +Equation 17, we use the fact that stars with the initial +mass M > 20M⊙ will convert to black holes. We ignore +the contribution of black holes generated from primordial +fluctuations in the early universe. +In order to estimate F1, we take the initial mass +function from the Besan¸con model (Robin et al. 2003, +2012), and assume that all lens objects are inside the +Galactic disk. This mass function is η(Ml) ∝ M −1.6 +l +for +0.08 ≤ Ml(M⊙) ≤ 1, and η(Ml) ∝ M −3 +l +for Ml(M⊙) ≥ 1. +The stars with Ml > 20M⊙ are converted to ISMBHs. +For 13MJ < Ml < 0.08M⊙ we take the Brown dwarf +mass function, i.e., M −0.7 +l +(Muˇzi´c et al. 2015; Luhman +2004). We do not include free floating planets, because +of their negligible contribution. The upper limit should +in reality be the mass due to the most massive star in +the Galactic disk. +We set this upper limit to infinity, +because the mass function for M > 1M⊙ decreases as +M −3, so it tends to zero fast. +Accordingly, we find +F1 = 0.019. +(ii) The microlensing event rate is proportional to +� +ϵ(tE) +� +tE +� +, which specifies the inverse of the average du- +ration of microlensing events. +Here, ϵ(tE) is the +Ro- + +2.2 +2.1 +Relative Error +2.0 +1.9 +1.8 +1.7 +1.6 +10 +20 +30 +40 +M[Mo]2.8 +2.6 +2.4 + Error +2.2 +2.0 +Relative I +1.8 +1.6 +1.4 +1.2 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +XIs2.25 +2.00 +1.75 +Error +1.50 +Relative +1.25 +(0m, / M[%] +1.00 +(αD. / Di[%]> +0.75 +(0μ/ / μi[%]) +(O πe / TE[%]) +0.50 +0 +2 +4 +6 +8 +10 +12 +Ds(kpc)3.5 +(om / M[%]) + +Relative Error +2.5 +2.0 +1.5 +1.0 +16 +17 +18 +19 +20 +21 +22 +23 +24 +mbase(mag)12 +Sajadian and Sahu +man efficiency for detecting a microlensing event with the +specified time scale tE, and was kindly provided by M. +Penny. Since ISMBHs make longer microlensing events +than usual events, we expect this factor for ISMBHs to +be smaller than that due to all detectable microlensing +events due to all potential lens objects. We define an- +other factor: +F2 = +�ϵ(tE) +tE +� +BHs +� �ϵ(tE) +tE +� +Total +. +(18) +To estimate this factor, we simulate the microlensing +events detectable by the Roman telescope, and by adopt- +ing a uniform mass function for ISMBHs. However, we +tune the ratio of the number of ISMBHs to the number +of total objects ≃ 0.0001, as expected. In the simulation, +the lens objects can be brown dwarfs, main-sequence +stars and ISMBHs, and we obtain F2 = 0.15, 0.11 with +and without considering sparse observations during the +time gap, respectively. We note that considering extra +observations enables us to detect ISMBHs in shorter mi- +crolensing events (the average tE changes from 329 days +to 303 days). +Therefore, the Roman telescope roughly will detect +Ne,BHs = Ne,tot × F1 × F2 ≃ 56-77 microlensing events +due to ISMBHs (under the assumption that their masses +are uniformly distributed in the range of [2, 50]M⊙, +and their contribution with respect to all lens objects +is 0.0001). In 2-4, 11-17, and 17-24 of these events the +physical parameters of ISMBHs (including their mass, +distance and proper motion) can be determined with the +relative errors less than 1%, 5%, and 10%, respectively, +as reported in the last column of Table 1. +For other mass functions, i.e., dN/dM ∝ M −α with +α = 0.5, 1, 2, we get F2 = 0.16-0.13, 0.17-0.16, 0.18- +0.0.15 (with and without adding extra observations dur- +ing the time gap), respectively. The corresponding num- +ber of ISMBHs that can be detected and characterized +through the Roman observations are reported in Table +1. +4. CONCLUSIONS +In this work, we studied detection and characterization +of ISMBHs through astrometric microlensing to be done +by the upcoming microlensing survey by the Roman tele- +scope. +This telescope has been planned to detect mostly short- +duration microlensing events due to exoplanets beyond +the snow line of main-sequence stars and free-floating +exoplanets. +Nevertheless, the duration of its mission is long enough +to detect and characterize long-duration microlensing +events, and its astrometric accuracy is high enough to +discern the astrometric trajectories (and the dimensional +lensing-induced shifts) of source stars. +Here, we have done a comprehensive simulation of as- +trometric microlensing events due to ISMBHs that can +be discerned by the Roman telescope. For each simu- +lated event we have calculated Fisher and Covariance +matrices for photometry and astrometry measurements +separately, and estimated the errors in observable param- +eters, and physical parameters of ISMBHs as well. +Since the long time gap between Roman’s first three +observing seasons and the other three seasons would limit +its efficiency and robustness for discerning and charac- +terizing ISMBHs, we considered a small amount of ad- +ditional observations when the Galactic bulge is visible +during this time gap, by adding one hour of observa- +tions (4 data points) every 10 days when the Galactic +bulge is detectable in our simulations. These additional +observations amount to a total of about one day of obser- +vations with Roman. We found that this small amount +of extra observations increases Roman’s efficiency of de- +tecting and characterizing ISMBHs by ∼ 1 − 2%, and, +more importantly, improve the robustness of the results +and help avoiding degenerate solutions. +We note that photometric follow-up of these microlens- +ing events with ground-based telescopes such as the Ru- +bin Observatory during the time gap should also be help- +ful.The ground-based images may suffer from blending, +but the higher-resolution images of Roman should help in +correctly estimating the blending factor, thus providing +useful data for better characterization of the microlens- +ing light curves. +For long-duration microlensing events due to ISMBHs, +the efficiency of Roman microlensing survey for measur- +ing the physical parameters of the lens by considering +different ISMBHs mass functions are summarized in Ta- +ble 1. +The efficiencies for measuring with better than 5% un- +certainty the lens mass, its distance, and its proper mo- +tion are 20-25%, 42-52%, and 19-24%, respectively, and +the efficiency of measuring all the three parameters with +better than 5% uncertainty is 16-21%. +ISMBHs produce long-duration microlensing events +which are appropriate for discerning the annual parallax. +On the other hand, the normalized parallax amplitudes +decrease with 1/√Ml. Therefore, πE can be measured +with the relative error less than 5% in only 21-26% of +these long-duration events. +The relative errors in the physical parameters of +ISMBHs increases with the source distance and xls = +Dl/Ds. The dependence of these relative errors to the +lens mass is relatively weak and by changing the lens +mass from 2 to 50 solar mass, these error changes less +than 1%. On the whole, the least relative errors in the +lens mass and its distance occurs when Ml ≃ 10-25M⊙, +xls ≲ 0.5, and Ds ≲ 6 kpc. +We also statistically estimated the total number +of microlensing events due to ISMBHs that can be +detected and characterized with the Roman telescope. +By assuming different mass functions for ISMBHs (given +in Equation 15) in the range of [2, 50]M⊙, we concluded +that this telescope will detect 56-77 long-duration +microlensing events due to ISMBHs during its mission. +Additionally, it can measure the physical parameters +of ISMBHs with the relative errors less than 1%, 5%, +and 10% in 3-4, 15-17, 22-24 of these events, respectively. +All simulations that have been done for this paper +are available at: +https://github.com/SSajadian54/ +AstrometryMicrolensing +Research efforts of KCS were supported by NASA +through grants from STScI, under proposal IDs 14783, +15318 and 16200. We thank the anonymous referee for +his/her careful and useful comments, which improved the + +Detecting stellar-mass black holes by Roman +13 +Fig. 6.— Figure shows the Galactic plane and two coordinate systems which are needed to project stellar velocities on the sky plane. +quality of the paper. +APPENDIX +TRANSFORMING COORDINATE SYSTEMS +In this section, we will review how to transform the stellar velocity from the Galactic coordinate frame to the observer +one and project them on the sky plane. +In this Figure, the horizontal and vertical black lines describe the Galactic plane and make a right-hand coordinate +system. We note that in this Figure the scales are not respected. +We consider a star in our galaxy with the galactic coordinate (l, b), i.e., the galactic longitude and latitude, respectively. +Three points of the Galactic center (GC), the star position projected on the Galactic plane (yellow star) and the observer +position (black filled point) make a triangle with the angles l, α, β, as shown in Figure 6. The length scales: Roc the +observer distance from the Galactic center, Ros the distance between the star position projected on the Galactic plane +and the observer, and Rsc which is the distance between the Galactic center and the projected stellar position on the +Galactic plane. Rsc can be given by: +Rsc = +� +R2oc + R2os − 2RosRoc cos(l). +(A1) +where, Ros = D⋆ cos(b), and D⋆ is the star distance from the observer. Using the sinuous law in a triangle, we can +derive the angle of β, as: +sin(β) = Ros +Rsc +sin(l). +(A2) +By having the Galactic longitude, we will calculate the angle of α as α = π − l − β. +In simulations, we determine the stellar velocities in the Galactic coordinate, i.e., (vU, vV, vW), which are toward the +Galactic center, in the direction of the Galactic rotation, and toward the Galactic north, respectively. These velocities +include the global rotational velocity which is a function of the stellar distance from the Galactic center (see, e.g., +Rahal et al. 2009), and velocity dispersion components which are functions of the stellar age, weakly mass, and the +Galactic latitude (Carlberg et al. 1985; Sajadian & Rahvar 2019; Sajadian et al. 2021). +In the lensing formalism, we need the projected components of stellar velocities on the sky plane. So we introduce +another coordinate frame, (x, y, z), which z-axis is parallel with W (toward the Galactic north), and (x, y) describes +the Galactic plane, as shown in Figure 6 with red vectors. We can easily convert the velocity components from Galactic +coordinate frame to this new coordinate system, (x, y, z), as: +vx =− cos(α) vU − sin(α) vV, +vy =+ sin(α) vU − cos(α) vV, +vz =vW, +(A3) +Note that stars are not in the Galactic disk and their line of sight (los) with respect to the Galactic plane make +the angle b, the Galactic latitude. So, we should apply another rotation around y-axis with −b angle to obtain the + +GC +y +1- V +B +α +Roc +-U +Observer14 +Sajadian and Sahu +components of stellar velocities projected on the sky plane normal to the line of sight toward the stellar position as: +vlos =cos(b) vx + sin(b) vz, +vn1 =vy, +vn2 =− sin(b) vx + cos(b) vz, +(A4) +n1 and n2 are two unit vectors describe the sky plane. For projection of the Sun velocity, α⊙ ≃ π − l, since β⊙ ≃ 0. +For the observer orbit around the Sun, we easily consider a circular orbit with the radius of the astronomical unit. +REFERENCES +Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, +Phys. Rev. Lett., 116, 241103, +doi: 10.1103/PhysRevLett.116.241103 +—. 2017, Phys. Rev. Lett., 118, 221101, +doi: 10.1103/PhysRevLett.118.221101 +Agol, E., Kamionkowski, M., Koopmans, L. V. E., & Blandford, +R. D. 2002, ApJ, 576, L131, doi: 10.1086/343758 +An, J. H., Albrow, M. D., Beaulieu, J. P., et al. 2002, ApJ, 572, +521, doi: 10.1086/340191 +Auffinger, J. 2022, arXiv e-prints, arXiv:2206.02672. +https://arxiv.org/abs/2206.02672 +Bagheri, F., Sajadian, S., & Rahvar, S. 2019, MNRAS, 490, 1581, +doi: 10.1093/mnras/stz2682 +Bailyn, C. D., Jain, R. K., Coppi, P., & Orosz, J. A. 1998, ApJ, +499, 367, doi: 10.1086/305614 +Bambi, C. 2018, Annalen der Physik, 530, 1700430, +doi: 10.1002/andp.201700430 +Bennett, D. P., Becker, A. C., Quinn, J. L., et al. 2002, ApJ, 579, +639, doi: 10.1086/342225 +Boutreux, T., & Gould, A. 1996, ApJ, 462, 705, +doi: 10.1086/177183 +Carlberg, R. G., Dawson, P. C., Hsu, T., & Vandenberg, D. A. +1985, ApJ, 294, 674, doi: 10.1086/163337 +Corral-Santana, J. M., Casares, J., Mu˜noz-Darias, T., et al. 2016, +A&A, 587, A61, doi: 10.1051/0004-6361/201527130 +Dominik, M., & Sahu, K. C. 2000, ApJ, 534, 213, +doi: 10.1086/308716 +Dong, S., M´erand, A., Delplancke-Str¨obele, F., et al. 2019, ApJ, +871, 70, doi: 10.3847/1538-4357/aaeffb +Einstein, A. 1936, Science, 84, 506, +doi: 10.1126/science.84.2188.506 +Fonseca, E., Cromartie, H. T., Pennucci, T. T., et al. 2021, ApJ, +915, L12, doi: 10.3847/2041-8213/ac03b8 +Fryer, C. L., & Kalogera, V. 2001, ApJ, 554, 548, +doi: 10.1086/321359 +Gao, S.-J., Li, X.-D., & Shao, Y. 2022, MNRAS, 514, 1054, +doi: 10.1093/mnras/stac1426 +Gaudi, B. S. 2012, ARA&A, 50, 411, +doi: 10.1146/annurev-astro-081811-125518 +Gould, A. 1994, ApJ, 421, L71, doi: 10.1086/187190 +Gould, A., & Salim, S. 1999, ApJ, 524, 794, doi: 10.1086/307843 +Greene, J. E., Strader, J., & Ho, L. C. 2020, ARA&A, 58, 257, +doi: 10.1146/annurev-astro-032620-021835 +Haiman, Z. 2013, in Astrophysics and Space Science Library, Vol. +396, The First Galaxies, ed. T. Wiklind, B. Mobasher, & +V. Bromm, 293, doi: 10.1007/978-3-642-32362-1_6 +Hawking, S. 1971, Monthly Notices of the Royal Astronomical +Society, 152, 75, doi: 10.1093/mnras/152.1.75 +Hawking, S. W. 1974, Nature, 248, 30, doi: 10.1038/248030a0 +Hog, E., Novikov, I. D., & Polnarev, A. G. 1995, A&A, 294, 287 +Jayasinghe, T., Stanek, K. Z., Thompson, T. A., et al. 2021, +MNRAS, 504, 2577, doi: 10.1093/mnras/stab907 +Johnson, S. A., Penny, M., Gaudi, B. S., et al. 2020, AJ, 160, 123, +doi: 10.3847/1538-3881/aba75b +Kains, N., Bramich, D. M., Sahu, K. C., & Calamida, A. 2016, +MNRAS, 460, 2025, doi: 10.1093/mnras/stw1137 +Kains, N., Calamida, A., Sahu, K. C., et al. 2018, ApJ, 867, 37, +doi: 10.3847/1538-4357/aae311 +Karolinski, N., & Zhu, W. 2020, MNRAS, 498, L25, +doi: 10.1093/mnrasl/slaa121 +Koliopanos, F. 2017, in XII Multifrequency Behaviour of High +Energy Cosmic Sources Workshop (MULTIF2017), 51. +https://arxiv.org/abs/1801.01095 +Lam, C. Y., Lu, J. R., Udalski, A., et al. 2022, ApJ, 933, L23, +doi: 10.3847/2041-8213/ac7442 +Lamberts, A., Garrison-Kimmel, S., Hopkins, P. F., et al. 2018, +MNRAS, 480, 2704, doi: 10.1093/mnras/sty2035 +Liebes, S. 1964, Phys. Rev., 133, B835, +doi: 10.1103/PhysRev.133.B835 +Lu, J. R., Sinukoff, E., Ofek, E. O., Udalski, A., & Kozlowski, S. +2016, ApJ, 830, 41, doi: 10.3847/0004-637X/830/1/41 +Luhman, K. L. 2004, ApJ, 617, 1216, doi: 10.1086/425647 +Malyshev, D., Moulin, E., & Santangelo, A. 2022, Phys. Rev. D, +106, 123020, doi: 10.1103/PhysRevD.106.123020 +Mao, S., Smith, M. C., Wo´zniak, P., et al. 2002, MNRAS, 329, +349, doi: 10.1046/j.1365-8711.2002.04986.x +Miyamoto, M., & Yoshii, Y. 1995, AJ, 110, 1427, +doi: 10.1086/117616 +Monet, D. G., Jenkins, J. M., Dunham, E. W., et al. 2010, arXiv +e-prints, arXiv:1001.0305. https://arxiv.org/abs/1001.0305 +Moniez, M., Sajadian, S., Karami, M., Rahvar, S., & Ansari, R. +2017, A&A, 604, A124, doi: 10.1051/0004-6361/201730488 +Montet, B. T., Yee, J. C., & Penny, M. T. 2017, PASP, 129, +044401, doi: 10.1088/1538-3873/aa57fb +Mroz, P., Udalski, A., & Gould, A. 2022, arXiv e-prints, +arXiv:2207.10729. https://arxiv.org/abs/2207.10729 +Muˇzi´c, K., Scholz, A., Geers, V. C., & Jayawardhana, R. 2015, +ApJ, 810, 159, doi: 10.1088/0004-637X/810/2/159 +Paczynski, B. 1986, ApJ, 304, 1, doi: 10.1086/164140 +Penny, M. T., Gaudi, B. S., Kerins, E., et al. 2019, ApJS, 241, 3, +doi: 10.3847/1538-4365/aafb69 +Poindexter, S., Afonso, C., Bennett, D. P., et al. 2005, ApJ, 633, +914, doi: 10.1086/468182 +Rahal, Y. R., Afonso, C., Albert, J. N., et al. 2009, A&A, 500, +1027, doi: 10.1051/0004-6361/200811515 +Refsdal, S. 1964, MNRAS, 128, 295, +doi: 10.1093/mnras/128.4.295 +Robin, A. C., Marshall, D. J., Schultheis, M., & Reyl´e, C. 2012, +A&A, 538, A106, doi: 10.1051/0004-6361/201116512 +Robin, A. C., Reyl´e, C., Derri`ere, S., & Picaud, S. 2003, A&A, +409, 523, doi: 10.1051/0004-6361:20031117 +Sahu, K. C., Anderson, J., Casertano, S., et al. 2017, Science, +356, 1046, doi: 10.1126/science.aal2879 +—. 2022, ApJ, 933, 83, doi: 10.3847/1538-4357/ac739e +Sajadian, S. 2014, MNRAS, 439, 3007, +doi: 10.1093/mnras/stu158 +—. 2015, AJ, 149, 147, doi: 10.1088/0004-6256/149/4/147 +—. 2021a, MNRAS, 508, 5991, doi: 10.1093/mnras/stab2942 +—. 2021b, MNRAS, 506, 3615, doi: 10.1093/mnras/stab1907 +Sajadian, S., & Poleski, R. 2019, ApJ, 871, 205, +doi: 10.3847/1538-4357/aafa1d +Sajadian, S., & Rahvar, S. 2012, MNRAS, 419, 124, +doi: 10.1111/j.1365-2966.2011.19671.x +—. 2015, MNRAS, 452, 2579, doi: 10.1093/mnras/stu1875 +—. 2019, IJPR, 19, 391, doi: 10.29252/ijpr.19.2.391 +Sajadian, S., Rahvar, S., & Kazemian, F. 2021, arXiv e-prints, +arXiv:2103.10593. https://arxiv.org/abs/2103.10593 +Sajadian, S., & Salehi, A. 2020, MNRAS, 498, 1298, +doi: 10.1093/mnras/staa2377 +Shapiro, S. L., & Teukolsky, S. A. 1983, Black holes, white +dwarfs, and neutron stars : the physics of compact objects +Sicilia, A., Lapi, A., Boco, L., et al. 2022, ApJ, 924, 56, +doi: 10.3847/1538-4357/ac34fb +Thompson, T. A., Kochanek, C. S., Stanek, K. Z., et al. 2019, +Science, 366, 637, doi: 10.1126/science.aau4005 + +Detecting stellar-mass black holes by Roman +15 +Tsapras, Y. 2018, Geosciences, 8, 365, +doi: 10.3390/geosciences8100365 +Udalski, A. 2003, AcA, 53, 291. +https://arxiv.org/abs/astro-ph/0401123 +Udalski, A., Szyma´nski, M. K., & Szyma´nski, G. 2015, +Acta Astronautica, 65, 1. https://arxiv.org/abs/1504.05966 +Volonteri, M. 2010, A&A Rev., 18, 279, +doi: 10.1007/s00159-010-0029-x +Volonteri, M., Habouzit, M., & Colpi, M. 2021, Nature Reviews +Physics, 3, 732, doi: 10.1038/s42254-021-00364-9 +Walker, M. A. 1995, ApJ, 453, 37, doi: 10.1086/176367 +Webster, B. L., & Murdin, P. 1972, Nature, 235, 37, +doi: 10.1038/235037a0 +Wise, J. H., Regan, J. A., O’Shea, B. W., et al. 2019, Nature, +566, 85, doi: 10.1038/s41586-019-0873-4 +Wyrzykowski, �L., Kostrzewa-Rutkowska, Z., Skowron, J., et al. +2016, MNRAS, 458, 3012, doi: 10.1093/mnras/stw426 +Zang, W., Dong, S., Gould, A., et al. 2020, ApJ, 897, 180, +doi: 10.3847/1538-4357/ab9749 + diff --git a/I9E2T4oBgHgl3EQfUQfh/content/tmp_files/load_file.txt b/I9E2T4oBgHgl3EQfUQfh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8cae6b08756190bd4dc51b0c1771cf693eac043b --- /dev/null +++ b/I9E2T4oBgHgl3EQfUQfh/content/tmp_files/load_file.txt @@ -0,0 +1,1330 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf,len=1329 +page_content='Draft version January 11, 2023 Preprint typeset using LATEX style emulateapj v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 12/16/11 DETECTING ISOLATED STELLAR-MASS BLACK HOLES BY THE Roman TELESCOPE Sedighe Sajadian1 Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran Kailash C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Sahu2 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA and Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Draft version January 11, 2023 Abstract Isolated Stellar-Mass BlackHoles (ISMBHs) are potentially discernible through microlensing obser- vations because they are expected to be long-duration microlensing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In this work, we study detecting and characterizing ISMBHs with the Roman observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We simulate a big ensemble of these events as seen by Roman and estimate the errors in the physical parameters of the lens objects, including their masses, distances, and proper motions through calculating Fisher and Covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Since the ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3-year time gap between Roman’s first three observing seasons and the others may lower the efficiency of realizing microlensing events and characterizing ISMBHs, we additionally consider a scenario where we add a small amount of additional observations –one hour of observations every 10 days when the Bulge is observable during the large time gap– which is equivalent to a total of about one additional day of observations with the Roman telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' These extra observations increase Roman’s efficiency for characterizing ISMBHs by ∼ 1-2% and, more importantly, improve the robustness of the results by avoiding possible degenerate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' By considering uniform, and power-law mass functions (dN/dM ∝ M −α, α = 2, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5) for ISMBHs in the range of [2, 50]M⊙, we conclude that the Roman telescope will determine the physical parameters of the lenses within < 5% uncertainty, with efficiencies of 21%, and 16-18%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' By considering these mass functions, we expect that the Roman telescope during its mission will detect and characterize 3-4, 15-17 and 22-24 ISMBHs through astrometric microlensing, with the relative errors for all physical parameters less than 1, 5, 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Microlensing events owing to ISMBHs with a mass ≃ 10-25M⊙ and located close to the observer with Dl ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5Ds while the source is inside the Galactic disk can be characterized with least errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Subject headings: (cosmology:) gravitational lensing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' astrometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' techniques: photometric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' methods: numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' INTRODUCTION A black hole (BH) refers to a massive object where the escape velocity from it exceeds the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Therefore, a BH can not reflect any light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' However, it radiates what is called the Hawking radiation (Hawking 1974), which is generally faint (Malyshev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Auffinger 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Their formation mechanisms are as follows: (a) BHs can be formed by the death of massive stars with an ini- tial mass higher than 20M⊙ (Bailyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Fryer & Kalogera 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Bambi 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (b) The interstellar gas at the centre of massive galaxies can directly collapse to form massive BHs (Volonteri 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Haiman 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Wise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (c) Initial spatial fluctuations in the early universe (during the first second after the Big Bang) could potentially lead to the formation of primordial BHs as proposed by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Hawking (Hawking 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' BHs are generally classified based on their mass into three categories: (i) Super-massive BHs, (ii) Intermediate-Mass BHs (IMBHs), and (iii) Stellar-Mass BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 1 Email: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='sajadian@iut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='ir 2 Email: ksahu@stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='edu The first class—the super-massive BHs—have masses M ≥ 105M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' These objects can be found at the centers of massive galaxies (such as the Milky Way Galaxy, and M87), bright quasars, and Active Galactic Nuclei (AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' These massive objects can be detected and characterized by tracking stars near massive galaxies’ centre (Volonteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The second class—the IMBHs—have masses in the range of 100-105 M⊙ and are thought to reside at cen- tres of globular clusters (Koliopanos 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Greene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' One method to indirectly detect these objects is through gravitational waves caused by the merging of these massive objects (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2016, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' At- tempts have also been made to detect IMBHs through astrometric microlensing of background stars caused by the IMBHs (Kains et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2016, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The third class—the stellar-mass BHs—form after the gravitational collapse of massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' These objects have masses as high as a few tens of solar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The num- ber of such BHs in our galaxy has been predicted to be more than 10 million (Shapiro & Teukolsky 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Lam- berts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The lowest-mass confirmed stellar- mass BHs have a mass in the range of 3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 M⊙ (Thomp- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Jayasinghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2021), whereas the most arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03812v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='GA] 10 Jan 2023 2 Sajadian and Sahu massive neutron stars (NSs) confirmed up to now have masses of ≲ 2M⊙ (Fonseca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2021), so there is a mass gap between confirmed NSs and stellar-mass BHs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Stellar-mass BHs in binary systems can be detected either through transient X-rays emitted by the accretion of matter (from companions or close objects) onto the BHs’ surface, or through observations of Doppler shifts in the spectra of stellar companions orbiting the BHs, or through both of them (Webster & Murdin 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In these systems, the Doppler shifts provide radial velocity measurements which are used to determine the dynamic masses of BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Up to now, more than 65 stellar-mass BHs have been discovered in binary systems and through X-ray transient observations, mostly in our galaxy 3 (Corral-Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This method is restricted only to cases where the stellar-mass BHs are in binary systems with luminous companion objects, thus ISMBHs cannot be detected by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' A unique and powerful method for discovering ISMBHs is gravitational microlensing, which refers to a temporary enhancement in the brightness of a background star while passing behind a massive foreground object (the so-called gravitational lens) (Einstein 1936;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Liebes 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Refsdal 1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In this phenomenon, the lens could be completely dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Hence, microlensing observations can reveal the existence of dark (or faint) and massive compact objects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', stellar-mass BHs, even ones located outside of our galaxy (Paczynski 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Sajadian & Rahvar 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Sahu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The important observing issue is that the photometric light curve alone is not sufficient to calculate the physi- cal parameters of the lens, such as its mass, distance and proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' However, by additionally measuring the parallax effect and astrometric shift in the source star position which is proportional to the angular Einstein radius, θE, a length-scale in the lensing formalism (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Walker 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Hog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Miyamoto & Yoshii 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Dominik & Sahu 2000)), the lensing degeneracy can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Instead of measuring the astrometric mo- tion of the source star, the interferometry observations by even ground-based telescopes can resolve the lensing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This leads to a direct measurement of θE, which also resolves the lensing degeneracy (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Zang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Measuring finite source effects in tran- sit, caustic-crossing and high-magnification microlensing events is another method to estimate θE and resolve the lensing degeneracy (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The first unambiguous detection of an ISMBH in the Galactic disk has been reported recently based on the combined observations by the Hubble Space Telescope (HST) and ground-based telescopes in the microlensing event OGLE-2011-BLG-0462 (Sahu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' There were some claims that this long-duration microlensing event could also be due to lower-mass objects (Lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2022), but recently Mroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (2022) have shown that the lower mass estimates come from systematic er- rors and the lens mass should be ≃ 7M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' There were other reports of microlensing events due to ISMBHs, but their lensing parameters or the nature of the lens objects were not determined uniquely (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Bennett 3 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='puc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='cl/BlackCAT/ et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Agol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Poindexter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='The Optical Gravitational Lensing Experi- ment group (OGLE) (Udalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Udalski 2003) has also found 13 long-duration microlensing events from observations in the years 2001-2009 which were due to white dwarfs, neutron stars, or black holes (Wyrzykowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In this work, we aim to study the possible detection and characterization of ISMBHs by the Roman mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The Nancy Grace Roman Telescope will observe the Galactic-bulge field during six 62-day seasons in its 5-year mission (Penny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Even though its observing strategy is aimed at detecting free-floating planets and exoplanets beyond the snow line, we expect that the Roman telescope will also detect microlensing events due to other lens objects (Sajadian 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Additionally, because of high photometric accuracy during microlensing observations, it can resolve some second-order perturbations (Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Sa- jadian & Salehi 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Roman is also expected to detect ISMBHs through observations of long-duration microlensing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The relatively long lifespan of the Roman mission is very appropriate for detecting long-duration microlensing events and measuring both annual parallax effects and astrometric trajectories of source stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The scheme of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Section 2, we explain all the details for simulating astrometric mi- crolensing events as seen by the Roman telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Sec- tion 3, we first explain how to calculate Fisher and Co- variance matrices for photometry and astrometry mea- surements by Roman from microlensing events due to ISMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Then, we illustrate the results of our simula- tions and statistical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Finally, in Section 4, we briefly review our results and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' FORMALISM Here we review the known formalism for astrometric microlensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We start with ignoring the parallax effect but add this at a later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The temporary enhance- ment in the stellar brightness due to the gravitational lensing of a point-like and massive object which is called the magnification factor versus time, t, is given by (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Gaudi 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Tsapras 2018): A(t) = u2 + 2 u √ u2 + 4 , u = � u2 0 + �t − t0 tE �2, (1) where, u is the lens-source distance projected on the sky plane and normalized to the Einstein radius (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', RE the radius of the image ring at the complete alignment), u0 is the lens impact parameter (the smallest lens-source distance), and t0 is the time of the closest approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The Einstein crossing time, tE, represents the lensing timescale which is given by: tE = θE µrel,⊙ = 1 µrel,⊙ � Ml πrel κ, (2) Here, Ml is the lens mass, κ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='14 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='M−1 ⊙ is a con- stant, and πrel = au � 1/Dl −1/Ds � is the relative parallax amplitude, and Dl, Ds are the lens and source distances Detecting stellar-mass black holes by Roman 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='— Two examples of simulated magnification curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The left panels show the magnification curves with (dashed curves) and without (dotted curves) the parallax effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The right panels show the corresponding astrometric motions of the source stars (blue curves), lens objects (magenta curves), and their relative motions (dark red curves) projected on the sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The synthetic data are taken with the Roman telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The observable parameters used to make them are mentioned at the top of their lightcurves and astrometric plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that θE = RE � Dl is an an- gular length-scale in the lensing formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' µrel,⊙ is the size of the relative lens-source angular veloc- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' If we ignore the observer’s motion around the Sun, the relative velocity vector (with respect to the Sun) is given by: µrel,⊙ = µs − µl = vs − v⊙ Ds − vl − v⊙ Dl , (3) where, vs, vl, and v⊙ are the source, lens and the Sun velocity vectors projected on the sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Appendix A, we explain how to convert the stellar velocities from the Galactic coordinate frame to the observer frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Parallax effect: We know that the observer (here, the Roman telescope) rotates around the Sun, so the real relative lens-source angular velocity will be a function of time and is given by: µrel(t) = µrel,⊙ + πrel au vo(t), (4) vo being the velocity vector of the observer with respect to the Sun projected on the sky plane as explained in Appendix A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Hence, the observer’s rotation around the Sun, which is a function of time, causes the relative lens- source angular velocity to be a function of time, and as a result, it makes a periodic perturbation in the magnifi- cation curve, the so-called parallax effect (Gould 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' By considering this effect in the lensing formalism, the normalized source-lens angular displacement (which de- termines the magnification factor) versus time is given by: u = u0 � − sin ξ cos ξ � + t − t0 tE � cos ξ sin ξ � + πE au � t t0 dt � vo,n1 vo,n2 � (5) where, πE = πrel/θE which is a dimensionless parameter, and ξ is the angle between the relative source-lens trajectory and the direction of increasing Galactic longitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' n1 (as defined in Appendix A) which is given by tan ξ = µrel,⊙,n2/µrel,⊙,n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 4 For projection of the observer orbit on the sky plane, first we should project the observer orbit on the Galactic plane by a rotation 60◦ around the intersection line of the orbital plane and the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' te(days) =134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='7, Q(mas) =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='77, TE =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='007 Magnification 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 Magnification + parallax 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='80 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='95 149 W1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='10 0 2 3 1 4 5 time(yrs)Uo =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75, mbase(mag) =20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11, to(years) =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 6 4 position(mas) 2 0 2 source(undeflected) + parallax source(deflected) + parallax 4 lens - source(undeflected) + parallax Lens ＋ parallax Deflection 6 20 10 0 10 20 30 x position(mas)te(days) =113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='9, Qe(mas) =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='64, TE =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='013 Magnification 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 Magnification + parallax 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 0 1 2 3 4 5 time(yrs)Uo =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='16, mbase(mag) =19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='25, to(years) =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 source(undeflected) + parallax source(deflected) + parallax 20 lens - source(undeflected) + parallax Lens + parallax 15 Deflection position(mas) 10 5 y 0 5 10 40 30 20 10 0 10 20 x position(mas)4 Sajadian and Sahu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='— Same as Figure 1, but by considering extra observations, one-hour observations of the Galactic bulge every 10 days when the Bulge is observable during the ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3-year time gap, with the Roman telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' These extra data points are depicted in green color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Uo =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='64, mbase(mag) =15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='9, to(years) =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 source(undeflected) + parallax source(deflected) + parallax 10 lens - source(undeflected) + parallax Lens + parallax Deflection 5 position(mas) 0 y 5 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 x position(mas)te(days) =249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4, Q(mas) =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='19, TE =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='013 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 Magnification Magnification + parallax 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 magnitude 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 149 M 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 0 1 2 3 4 5 time(yrs)Uo =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='17, mbase(mag) =18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='85, to(years) =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='7 30 source(undeflected) + parallax source(deflected) + parallax lens - source(undeflected) + parallax 20 Lens + parallax Deflection position(mas) 10 0 y 10 20 0 5 10 15 x position(mas)te(days) =125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2, Qe(mas) =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03, Te =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='024 Magnification 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 Magnification + parallax magnitude 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='7 W149 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='9 0 2 3 4 5 time(yrs)Uo =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='41, mbase(mag) =19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='87, to(years) =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='1 source(undeflected) + parallax 6 source(deflected) + parallax lens - source(undeflected) + parallax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 : Lens + parallax Deflection position(mas) 2 0 2 4 6 20 10 0 10 x position(mas)te(days) =275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5, Qe(mas) =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99, TE =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='009 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3 Magnification Magnification + parallax 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='9 0 1 2 3 4 5 time(yrs)Detecting stellar-mass black holes by Roman 5 According to the literature, we could define πE as a vec- tor which is parallel with the relative lens-source proper motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', πE = � πn1, πn2 � = πE � cos ξ, sin ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (6) The initial parameters that can be derived from the simple form of microlensing lightcurves (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 1) are t0, u0, and tE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In observations toward the Galactic bulge, most of the source stars are located in the Galactic bulge, at a distance Ds = 8 kpc from us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Measuring tE gives us only a relation between the lens mass, the lens distance, and the relative lens-source angular velocity, even by fixing the source distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' However, discerning the parallax effect in the lightcurve allows us to measure the vector of the parallax amplitude, πE, which is still not enough to resolve the lensing degeneracy completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Astrometric microlensing: One way to resolve this degeneracy and determine these parameters specially for long-duration microlensing events due to ISMBHs is re- solving the source angular trajectory projected on the sky plane: θs(t) = θs,0(t) + u u2 + 2θE, (7) where, the last term is the astrometric shift in the ap- parent brightness center of the source star which is an- other result of the lensing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In the lensing formal- ism where a background star is lensed by a point-like and massive lens object, two distorted images are formed whose brightness center does not coincide with the source center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that this astrometric shift is propor- tional to the Einstein angular radius which is a function of the lens mass and its distance (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Miyamoto & Yoshii 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Dominik & Sahu 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Equation 7, θs,0(t), is the position vector of the source star projected on the sky plane as a function of time as seen by the observer, which is: θs,0(t) = θs,0(t0) + µs(t − t0) − 1 Ds � t t0 vo(t)dt, (8) where, the first term, θs,0(t0) = u0 θE � − sin ξ, cos ξ � , is the source position on the sky plane at the time of the closest approach with respect to the lens position (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', the coordinate center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The second term specifies a straight line over the sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The last term, which is related to the effect of the observer’s motion around the Sun on the source position, is mostly very small because of the large source distance from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This can be clearly seen by comparing the blue dotted lines (which do not take the parallax effect into account) and the blue dashed lines (which take the parallax effect into account) in the right panels of Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This term makes a periodic perturbation on the source trajectory projected on the sky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The lens also has a similar angular trajectory projected on the sky plane, given by θl(t) = µl(t − t0) − 1 Dl � t t0 vo(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (9) Here, we have set the lens location at the coordinate center at the time of the closest approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' However, in most of the gravitational microlensing events the lens objects are dark and their angular trajectories cannot be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that u(t) = θs(t) − θl(t) θE Let’s come back to Equation 7, which describes the source angular trajectory projected on the sky plane ver- sus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In the case of astrometric observations where we discern this source trajectory, the observables that we can measure are: (a) θE, which is the angular size of the Einstein ring radius, (b) µs, the angular source velocity projected on the sky plane with respect to the observer, and (c) the sign of the lens impact parameter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Sajadian & Rahvar 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' However, for discerning the second one, observations are necessary either long after or long before the lensing event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Additionally, the astrometric shift due to lensing effect has longer timescale than tE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' It tends to zero as u−1, while the magnification factor is proportional to ∝ u−4 for u ≫ 1 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Dominik & Sahu 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Its long timescale helps to resolve the time dependent perturbations, such as the orbital-motion effect in binary lensing (Sajadian 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' By measuring both astrometric shift due to microlens- ing and the parallax effect in the magnification curve, we determine tE, θE, πE, ξ, and µs, which allows us to completely resolve the lensing degeneracy and determine Dl, Ml, µrel,⊙, and µl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that u0, and t0 are measurable from magnification curve and are necessary while modeling the astrometric motion of the source star, but they are not directly involved in extracting the physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' One class of microlensing events that are specially interesting are the long-duration events caused by ISMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In these events, the astrometric shift in the source angular position is considerable, because of the large angular Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Additionally the paral- lax effect potentially could be measured, because of long duration of such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that in most of the mi- crolensing events due to ISMBHs, the finite source effect is negligible, unless the lens passes over the source sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This is is rare since the impact parameter has to be less than the normalized angular source radius, u0 < ρs, ρs = θs/θE, where θs is the angular source radius, and the large value of θE decreases ρs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Using the introduced formalism, we simulate the astro- metric microlensing events due to ISMBHs toward the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We also make the synthetic data points according to the Roman observing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In this re- gard, the observing cadence is fixed at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='16 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The observations include six 62-day seasons, three of them at the first part of the Roman mission with a time interval 110-day between seasons, and three other seasons at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The photometric observations are mostly done in the W149 filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This filter roughly corresponds to W149 = (K + H + J) � 3 (Montet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Its photometric precision, σm, is a function of the apparent magnitude (Penny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The astrometric precision of the Roman observations also 6 Sajadian and Sahu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='— The normalized (fractional) distributions of tE, mbase, t0, and u0 for all the detected microlensing events by Roman are depicted in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Also, the normalized distributions of the events for which the physical parameters of the lenses are measurable with ≤ 5% relative errors (after considering the extra observations during ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3-year time gap) are shown as black stepped curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The average values of these parameters calculated from related distributions are mentioned in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' strongly depends on the apparent stellar brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Calchi Novati (private communication) has modelled the Roman astrometric precisions for stars of different magnitudes through Jitter simulations and in this work we use his simulations to determine the Roman astrometric precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' He has used the Roman observing strategy described by Penny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (2019), and calculated the astrometry precisions through simulations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Monet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Two examples of simulated astrometric microlensing events are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The left panels show the magnification curves with (dashed curves) and with- out (dotted curves) the parallax effect and their cor- responding right panels show the related astrometric motions of the source stars (blue curves), lens objects (magenta curves), and their relative motions (dark red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The observable parameters which characterize these events are specified at the top of the light curve and astrometric motion plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' There is a large time gap of ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3 years between the first three and the last three observing seasons of Roman5, 5 https://roman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='gov/galactic_bulge_time_ which lowers the detection efficiency of ISMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' If the peak of the light curve happens during this large time gap (which lasts ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3 years), discerning such events will have large uncertainties, and several degenerate models will fit the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For instance, the peak of the first lightcurve in the top panel of Figure 1 was not covered by Roman data which would have been useful in correctly determining the microlensing parameters, including the parallax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Hence, for a robust determination of the microlensing pa- rameters, we additionally consider a case where the Ro- man telescope observes the seven Galactic-bulge fields for a total of one hour every 10 days when the Galac- tic bulge is observable during the ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3-year time gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Although these observations are sparse and use a total of ∼1-day of Roman time, they are very helpful in dis- cerning the source trajectories during the Roman mission (see the first astrometry microlensing event in Figure 1), and fully characterizing the microlensing lightcurves with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Figure 2, we show three more simu- lated astrometric microlensing events due to ISMBHs as detected by Roman, by assuming additional sparse obser- domain_survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='html 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='20 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='17 556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 log1o[te(days)]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='10 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08 Distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 16 17 18 19 20 21 22 23 24 mbase(mag)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 Distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 Normalized [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 0 3 2 4 5 to(years)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 Normalized Distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 UoDetecting stellar-mass black holes by Roman 7 vations as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In these plots the extra data points are depicted in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that the astrom- etry data points during the time gap (green points) can jump to the observing seasons (shown by the red points) because of the added noise in the simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In the next section, we evaluate the expected errors in the physical parameters of ISMBHs detected through astrometric microlensing by the Roman telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' OBSERVATIONS OF ASTROMETRIC MICROLENSING To study detection and characterization of the ISMBHs by microlensing observations during the Roman mission, we extend our simulation and make a big ensemble of detectable astrometric microlensing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Since the mass function for ISMBHs are not well de- termined, so we consider several different mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' A simple form for ISMBHs’ mass function is a uniform function versus mass in the range of Ml ∈ [2, 50]M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Through modeling of black holes, Sicilia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (2022) have found that the mass function of ISMBHs is almost flat up to 50M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Additionally, we examine three more mass functions, which are log-uniform (dN/dM ∝ 1/M) and power-law (dN/dM ∝ M −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5, and dN/dM ∝ M −2) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Other parameters are determined according to their distribution functions, as explained in the previous pa- pers (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Sajadian & Poleski 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Moniez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For each mass function, we perform the simula- tions two times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', with and without considering sparse observations during the ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3-year time gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We choose the discernible events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Our criteria for de- tectability are (i) ∆χ2(= ��χ2 base − χ2 real ��) > 800 for pho- tometry data points, and (ii) at least three photome- try data points above the baseline by 4σm, where σm is the photometric accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Figure 3, we show the normalized (fractional) distributions for four observing parameters including tE, mbase, t0, u0 of detectable mi- crolensing events due to ISMBHs (by considering a uni- form mass function and sparse observations during the large time gap) in green color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In order to study for which kind of these microlensing events the physical parame- ters of their lens objects are measurable with reasonable accuracy, we also plot the corresponding normalized dis- tributions of events with the relative errors in the lens mass, distance, and proper motion ≤ 5% (black stepped curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Accordingly, detectable microlensing events due to ISMBHs have the average timescale of ⟨tE⟩ = 303 days and their average source magnitude at the baseline is ⟨mbase⟩ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Discerning these microlensing light curves (by adding extra observations during the large time gap) does not highly depend on the time of the closest approach and the lens impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The events with measurable physical parameters of their lens objects have on average smaller lens impact parameters (by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='13), and mostly happen during either three first or three last observing seasons of the Roman telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For each discernible event, we determine the errors in the physical parameters of microlenses through calculat- ing Fisher and Covariance matrices (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', Boutreux & Gould 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Gould & Salim 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Sajadian 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In this regard, we make several simple assumptions which are listed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (i) We separate the photometry and astrometry measurements completely and calculate two Fisher matrices corresponding to these measurements, A, and B for each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (ii) We assume that the lens- ing parameters such as t0, u0, tE, and ξ are determined through photometry observations well and their real val- ues are used for astrometric modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In fact, the photo- metric accuracy is better than the astrometric accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (iii) We ignore the parallax effect on the source trajec- tories, which are too small to be measured (compare the dotted and dashed blue lines in right panels in Figures 1, and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (iv) We ignore the finite source effects on both microlensing lightcurves and astrometric shifts in the source position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (v) We assume that the source dis- tances from the observer, Ds, are determined by other observations, and we do not need to tune them through microlensing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For instance, the Gaia obser- vations provide stellar parallax distances for some source stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Photometry and astrometry Fisher matrices are: Aij = N � k=1 1 σ2m(tk) ∂2ms(tk) ∂pi∂pj , Bij = N � k=1 1 σ2a(tk) �∂2θs,n1(tk) ∂qi ∂qj + ∂2θs,n2(tk) ∂qi ∂qj � , (10) where, ms(tk) = mbase − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 log10 � fbA(tk) + 1 − fb � is the apparent source magnitude at the given time tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' fb is the blending factor in W149 filter, mbase is the base- line magnitude without lensing effect in that filter (its distribution for detectable events is shown in the second panel of Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' pis, and qis are observable parameters that affect on photometry and astrometry measurements (ms, θs), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Observable parameters: A microlensing light curve by considering the parallax effect can be modeled with 7 parameters which are: pi ∈ t0, u0, tE, ξ, fb, mbase, πE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The finite source effect can be ignored in long-duration microlensing events due to ISMBHs, so we put aside this effect while calculating A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The source apparent trajec- tory on the sky plane can be modeled with 3 parameters: qi ∈ θE, µs,n1, µs,n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We calculate Fisher matrices numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Their in- verses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', covariance matrices, A−1 and B−1) are de- rived using the Python module Numpy 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The square roots of diagonal elements are the errors in the observ- able parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', σpi = � A−1 ii and σqi = � B−1 ii , and non-diagonal elements are the correlation coefficients between errors in the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Taking these errors into account, we determine the errors in the physical parameters of ISMBHs, which is explained in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Errors in the physical parameters According to Equation 2, the lens mass and its error as a function of observable parameters are: 6 https://numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='org/ 8 Sajadian and Sahu Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='— The fractional distributions of the relative errors in the normalized parallax amplitude, the lens mass, the lens distance, and the lens proper motion for a big samples of microlensing events due to ISMBHs detectable by the Roman telescope with (green distributions) and without (black step ones) considering sparse observations when the Galactic bulge is observable during the large time gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The vertical (solid, dashed and dotted) lines show the thresholds of the relative errors 10%, 5%, and 1%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The samples due to both distributions have the same entrances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Ml = θE κ πE , σMl =Ml ��σθE θE �2 + �σπE πE �2 , (11) where σMl, σθE, and σπE are the error in the lens mass, error in the angular Einstein radius, and the error in normalized parallax amplitude, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that there is no correlation between σπE and σθE, because these two parameters are determined from photometry and astrometry Fisher matrices independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The next parameter is the lens distance which is given by: 1 Dl = 1 Ds + πE θE au , σDl =Dl Ds − Dl Ds σMl Ml , (12) Here, we assume that the error in source distance is very small and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The last parameter is the lens angular velocity components which are: µl,n1 =µs,n1 − θE tE cos ξ, µl,n2 =µs,n2 − θE tE sin ξ, (13) Accordingly, the errors in the lens angular velocity com- ponents are given by: σ2 l,n1 = σ2 s,n1 +µ2 rel,⊙ cos2 ξ ��σθ θE �2 + �σt tE �2 + � σξ cot ξ �2 − 2σt tE σξ cot ξ ˆ A−1 ij � , σ2 l,n2 = σ2 s,n2 +µ2 rel,⊙ sin2 ξ ��σθ θE �2 + �σt tE �2 + � σξ tan ξ �2 − 2σt tE σξ tan ξ ˆ A−1 ij � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (14) where, σl,i, σs,i are the errors in ith component of the lens and source angular velocity projected on the sky plane, and ˆ A−1 ij = A−1 ij / � A−1 ii A−1 jj is the correlation coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='09 Distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='. Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 0 2 3 5 0g10l0E / TE(%))0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08 istribu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='07 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 0 3 5 [(%)W / W0]0160l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='12 ≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='10 ibutior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08- Distril D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='07 lormalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 2 log10[g D// Di(%)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='12 ution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='10 istribu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08 D 8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='07 Normalize 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 0 3 5 l0g10[0μ/μ(%)]Detecting stellar-mass black holes by Roman 9 TABLE 1 Statistical information about simulated microlensing events due to ISMBHs detectable with the Roman telescope by assuming different ISMBHs mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' σtE � tE σπE � πE σθE � θE σMl � Ml σDl � Dl σµs � µs σµl � µl ϵm(%) Ne,BHs dN/dM = const No observations during the time gap ≤ 1% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='50 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='15 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='21 2 ≤ 5% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='26 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='35 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='32 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='59 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='32 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='37 11 ≤ 10% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='91 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='86 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='88 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='77 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='29 17 Sparse observations during the time gap ≤ 1% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='81 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='32 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='93 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='66 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='15 4 ≤ 5% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='72 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='66 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='85 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='40 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='37 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='27 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='48 17 ≤ 10% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='14 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='26 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='54 24 dN/dM ∝ M−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 No observations during the time gap ≤ 1% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='52 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='34 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='43 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='68 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='64 2 ≤ 5% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='88 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='52 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='29 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='34 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='57 12 ≤ 10% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='84 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='65 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='66 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='07 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='94 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='30 18 Sparse observations during the time gap ≤ 1% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='77 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='70 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='64 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='65 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='49 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='22 3 ≤ 5% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='57 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='29 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='40 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='82 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='21 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='81 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='25 15 ≤ 10% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='18 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='33 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='32 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='15 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='54 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='05 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 22 dN/dM ∝ M−1 No observations during the time gap ≤ 1% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='89 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='52 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='11 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='85 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='51 3 ≤ 5% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='83 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='82 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='34 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='79 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 14 ≤ 10% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='02 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='20 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='56 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='97 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='68 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='15 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='56 21 Sparse observations during the time gap ≤ 1% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='48 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='55 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='30 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='56 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='92 3 ≤ 5% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='23 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='38 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='81 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='22 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='85 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='33 15 ≤ 10% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='95 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='17 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='79 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='42 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='79 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='61 22 dN/dM ∝ M−2 No observations during the time gap ≤ 1% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='15 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='51 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='60 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='83 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='34 3 ≤ 5% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='50 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='65 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='20 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='83 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='16 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='99 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='93 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='53 12 ≤ 10% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='21 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='56 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='69 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='24 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='89 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='30 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='07 18 Sparse observations during the time gap ≤ 1% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='57 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='46 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='91 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='31 3 ≤ 5% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='86 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='33 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='58 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='35 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='25 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='81 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08 15 ≤ 10% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='57 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='28 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='61 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='68 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='93 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='39 23 Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' — Each entry represents the persentage of simulated events with the desired relativel error (specified in its row) be less than the given threshold (determined in its column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' ϵm is the Roman efficiency for measuing the lens mass, distance, and its proper motion with the relative errors less than the given threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The last column reports the estimated number of ISMBHs that can be detected in the Roman observations by considering different mass functions, as explained in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' between errors in tE, and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The errors in the lens and source proper motion can be determined using the errors in their components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Results The normalized distributions for four relevant param- eters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', tE, mbase, t0, and u0) for simulated events whose relative errors in the lens mass, distance and proper motion are ≤ 5%, are shown in Figure 3 with black step lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Accordingly, longer microlensing events from brighter source stars, whose times of the closest ap- proach happen during either the first three or the last three observing seasons are more favourable for the mea- surement of the physical parameters of the lens objects with reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Figure 4, we show the normalized distributions of the relative errors in the physical parameters of ISMBHs (as microlenses), resulting from Monte Carlo simulations, by considering a uniform mass function for ISMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Green and black distributions are related to detectable events by the Roman telescope with and without considering sparse data points during the time gap, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' These parameters are the normalized parallax amplitude, the lens mass, the lens distance and the lens proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The threshold amounts of the relative errors in the given parameters of 10%, 5%, and 1% are depicted with solid, dashed, and dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Accordingly, adding extra observations during the time gap (one hour of observations every 10 days when the Galactic bulge is observable) improves the relative errors in all physical parameters, especially the lens distance from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For numerical evaluation, in Table 1 we give the per- 10 Sajadian and Sahu centages of simulated detectable events with the rela- tive errors (specified in the first row) less than the given thresholds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', 1, 5, 10% as mentioned in the first col- umn) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Hence, sparse observations during the time gap improve the Roman efficiencies by ∼ 1%, ∼ 2%, and ∼ 2% for measuring the physical parameters by the relative errors less than 1, 5, 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In 20-25% detectable events, the lens mass can be de- termined with the relative error less than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' These events have smaller relative errors in the lens distance, because the factor (Ds − Dl)/Ds is less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The source proper motion can be determined by monitoring the source positions during 6 observing seasons (with a 15 min cadence) of the Roman mission even without taking sparse data points during the ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3-year time gap very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Nevertheless, the lens proper motion can be determined with the relative error less than 5% in 19-24% of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Even though ISMBHs produce long-duration mi- crolensing events, which are suitable for discerning the annual parallax effects, the normalized parallax amplitude, πE, decreases with increasing the lens mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Hence, the parallax effect can be discerned in these long-duration microlensing events with the relative errors less than 5% only in 21-26% of all detectable events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In order to determine which kinds of ISMBHs might be well characterized through astrometric microlensing observations with the Roman telescope, we show the de- pendence of the relative errors in the lens mass, the lens distance, its proper motion, and the parallax amplitude to Ml, xls, Ds, and mbase in Figure 5, in different panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For these plots, we only use the events with the relative errors less than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' There are several factors which determine their dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' According to the first panel, the relative error in the lens mass minimize when Ml ≃ 10-25M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Increasing the lens mass has two against effects: (i) The lens mass enhances the Einstein crossing time and decreases the average photometry errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Because more data points are taken while the source is being lensed, and less data points are recorded over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (ii) Enhancing the lens mass decreases the normalized parallax amplitude πE significantly, and makes hard measure it (see the dot- ted red step line in the top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This point was also expressed by Karolinski & Zhu (2020) and while model- ing OGLE-2006-BLG-044 microlensing event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For that reason, the optimum value for the lens mass with least errors is neither the least (2-3 solar mass), nor the most (40-50 solar mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The relative error in the lens distance decreases with the lens mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In fact, by increasing the lens mass xls enhances to keep the Einstein crossing times close to reasonable values for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The relative error in the lens proper motion weakly de- pends on the lens mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In fact, σtE/tE is an increasing function versus the lens mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' By fixing the observing time and cadence (considering a determined observing platform) and increasing tE, its error increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In to- tal, the relative errors in the lens physical parameters enhance with the lens mass slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The second panel of Figure 5 shows the relative errors in the lens mass, lens distance, its proper motion, and the parallax amplitude versus xls = Dl/Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The smaller xls make larger πE and θE, with smaller observing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' That increases the relative error in the lens mass versus xls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' However, this enhancement is slower in the relative error in the lens distance, because of the factor (Ds − Dl)/Ds in Equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In the next panel of Figure 5, we show the depen- dence of the relative errors with the source distance from the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The source distance decreases πE, and θE, which increases the relative errors in the lens mass and its distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We note that decreasing the parallax am- plitude increases both errors in the parallax amplitude, and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Comparing these panels, we find that the effect of the source distance and the lens relative position (xls) on the errors is higher than the effect of the lens mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In the last panel, the relative errors versus the apparent magnitude of the source star at the baseline are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' As shown here, they enhance with the source magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Both Roman photometric and astrometric errors increase with the apparent magnitude of source stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Worse ac- curacies cause higher relative errors in the lens physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Therefore, long-duration microlensing events due to ISMBHs with the mass Ml ≃ 10-25M⊙, close to the ob- server (xls ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5) while the source is inside the Galactic disk (Ds ≲ 6kpc) can be characterized with the least errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Different mass function for ISMBHs We know that there is no accurate mass function for ISMBHs based on observations yet, so we perform the simulation by considering several mass functions for ISMBHs, which are given in the following: dN dM =const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', dN dM ∝1 �√ M, dN dM ∝M −1, dN dM ∝M −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (15) The results from simulations based on each of these mass functions are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Accordingly, by chang- ing ISMBHs mass function, the Roman efficiency to mea- sure the lens physical parameters can change up to 2-7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Also, the first mass function makes more ISMBHs with mass Ml ∈ [10, 25]M⊙ than other mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' So it has larger efficiencies to measure the physical parameters of lens objects than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In the next subsection, we do some statistical estima- tions about detecting and characterizing such events dur- ing the Roman mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Statistical estimations The number of microlensing events that the Ro- man telescope will detect is Ne,tot = 27000, which were estimated in Penny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Here, we want to evaluate what fraction of this total number of microlensing events detectable by the Ro- man telescope are due to ISMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In this regard, there are two factors: (i) the optical depth, and (ii) the av- erage microlensing duration which are discussed in the Detecting stellar-mass black holes by Roman 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='— The dependence of the average relative errors in the lens mass (solid green lines), the lens distance (dashed blue lines), its proper motion (dot-dashed magenta lines), and the normalized parallax amplitude (dotted red lines) versus the lens mass, the ratio of the lens distance to the source distance from the observer (xls), the source distance, and the source apparent magnitude at the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (i) The number of detectable microlensing events is pro- portional to the optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The microlensing optical depth at a given line of sight (l, b) and one specified dis- tance from the observer, (D), is proportional to the lens mass Ml, because it is given by: dτ(l, b, D) dD = π θ2 E n(l, b, D) D2, (16) where, (l, b) are the Galactic longitude and latitude, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' n(l, b, D) is the number density of stars in our galaxy which is the Galactic mass density divided by the average stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Accordingly, the ratio of the optical depth (and as a re- sult the number of microlensing events) due to ISMBHs to the overall optical depth due to all potential lens ob- jects can be estimated by: F1 = � ∞ 20M⊙ Ml η(Ml) dMl � � ∞ 13MJ Ml η(Ml) dMl,(17) where,MJ is the Jupiter mass, η(Ml) is the initial mass function in the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In fact, F1 determines the contribution of the ISMBHs in producing the effec- tive lensing surface in comparison with the total lens- ing surfaces covered by all possible Einstein rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In Equation 17, we use the fact that stars with the initial mass M > 20M⊙ will convert to black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We ignore the contribution of black holes generated from primordial fluctuations in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' In order to estimate F1, we take the initial mass function from the Besan¸con model (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2003, 2012), and assume that all lens objects are inside the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' This mass function is η(Ml) ∝ M −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 l for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08 ≤ Ml(M⊙) ≤ 1, and η(Ml) ∝ M −3 l for Ml(M⊙) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The stars with Ml > 20M⊙ are converted to ISMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' For 13MJ < Ml < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='08M⊙ we take the Brown dwarf mass function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=', M −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='7 l (Muˇzi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Luhman 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We do not include free floating planets, because of their negligible contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' The upper limit should in reality be the mass due to the most massive star in the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' We set this upper limit to infinity, because the mass function for M > 1M⊙ decreases as M −3, so it tends to zero fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Accordingly, we find F1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' (ii) The microlensing event rate is proportional to � ϵ(tE) � tE � , which specifies the inverse of the average du- ration of microlensing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' Here, ϵ(tE) is the Ro- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='1 Relative Error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 10 20 30 40 M[Mo]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 Error 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 Relative I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='9 XIs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 Error 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='50 Relative 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='25 (0m, / M[%] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='00 (αD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content=' / Di[%]> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='75 (0μ/ / μi[%]) (O πe / TE[%]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='50 0 2 4 6 8 10 12 Ds(kpc)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E2T4oBgHgl3EQfUQfh/content/2301.03812v1.pdf'} +page_content='5 (om / M[%]) 40. These results are consistent with the earlier study [17]. +Appendix B +νr dependence of the effective aspect ratio +As mentinoed in Sec. 3, the ratio ref/rp depends on νr. The results are shown in Fig. 8. As νr increases, +ref/rp monotonically decreases. Thus, one may optimize νr to have ref/rp that is consistent with a specific +experimental system. To be fair, we note that the conditions νr < 0.5 are not suitable to our numerical +method, and we cannot attain ref/rp value larger than 0.7, because the torque exerted to solid particles +becomes comparable to the discretization error. +References +[1] M. B. Liu and G. R. Liu. Arch. Comput. Methods Eng., 17(1):25–76, mar 2010. +[2] Hitoshi Gotoh and Abbas Khayyer. J. Ocean Eng. Mar. Energy, 2(3):251–278, apr 2016. +[3] Hitoshi Gotoh, Abbas Khayyer, and Yuma Shimizu. Appl. Ocean Res., 115:102822, oct 2021. +[4] S. Koshizuka and Y. Oka. Nucl. Sci. Eng., 123(3):421–434, 1996. +[5] R. A. Gingold and J. J. Monaghan. Mon. Not. R. Astron. Soc., 181(3):375–389, dec 1977. +[6] J. J. Monaghan. J. Comput. Phys., 110(2):399–406, feb 1994. +[7] S. Koshizuka, Atsushi Nobe, and Y. Oka. Int. J. Numer. Methods Fluids, 26(7):751–769, 1998. +[8] Rui Xu, Peter Stansby, and Dominique Laurence. J. Comput. Phys., 228(18):6703–6725, oct 2009. +[9] Abbas Khayyer and Hitoshi Gotoh. J. Comput. Phys., 230(8):3093–3118, apr 2011. +[10] Tasuku Tamai and Seiichi Koshizuka. Comput. Part. Mech., 1(3):277–305, sep 2014. +[11] Antonio Souto-Iglesias, Fabricio MacI`a, Leo M. Gonz´alez, and Jose L. Cercos-Pita. Comput. Phys. +Commun., 184(3):732–745, mar 2013. +8 + +Fig. 6: +Rotation orbits of a single fiber. Solid curves show our simulation results of (a) θ0 = π/6 for +12 ≤ γ ≤ 50 and (b) θ0 = π/3 for 0 ≤ γ ≤ 50. Other parameters are the same as Fig. 3 (a). Dashed curves +are the Jeffery orbits with ref = 0.36rp. +[12] Guangtao Duan, Akifumi Yamaji, Seiichi Koshizuka, and Bin Chen. Comput. Fluids, 190:254–273, +aug 2019. +[13] Gen Li, Jinchen Gao, Panpan Wen, Quanbin Zhao, Jinshi Wang, Junjie Yan, and Akifumi Yamaji, +aug 2020. +[14] G. B. Jeffery. Proc. R. Soc. London, Ser. A, 102(715):161–179, nov 1922. +[15] Nils Meyer, Oleg Saburow, Martin Hohberg, Andrew N. Hrymak, Frank Henning, and Luise K¨arger. +J. Compos. Sci., 4(2):77, jun 2020. +[16] S. Yashiro, T. Okabe, and K. Matsushima. Adv. Compos. Mater., 20(6):503–517, 2011. +[17] S. Yashiro, Hideaki Sasaki, and Yoshihisa Sakaida. Compos. Part A Appl. Sci. Manuf., 43(10):1754– +1764, oct 2012. +[18] A. Eringen. Indiana Univ. Math. J., 16(1):1–18, 1966. +[19] A. Souto-Iglesias, J. Bonet Avalos, M. Antuono, and A. Colagrossi. Phys. Rev. E, 104(1):015315, +jul 2021. +[20] M. Oochi, S. Koshizuka, and M. Sakai. +Trans. Japan Soc. Comput. Eng. Sci., 2010:20100013– +20100013, 2010. +[21] Ahmad Shakibaeinia and Yee Chung Jin. Int. J. Numer. Methods Fluids, 63(10):1208–1232, aug +2010. +[22] Satoru Yamamoto and Takaaki Matsuoka. J. Chem. Phys., 98(1):644–650, 1993. +[23] Vitaly A. Kuzkin and Igor E. Asonov. Phys. Rev. E, 86(5):051301, nov 2012. +[24] Andrea Colagrossi, B. Bouscasse, M. Antuono, and S. Marrone. +Comput. Phys. Commun., +183(8):1641–1653, aug 2012. +[25] J. Einarsson, F. Candelier, F. Lundell, J. R. Angilella, and B. Mehlig. Phys. Fluids, 27(6):063301, +jun 2015. +9 + +Fig. 7: +(a) Flow velocity profile generated by moving walls in our numerical method for a fluid without +a fiber. The gap length is 55. (b) The gap length dependence of the average shear rate. +Fig. 8: +The rotational kinematic viscosity dependence of the effective aspect ratio in our model. ref and +νr are normalized by rp and ν, respectively. +[26] F. P. Bretherton. J. Fluid Mech., 14(2):284–304, 1962. +[27] B. J. Trevelyan and S. G. Mason. J. Colloid Sci., 6(4):354–367, aug 1951. +[28] P. G. Saffman. J. Fluid Mech., 1(5):540–553, 1956. +[29] Xijun Fan, N. Phan-Thien, and Rong Zheng. J. Nonnewton. Fluid Mech., 74(1-3):113–135, jan 1998. +[30] Satoru Yamamoto and Takaaki Matsuoka. J. Chem. Phys., 100(4):3317–3324, 1994. +[31] Paal Skjetne, Russell F. Ross, and Daniel J. Klingenberg. J. Chem. Phys., 107(6):2108–2121, aug +1997. +[32] Stefan B. Lindstr¨om and Tetsu Uesaka. Phys. Fluids, 21(8):083301, aug 2009. +[33] Toshiki Sasayama, Hirotaka Okamoto, Norikazu Sato, and Jumpei Kawada. +Powder Technol., +404:117481, may 2022. +10 + diff --git a/KdE2T4oBgHgl3EQfVAc5/content/tmp_files/load_file.txt b/KdE2T4oBgHgl3EQfVAc5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0313ee06593a07602c3056faa1af706e27d25d4d --- /dev/null +++ b/KdE2T4oBgHgl3EQfVAc5/content/tmp_files/load_file.txt @@ -0,0 +1,522 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf,len=521 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='03818v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='flu-dyn] 10 Jan 2023 Extension of Moving Particle Simulation including rotational degrees of freedom for dilute fiber suspension Keigo Enomoto1, Takato Ishida1, Yuya Doi1, Takashi Uneyama1, and Yuichi Masubuchi1 1Department of Materials Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya 464–8603, Japan Abstract We develop a novel Moving Particle Simulation (MPS) method to accurately reproduce the motion of fibers floating in sheared liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In conventional MPS schemes, if a fiber suspended in a liquid is represented by a one-dimensional array of MPS particles, it is entirely aligned to the flow direction due to the lack of shear stress difference between fiber-liquid interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To address this problem, we employ the micropolar fluid model to introduce rotational degrees of freedom into the MPS particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The translational motion of liquid and solid particles and the rotation of solid particles are calculated with the explicit MPS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The fiber is modeled as an array of micropolar fluid particles bonded with stretching and bending potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The motion of a single rigid fiber is simulated in a three-dimensional shear flow generated between two moving solid walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We show that the proposed method is capable of reproducing the fiber motion predicted by Jeffery’s theory being different from the conventional MPS simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 1 Introduction Fluid particle methods have been developed for simulations of multi-phase flows [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In the simulations of liquid-solid systems, the particles represent the included liquid and solid to possess local quantities such as velocity and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The motion of each particle is calculated according to interactions based on its discretized governing equation with neighboring particles within a certain distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The Moving Particle Simulation (MPS) method, developed by Koshizuka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [4], is one of such methods along with Smoothed Particle Hydrodynamics (SPH) [5, 6] and has been actively developed in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Following the original MPS, which employs a semi-implicit scheme [7], high-precision schemes such as particle regularization schemes [8] and improvements of the differential operator models [9,10] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Further developments for MPS have been being attempted for various issues including variable resolution schemes, theoretical error analysis, momentum conservation at interfaces, etc [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' A possible direction for further improvement of MPS is the inclusion of rotational degrees of freedom for particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Such an aspect is necessary for fiber suspensions when the fiber is represented by a one- dimensional array of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Let us consider a rotational motion of a fiber oriented in the flow direction under shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In conventional MPS schemes, this fiber is trapped in the fully aligned state due to the balance of particle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' However, in reality, due to the difference of the shear stress between the interfaces in the shear gradient direction, the fiber exhibits periodic rotation as theoretically argued by Jeffery [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Although this problem has been known [15], it has not been properly considered in most of the simulations for fiber suspensions with MPS [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In the conventional fluid particle method, viscous torque exerted by the fluid cannot be transferred to the motion of solid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In this study, we propose a novel MPS method for fiber suspensions to reproduce the rotational motion of fibers in a correct manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To achieve this objective, we employ the micropolar fluid model to introduce an angular velocity field through the rotational degrees of freedom of the constituent particles [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To evaluate our method, we performed simulations of a single fiber suspended in the sheared Newtonian liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The fiber is represented as an array of micropolar fluid particles connected with each other with stretching, bending, and torsional potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We compare the fiber motion with Jeffery’s theory [14] to confirm that the fiber motion is correctly captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Details are shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 1 2 Model and Simulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='1 Explicit MPS with rotational degrees of freedom In the MPS model, the dynamics of fluid velocity obey the continuum Navier-Stokes equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To incorporate the rotational degrees of freedom into the dynamics model, we employ the micropolar fluid model [18] in which the angular velocity field is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The conservation laws of linear and angular momentum are written as follows: Du(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) Dt = −1 ρ∇P(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) + ν∇2u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) + νr∇ × Υ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) + f(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (1) I DΩ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) Dt = G(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) − νrΥ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (2) where D/Dt is the time material derivative,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' r is the position vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' t is time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) is the fluid velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' ρ is the mass density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' P(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) is the pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' ν is the kinematic viscosity coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' νr is the rotational kinematic coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Ω(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) is the angular velocity field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Υ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) = 2Ω(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) − ∇ × u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' f(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) is the external volume force,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' I is the micro-inertia coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' and G(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='t) is the torque density due to the external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' According to the second law of thermodynamics, νr is a parameter properly chosen in the following range [19]: 0 ≤ νr ≤ (1 + 2 d)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (3) Here, d is the spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' For a normal fluid without micropolar degrees of freedom, Ω is given as Ω = (∇ × u)/2 which guarantees that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (1) reduces the standard Navier-Stokes equation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In this work, we simply set Ω = (∇ × u)/2 for liquid region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In this study, we employ the explicit MPS (EMPS) method [20,21] to discretize Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The equations for the constituent particle i are as follows: dui(t) dt = − 1 ρi ⟪∇P⟫i(t) + 1 Re ⟪∇2u⟫i(t) + 1 Rer ⟪∇ × Υ⟫i(t) + fi(t), (4) dΩi(t) dt = αGi(t) − 2α Rer Υi(t), (5) Υi(t) = 2Ωi(t) − ⟪∇ × u⟫i(t), (6) where ⟪⟫ indicates the quantity evaluated by the operator model in MPS at the position of particle i mentioned in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The equations are non-dimensionalized using the following quantities: the fluid mass density ρ0, the reference kinematic viscosity coefficient ν0, and the size of the fluid particle l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' ν0 is a reference value, and l0 can be interpreted as the characteristic length scale of the discretized system (which may be interpreted as the grid size in the finite difference scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These quantities define units of length, time, and energy, and the quantities discussed below are normalized according to these units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Re = ν0/ν is the Reynolds number, Rer = ν0/νr is the rotational Reynolds number, and α is defined as α = l2 0/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' As the case of the integration of the micropolar fluid model to the SPH model [19], translational and rotational velocities are mapped onto constituent (liquid and solid) particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In our model, solid particles are micropolar fluid particles, and their motion follows Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The motion of liquid particles follows the standard Navier-Stokes equation plus the reaction force based on the third term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (4) exerted by the surrounding solid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To calculate the physical quantities and their differentials at the position of particle i, we need the weighting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We employ the following weighting function: w(r) = ⎧⎪⎪⎨⎪⎪⎩ lc/r − 1 (0 < r < lc) 0 (r ≥ lc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (7) Here, lc is the cutoff radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The local density is evaluated by the local number density of the constituent particles defined as ni = ∑ j≠i w (∣rj − ri∣) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (8) 2 The differential operators in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (4) and (5) are calculated by the following operator models: ⟪∇ψ⟫i = d n0 ∑ j≠i [ ψi + ψj ∣rj − ri∣2 (rj − ri)w (∣rj − ri∣)], (9) ⟪∇ × b⟫i = d n0 ∑ j≠i [(bj − bi) × (rj − ri) ∣rj − ri∣2 (rj − ri)w (∣rj − ri∣)], (10) ⟪∇2b⟫i = 2d λn0 ∑ j≠i [(bj − bi)w (∣rj − ri∣)], (11) λ = ∑j≠i (rj − ri)2w (∣rj − ri∣) ∑j≠i w (∣rj − ri∣) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (12) Here, ψi and bi are scalar and vector variables on the particle i, n0 is the initial particle number density, and λ is the parameter defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (12) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (9) we use ψi + ψj instead of ψj − ψi, as proposed by Oochi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [20], for better momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='2 Fiber model Ωi ui ti si uj Ωj = 1 2 (∇ × uj) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' "#$"%&\'()*"+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' "%&\'()*"+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', /&0"+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content="'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' ()&1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='$"%&\'()*"+!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=',&2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' "#$% &"\'(") y z x ri Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 1: Schematic of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The fiber is composed of micropolar fluid particles which possess the velocity ui and angular velocity Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The liquid particles are represented as a micropolar fluid particle with Ωj = (∇ × uj)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The fiber is modeled as an array of solid particles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The solid particles are connected with stretching, bending, and torsional potential energies, in a similar manner proposed by Yamamoto and Matsuoka for the other simulation scheme [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These potential forces should be a function of the bond vector of neighboring particles and the orientation of each solid particle [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To describe the orientation of the solid particles, we introduce two directors si and ti on each solid particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' si and ti are unit vectors for which directions are parallel and perpendicular to the bond vector, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 1 The time derivative of directors is related to the angular velocity as follows: dsi(t) dt = (1 − sisi) ⋅ (Ωi × si), dti(t) dt = (1 − titi) ⋅ (Ωi × ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (13) Here, 1 is the unit tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The projection tensors (1 − sisi) and (1 − titi) maintain si ⋅ ti = 0 within numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The stretching potential Us, bending potential Ub, and torsional potential Ut are defined as Us ({ri}) = ∑ ⟨i,j⟩ ks 2 (∣rj − ri∣ − 1)2, (14) Ub ({ri},{si}) = ∑ ⟨i,j⟩ ⎡⎢⎢⎢⎢⎣ kb 2 (sj − si)2 − kr 2 ⎧⎪⎪⎨⎪⎪⎩ (si ⋅ rj − ri ∣rj − ri∣) 2 + (sj ⋅ ri − rj ∣ri − rj∣) 2⎫⎪⎪⎬⎪⎪⎭ ⎤⎥⎥⎥⎥⎦ , (15) Ut ({ti}) = ∑ ⟨i,j⟩ kt 2 (tj − ti)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (16) 3 Here, ks,kb,kr,kt are the spring constants and ⟨i,j⟩ represents a pair of two adjacent solid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The potential force fi and torque Gi are calculated as fi = −∂ (Us + Ub) ∂ri , Gi = si × (−∂Ub ∂si ) + ti × (−∂Ut ∂ti ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (17) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (14), if ks is large sufficiently, the fiber length L corresponds to the number of solid particles in the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Since the unit length of the system is the size of the fluid particle, the aspect ratio of the fiber rp corresponds to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='3 Numerical algorithms In the EMPS method, the fractional step algorithm is applied for time integration as in the original semi-implicit scheme for MPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Each integration step is divided into prediction and correction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In the prediction step, predicted velocity u∗ i is calculated by using terms other than the pressure gradient term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (4), and the angular velocity of the solid particles is also updated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (5) as follows: u∗ i = uk i + ∆t[ 1 Re ⟪∇2u⟫ k i + 1 Re r ⟪∇ × Υ⟫k i + f k i ] , Ωk+1 i = Ωk i + ∆tα [Gk i − 2 Rer Υk i ], Υk i = 2Ωk i − ⟪∇ × u⟫k i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (18) Here, ∆t is the step size, and the upper indexes k represent the step number: bk i = bi(t = k∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The predicted position r∗ i and directors are updated as r∗ i = rk i + ∆tu∗ i , sk+1 i = sk i + ∆t(1 − sk i sk i ) ⋅ (Ωk+1 i × sk i ), t∗ i = tk i + ∆t(1 − tk i tk i ) ⋅ (Ωk+1 i × tk i ), (19) where t∗ i is the predicted torsional director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To maintain the relation si ⋅ ti = 0, we adjust t as follows: tk+1 i = (1 − sk+1 i sk+1 i ) ⋅ t∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (20) In the correction step, the velocity and position are calculated as uk+1 i = u∗ i − ∆t ρi ⟪∇P⟫k+1 i , rk+1 i = r∗ i + (uk+1 i − u∗ i )∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (21) In the EMPS, the pressure is calculated by the following explicit form [20]: P k+1 i = ρics n0 (n∗ i − n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (22) Here, cs is the sound speed, and n∗ i is the number density at r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' This cs is optimized concerning reasonable incompressibility and numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='4 Simulations We apply shear flows in the following boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Hereafter, we refer to flow, shear gradient, and vorticity directions as x, y, and z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We employ periodic boundary conditions for x and z directions, whereas we place solid walls at y = 0 and h perpendicular to the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These walls consist of three layers of liquid particles, which are fixed on a squared lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Following the earlier study [17], we move the walls toward the x direction with the speed of uwall = ±˙γh/2, where ˙γ is the apparent shear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We have confirmed that the actual shear rate is equal to ˙γ and uniform throughout the system within a numerical error, in simulations without fibers, as shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Simulations of a single fiber in a simple shear flow were carried out, and the rotational motion of the fiber was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To describe the fiber motion, we use the orientation angles φ and θ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The number of MPS particles was N = 64000 in total including those for walls and the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The simulation box dimension was 40 ×40 ×40 in x-y-z directions, respectively, and the distance between the walls was 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The kinematic viscosity coefficient ν and the strain rate ˙γ were chosen so that the 4 θ φ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 2: Schematic of a fiber (an array of blue particles) at orientation angles φ and θ in a shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The dashed curve shows the orbit of the head of the fiber (Jeffery orbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' fiber-based Reynolds number was Ref = L2 ˙γ/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='1 to realize a viscous dominant condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The sound speed of the fluid cs was set so that the Mach number became Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='5h˙γ/cs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The numerical step size ∆t was chosen to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='01 according to the Courant condition, the viscous constraint, and the relation to the spring constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Other model parameters were set as lc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='1, νr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='5ν, I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='8 unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The mass density of the solid particles is the same as that of liquid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The aspect ratio of the fiber rp was varied in the range from 2 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The spring constants were chosen at ks = 1000 and kb = kt = kr = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These values realized a rigid fiber, for which the effect of fiber deformation is negligible in the result as shown later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We performed the simulations with a house-made code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In the initial condition, we placed the fiber at the center of the simulation box to overlap the center of mass of the fiber and the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The initial fiber orientation angle to x-direction, φ0, was fixed at π/2, whereas the initial angle to z-direction, θ0, was chosen at π/6, π/3, or π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Surrounding liquid particles were randomly arranged by the particle packing algorithms proposed by Colagrossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 3 Results and Discussion Typical snapshots of a single rigid fiber in a shear flow with θ0 = π/3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These figures clearly demonstrate that the fiber rotates as expected, even after it experiences the configuration aligned to the flow direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Snapshots of another fiber aligned to the vorticity direction (θ0 = 0) are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The fiber exhibits the rolling motion around the vorticity axis induced by the flow velocity difference between shear planes above and below the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' This behavior is known as the log-rolling motion [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In principle, we cannot reproduce this log-rolling motion of the fiber using MPS without introducing rotational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To analyze rotational behavior in the vorticity plane quantitatively, we show the time evolution of the rotation angle φ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We observe that the fiber rotates and approaches to φ = 0 in the MPS without rotational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' This is not consistent with Jeffery’s theory which predicts the periodic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In contrast, in our model, we observe the clear periodic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' This fact demonstrates the importance of the rotational degrees of freedom integrated into our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We compare the time evolution of φ with Jeffery’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' According to Jeffery’s theory, the periodic orbit depends on the aspect ratio of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The aspect ratio can be defined as the ratio of two axes of hydrodynamically equivalent ellipsoid for the fiber [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Here, one may argue that the fiber in our simulation model is not a rigid body and thus the aspect ratio is not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We found that with the employed simulation parameters, the fiber almost keeps its length and shape under the flow, and thus it can be approximately treated as a rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We use the effective aspect ratio ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36rp to achieve the best agreement between our model and Jeffery’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We performed the simulation with various aspect ratios to examine its effect on the rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 5 ux/uwall = ˙γy y z x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' " # $%& \'%& (%& )%\' %* $$%+ $,%* y z x !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' " # \'%- )%\' $-%( !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' "# !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='$# Ω !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='" Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 3: Typical snapshots of a fiber with rp = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Light blue spheres represent solid particles that compose the fiber, and red arrows show directors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Background colors correspond to the velocity of fluid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (a) The case of φ0 = π/2 and θ0 = π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The red arrows show si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (b) The case of φ0 = π/2 and θ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The red arrows show ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' According to Jeffery [14], the rotation period of the fiber T is described as T = 2π ˙γ (ref + 1 ref ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (23) As mentioned above, Jeffery’s theory with the effective aspect ratio ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36rp agrees with our simulation data for rp = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We use the same relation for other rp values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' As observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 5, our simulation data agree well with Jeffery’s theory with ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36rp within the examined rp range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The ratio ref/rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36 is not close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Here, we briefly discuss the validity of this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' A typical value in experiments is ref/rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='7 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' This value is larger than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' If we calculate the ratio of these two values, we have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='7/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' One interpretation of this result is that the fiber width in our model is twice larger than the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Intuitively, we expect that the motion of fluid particles around the fiber is somewhat synchronized and increases the effective width of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Note that this ratio ref/rp depends on νr as shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We further examine pivoting motion of fibers that tilt from the vorticity plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 6 shows typical rotation orbits of the head of fibers for (a) θ0 = π/3 and (b) θ0 = π/6 with Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These orbits are characterized by Cb defined as Cb = ∣CJ∣/(1 + ∣CJ∣), (24) CJ = 1 ref tanθ0(r2 ef sin2 φ0 + cos2 φ0) 1 2 , (25) where CJ is the orbit constant determined only by the initial configuration of the fiber φ0 and θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The examined cases correspond to Cb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='31 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='63, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Although there are small fluctuations due to discretization errors, the fibers reasonably follow closed trajectories, which are consistent with the Jeffery orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To be fair, we note that the fiber in our method eventually falls out of the Jeffery orbit if we continue the simulation for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Such behavior would be attributed to the properties of the Jeffery orbit and our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The Jeffery orbit is not stable against a perturbation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' If a fiber motion or flow field is slightly perturbed, the orbit moves to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' In our model, due to the discretization by using particles, 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='00Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 4: Time evolution of φ by our model (circle) and the MPS without rotational degrees of freedom (triangle) in dilute regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (rp = 10,φ0 = π/2,θ0 = π/2) Solid curves represent Jeffery’s theory with ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' both the fiber motion and flow field contain fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These fluctuations drive the orbit away from the original Jeffery orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We also note that the solid walls in our system and fluid inertia may probably play some roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Nevertheless, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 6, our scheme reasonably reproduces the Jeffery orbit in a similar manner to the other numerical studies [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Since our method is capable of reproducing the motion of single fibers in the dilute regime, extensions to the concentrated regime or real industrial application would be readily achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 4 Conclusion We have developed a new MPS method to accurately reproduce fiber motion in shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We employ the micropolar fluid model to introduce rotational degrees of freedom into constituent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To validate our method, we simulated the single fiber motion suspended in the sheared liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The fiber is represented by a single array of micropolar fluid particles bonded with stretching, bending, and torsional potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We demonstrated that the simulated rotation period and rotation orbits of the fiber are in good agreement with Jeffery’s theory given that the effective aspect ratio is tuned as a fitting parameter of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' As an application of the proposed method, we are conducting simulations for dense fiber suspensions since fiber rotation possibly plays some roles as argued by Lindstr¨om and Uesaka [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The proposed method is also capable of representing solids of arbitrary shape such as plate-shaped particles [33], not just fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' We are aware that the micropolar fluid model can be implemented to other fluid particle methods such as SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Studies toward such directions are ongoing and the results will be reported elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Acknowledgement The authors would like to express their gratitude to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Satoru Yamamoto at Center for Polymer Interface and Molecular Adhesion Science, Kyushu University for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Appendix A Calculation of a simple shear flow using EMPS We have conducted EMPS simulations without solid particles to test the method and the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The system settings are the same as simulations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 3 except for the gap size h and the absence of a fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 5: The aspect ratio dependence of the rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Symbols show our simulation data and the dashed curve shows the prediction by Jeffery’s theory (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (23)) with ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' An example of the steady-state flow profile of a shear flow is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Here, u∗ x = ux/uwall is the normalized fluid velocity in the flow direction (x– direction) where the wall velocity uwall, and y∗ = y/h is the normalized distance from the moving wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The Reynolds number of the flow is Reh = huwall/ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='5 for h = 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The result is in good agreement with the analytical solution u∗ x = 2(y/h − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The gap size dependence of the shear rate is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Here, ˙γ∗ is the average slope of the velocity profile divided by the shear rate expected from the wall velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' This result shows that the numerical error of the shear rate is less than 1% for h > 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' These results are consistent with the earlier study [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Appendix B νr dependence of the effective aspect ratio As mentinoed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 3, the ratio ref/rp depends on νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' As νr increases, ref/rp monotonically decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Thus, one may optimize νr to have ref/rp that is consistent with a specific experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' To be fair, we note that the conditions νr < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='5 are not suitable to our numerical method, and we cannot attain ref/rp value larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='7, because the torque exerted to solid particles becomes comparable to the discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Liu and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Methods Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 17(1):25–76, mar 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [2] Hitoshi Gotoh and Abbas Khayyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Ocean Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Energy, 2(3):251–278, apr 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [3] Hitoshi Gotoh, Abbas Khayyer, and Yuma Shimizu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Ocean Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 115:102822, oct 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Koshizuka and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Oka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 123(3):421–434, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Gingold and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Monaghan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 181(3):375–389, dec 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Monaghan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 110(2):399–406, feb 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Koshizuka, Atsushi Nobe, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Oka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Methods Fluids, 26(7):751–769, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [8] Rui Xu, Peter Stansby, and Dominique Laurence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 228(18):6703–6725, oct 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [9] Abbas Khayyer and Hitoshi Gotoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 230(8):3093–3118, apr 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [10] Tasuku Tamai and Seiichi Koshizuka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 1(3):277–305, sep 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [11] Antonio Souto-Iglesias, Fabricio MacI`a, Leo M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Gonz´alez, and Jose L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Cercos-Pita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 184(3):732–745, mar 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 6: Rotation orbits of a single fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Solid curves show our simulation results of (a) θ0 = π/6 for 12 ≤ γ ≤ 50 and (b) θ0 = π/3 for 0 ≤ γ ≤ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Other parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Dashed curves are the Jeffery orbits with ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content='36rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [12] Guangtao Duan, Akifumi Yamaji, Seiichi Koshizuka, and Bin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fluids, 190:254–273, aug 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [13] Gen Li, Jinchen Gao, Panpan Wen, Quanbin Zhao, Jinshi Wang, Junjie Yan, and Akifumi Yamaji, aug 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Jeffery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' London, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' A, 102(715):161–179, nov 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [15] Nils Meyer, Oleg Saburow, Martin Hohberg, Andrew N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Hrymak, Frank Henning, and Luise K¨arger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 4(2):77, jun 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Yashiro, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Okabe, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Matsushima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 20(6):503–517, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Yashiro, Hideaki Sasaki, and Yoshihisa Sakaida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Part A Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Manuf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 43(10):1754– 1764, oct 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Eringen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 16(1):1–18, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Souto-Iglesias, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Bonet Avalos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Antuono, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Colagrossi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' E, 104(1):015315, jul 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Oochi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Koshizuka, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Sakai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Japan Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 2010:20100013– 20100013, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [21] Ahmad Shakibaeinia and Yee Chung Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Methods Fluids, 63(10):1208–1232, aug 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [22] Satoru Yamamoto and Takaaki Matsuoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 98(1):644–650, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [23] Vitaly A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Kuzkin and Igor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Asonov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' E, 86(5):051301, nov 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [24] Andrea Colagrossi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Bouscasse, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Antuono, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Marrone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 183(8):1641–1653, aug 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Einarsson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Candelier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Lundell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Angilella, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Mehlig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fluids, 27(6):063301, jun 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 7: (a) Flow velocity profile generated by moving walls in our numerical method for a fluid without a fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' The gap length is 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' (b) The gap length dependence of the average shear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 8: The rotational kinematic viscosity dependence of the effective aspect ratio in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' ref and νr are normalized by rp and ν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [26] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Bretherton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 14(2):284–304, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [27] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Trevelyan and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Mason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Colloid Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 6(4):354–367, aug 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [28] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Saffman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 1(5):540–553, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [29] Xijun Fan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phan-Thien, and Rong Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Nonnewton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 74(1-3):113–135, jan 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [30] Satoru Yamamoto and Takaaki Matsuoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 100(4):3317–3324, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [31] Paal Skjetne, Russell F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Ross, and Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Klingenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 107(6):2108–2121, aug 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [32] Stefan B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Lindstr¨om and Tetsu Uesaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Fluids, 21(8):083301, aug 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' [33] Toshiki Sasayama, Hirotaka Okamoto, Norikazu Sato, and Jumpei Kawada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' Powder Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=', 404:117481, may 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE2T4oBgHgl3EQfVAc5/content/2301.03818v1.pdf'} diff --git a/LdE1T4oBgHgl3EQfGwON/content/tmp_files/2301.02918v1.pdf.txt b/LdE1T4oBgHgl3EQfGwON/content/tmp_files/2301.02918v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1d761f156456feaf7e8b944601e92667eaa475d9 --- /dev/null +++ b/LdE1T4oBgHgl3EQfGwON/content/tmp_files/2301.02918v1.pdf.txt @@ -0,0 +1,947 @@ +Statistical Power Analysis for Designing Bulk, +Single-Cell, and Spatial Transcriptomics +Experiments: Review, Tutorial, and Perspectives + +Hyeongseon Jeon1,2,*, Juan Xie1,2,3,*, Yeseul Jeon1,4,5,*, Kyeong Joo Jung6, Arkobrato Gupta1,2,3, +Won Chang7, Dongjun Chung1,2,# +1: Department of Biomedical Informatics, The Ohio State University, Columbus, OH, U.S.A. +2: Pelotonia Institute for Immuno-Oncology, The James Comprehensive Cancer Center, The +Ohio State University, Columbus, OH 43210, USA. +3: The Interdisciplinary PhD program in Biostatistics, The Ohio State University, Columbus, +Ohio, U.S.A. +4: Department of Statistics and Data Science, Yonsei University, Seoul, South Korea +5: Department of Applied Statistics, Yonsei University, Seoul, South Korea +6: Department of Computer Science and Engineering, The Ohio State University, Columbus, +Ohio, U.S.A. +7: Division of Statistics and Data Science, University of Cincinnati, Cincinnati, Ohio, U.S.A. +*: Joint first authors +#: Correspondence (chung.911@osu.edu) + +Abstract + +Gene expression profiling technologies have been used in various applications such as cancer +biology. The development of gene expression profiling has expanded the scope of target +discovery in transcriptomic studies, and each technology produces data with distinct +characteristics. In order to guarantee biologically meaningful findings using transcriptomic +experiments, it is important to consider various experimental factors in a systematic way through +statistical power analysis. In this paper, we review and discuss the power analysis for three types +of gene expression profiling technologies from a practical standpoint, including bulk RNA-seq, +single-cell RNA-seq, and high-throughput spatial transcriptomics. Specifically, we describe the +existing power analysis tools for each research objective for each of the bulk RNA-seq and +scRNA-seq experiments, along with recommendations. On the other hand, since there are no +power analysis tools for high-throughput spatial transcriptomics at this point, we instead +investigate the factors that can influence power analysis. + + +Keywords +Transcriptomics, gene expression analysis, power analysis, RNA-seq, scRNA-seq, high- +throughput spatial transcriptomics + +1. Introduction + +Transcriptomics refers to either gene expression profiling or the study of the transcriptome +using gene expression profiling technologies, where transcriptome refers to the collection of all +the ribonucleic acid (RNA) molecules expressed in a cell, cell type, or organism [1]. According to +the central dogma, RNA transcripts are generated by the cellular transcription process, play a role +in protein-coding, and connect the genome, proteome, and cellular phenotype [2]. Therefore, as +a proxy for proteome analysis, numerous transcriptomic studies have analyzed messenger RNA +(mRNA) molecules encoding proteins [3]. In addition, transcriptomic approaches have contributed +to the advancement of various biological and medical studies, such as cancer biology by +identifying possible prognostic biomarkers [4]. +Transcriptomic studies can be categorized by underlying gene expression profiling technology, +and technological advancements have increased the scope of target discovery. Figure 1 provides +a summary of three types of gene expression profiling technologies in terms of their profiling +resolution, data structure, and potential target discoveries. Hong et al. [4] illustrate the evolution +of RNA sequencing technology. Unlike microarrays, which profile predefined transcript through +hybridization, bulk RNA sequencing (bulk RNA-seq) allows genome-wide analysis across the +whole transcriptome within a cell population by employing next-generation sequencing (NGS) +technology [5]. In contrast to bulk RNA-seq, single-cell RNA sequencing (scRNA-seq) enables +the comparison of the transcriptomes of individual cells and the analysis of heterogeneity within +a cell population [3]. The high-throughput spatial transcriptomics (HST) technology permits gene +expression profiles at the cell or close-to-cell level while also preserving spatial tissue context +information [6]. We note that the characteristics of the transcriptomic data are contingent on the +underlying technology. Bulk RNA-seq data are highly reproducible, indicating that technical +replicates display minimal systemic changes and are thus unnecessary [7]. Bacher and +Kendziorski [8] demonstrate that scRNA-seq data has a greater proportion of zeros, more +variability, and a more complex distribution than bulk RNA-seq data. + + + +Figure 1: Comparison of bulk RNA-seq, single-cell RNA-seq, and high-throughput spatial transcriptomics +technologies in terms of the profiling resolution (level), data structure, and target discoveries. + +When designing a transcriptomic experiment, it is crucial to determine the experimental +factors, such as the number of biological replicates, the number of cells and sequencing depth, +to guarantee sufficient power. In the statistical framework, power refers to the probability of +detecting target discoveries, also known as sensitivity. In bulk RNA-seq analysis, Schurch et al. +[9] provided an empirical guideline for the number of biological replicates to guarantee sufficient +power, and Liu et al. [10] demonstrated that the number of biological replicates has a greater +influence on power than sequencing depth. Pollen, et al. [11] demonstrated that low-coverage +scRNA-seq is sufficient for cell-type classification. Despite the existence of basic guidelines, there +exists no unifying rule due to the complexity of power. For example, biological factors of the +experimental unit, such as sex and breeding type, may impact power and should be considered +when selecting experimental parameters more systematically. +Therefore, to determine experimental factors in transcriptomic experiments in a systematic +way, a power analysis can be conducted. Cohen [12] pioneered the concept of power analysis, +which refers to the examination of the relationship between power and all parameters influencing +power. The parameters include desired error rate and size of the experimental effect of interest +(effect size). In practice, power analysis aims to identify a parameter under the assumption that +all other parameters remain constant, with power itself being considered a parameter. In power +analysis, sample size or power itself is a common target parameter [13]. In this review paper, the +sample size refers to either the number of biological replicates or the number of cells. In an +experimental study, power analysis provides crucial information at each stage of the experiment. + +Bulk RNA-Seq +Single-cell RNA-seq +High-throughputSpatial Transcriptomics +Samples +Bulk Expression Profile +Single-cell +Single-cell/spot +Cell/Spot Coordinates +Level +Cell/Spot +Sample +Data +Cell/SpotxGeneExpressionCountData +SubjectxGeneExpression +CellxGeneExpressionCountData +Structure +CountData +Cell/Spot2-dimensionalCoordinates +SpatiallyVariableGenes +DifferentiallyExpressedGenes +Detection +DifferentiallyExpressedGenes +TissueArchitecture +Target +Cell Sub-populations +Cell-CellCommunicationBefore the study, prospective power analysis helps determine the experimental factors that will +provide sufficient power for detecting target discoveries. Researchers can conduct a retrospective +power analysis to evaluate the experiment, despite differing opinions regarding how to use the +collected data for the power analysis, as discussed in Thomas [14]. +Power analysis varies according to the underlying objectives of the study and how the +data will be analyzed to achieve the research objective [15]. As previously discussed, the +employed technology affects the scope of target discoveries and transcriptomic data +characteristics. In this context, the power analysis for three distinct transcriptomic technologies +will be examined, including bulk RNA-seq, scRNA-seq, and HST technologies. From Sections 2 +through 4, each transcriptomic technology is covered in a separate section. For a given +technology, we examine the power analysis for transcriptomic experiments with respect to +experimental factors, research objectives, and explanations of existing power analysis tools. If +there are power analysis tools for a particular technology and research objective, we provide +recommendations. + +2. Power analysis for bulk RNA-seq experiments + +2.1 Bulk RNA-seq experiment + +Sequencing technologies originate from Sanger sequencing, first introduced by Sanger et +al. [16]. In 2005, the introduction of Next-Generation Sequencing (NGS), also known as massively +parallel sequencing, improved sequencing in terms of high throughput, scalability, and speed. +Especially, NGS technology enables the bulk RNA-seq profiling of gene expression levels in over +ten thousand genes simultaneously in a specific tissue or cell population, where the gene +expression is characterized by an abundance of messenger RNA (mRNA). Typical bulk RNA-seq +protocol includes sample preparation, mRNA fragmentation, reverse transcription to +complementary DNA (cDNA), and mapping of cDNA fragments to a reference genome. A gene's +expression level is ultimately determined by counting the cDNA fragments, called reads, that are +mapped to the gene. See Stark et al. [17] and Van den Berge et al. [18] for more details. +Sequencing depth is defined as the total number of reads, influencing the sequencing's technical +precision [19]. The bulk RNA-seq profiling platforms include Illumina's HiSeq and MiSeq and ABI's +SOLID. Hong et al. [4] illustrate the RNA sequencing technological evolution over time and in- +depth explanations of the related platforms. +Bulk RNA-seq transcriptomic experiments typically aim to identify differentially expressed +genes (DEGs) across various experimental conditions, where multiple biological replicates are + +expected in each condition. DEGs are the bulk RNA-seq experiment’s detection target, with their +detection probability determining the associated power. Specifically, the power of the bulk RNA- +seq gene expression analysis is defined by the expected proportion of DEGs detected among all +DEGs, following a prespecified statistical procedure. Unlike conventional microarray technology +that generates continuous data, bulk RNA-seq generates count data. Due to the discrete nature, +the Poisson distribution was originally employed to model the bulk RNA-seq data. However, due +to its one-parameter nature, the Poisson distribution cannot account for extra-biological variation +in bulk RNA-seq data. Therefore, the negative binomial (NB) distribution, which can be viewed as +a Poisson-gamma mixture, has gained popularity. Under a model assumption, a DEG is +characterized as a gene whose mean expression ratio (i.e., fold change) deviates from 1 for any +pair of experimental conditions. The difference or ratio can be understood as a measure of the +effect size that characterizes DEGs. Bioconductor packages of edgeR [20], DESeq [21], DESeq2 +[22], and baySeq [23] employ the NB model to identify DEGs. While NB-based methods generally +have a higher detection power, there are also reports indicating its FDR inflation [24,25] due to +ignoring the uncertainty of the estimated dispersion parameters [26]. Alternatively, the voom +method [27] can be used to detect DEGs by applying normal-based theory to the log-transformed +count data, which is implemented in the limma Bioconductor package. Even though count data is +not directly modeled, the voom method adjusts heterogeneous variances across all observations +concurrently by utilizing an adequate mean and variance relationship. Additional software tools +for DEG analysis are described in Schurch et al. [9] and Stark et al. [17]. +In the case of a bulk RNA-seq experiment, it is essential to determine the number of +biological replicates that will provide sufficient DEG detection power, a type of power analysis. +Consider the factors that may affect the power. Note that the power depends on the assumed +model's parameters and the software tools that provide the p-value for each gene under +consideration. Additionally, the power is affected by the considered error rate and the target level. +Bulk RNA-seq gene expression analysis typically considers multiple genes. When multiple genes +are simultaneously inferred, it is common to control the false discovery rate (FDR) rather than the +type 1 error rate, which is appropriate for inferring a single gene. By controlling FDR, it is possible +to regulate the proportion of non-DEGs among genes declared to be DEGs on average. +Consequently, when inferring multiple genes and conducting power analysis, it is necessary to +consider the target FDR level. + + + + +2.2 Bulk RNA-seq power analysis tools + +Numerous power analysis software tools calculating the number of biological replicates, +alternatively sample size, for bulk RNA-seq experiments have been developed according to the +factors affecting the power: model assumptions, the testing type employed for each gene, and +desired error rates to be controlled. Model parameters are often estimated using pilot data, and +some tools provide stored data for this purpose. As demonstrated by data analysis in Poplawski +and Binder [28], if the stored data are utilized carelessly, a highly inappropriate sample size can +be suggested. In addition to sample size, some software tools consider sequencing depth to be +an experimental factor that influences the power to be chosen during experimental design. Liu et +al. [10] demonstrated the tradeoff between biological replicates and sequencing depth in the +context of statistical power. +Hart et al. [19] suggested a flexible power analysis approach that calculates the sample +size for a single gene expression analysis using the NB model, which is implemented in the +‘RNASeqPower’ Bioconductor package. Due to the asymptotic normality of the score test statistic, +a closed-form power function is obtained as a function of all possible parameters, including sample +size, fold change, average sequencing depth, target type 1 error rate, and coefficient of variation. +Due to the simplicity of the inference situation and the closed-form power function, it is possible +to perceive the relationship between all parameters affecting the detection power. Hart et al. [19] +also suggested a sequencing depth motivated by the parameters' relationship and demonstrated +that although the method does not assume FDR control, it can be extended to multiple gene +inference by setting the p-value threshold α to a small value, such as 0.001. +Li et al. [29] proposed a tool for calculating sample size based on the NB model and FDR +control via a gene-specific power function. The approach is effectively implemented in the +‘RnaSeqSampleSize’ Bioconductor package, with an additional parameter estimation procedure +supported by data. However, the ‘RnaSeqSampleSize’ tool tends to overestimate sample size in +the data analysis and data-based simulation study of Poplawski and Binder [28]. To overcome +this overestimation, Bi and Liu [30] suggested a method that assumes the NB model but uses the +normal-based test statistic via the voom method to assess the power function partially analytically, +implemented in the ‘ssizeRNA’ R package. According to the data-driven simulation study of +Poplawski and Binder [28], this approach is faster and provides the sample size closer to the +actual number required to achieve the desired power, compared to other approaches. Additionally, +Wu et al. [31] proposed a simulation-based FDR controlling approach, implemented in the +‘PROPER’ tool. Table 1 provides a summary of the information from different power analysis tools. + +The tools are chosen from the methods with relevant literature described in Poplawski and Binder +[28]. + +Table 1: A table shows six software tools for statistical power analysis for bulk RNA-seq +experiments. Each tool is presented along with the citation and the software environments that +have been implemented. + +Tool Name [Citation] (Implementation) +Pilot Data +Pilot Data with Stored Data +Type 1 +Error +Poisson +Lognormal +- +‘Scotty’ + [32] (Web Interface) + + + + +Negative +Binomial +‘RNASeqPower’ + [19] (R package) + +- +FDR +‘ssizeRNA’ + [30] (R package) +‘RnaSeqSampleSize’ + [33] (R package) + +‘RNASeqPowerCalculator’ + [34] (R package) +‘PROPER’ [31] (R package) + +2.3 Bulk RNA-seq power analysis tool recommendation + +The ‘ssizeRNA’ R package was chosen based on the outcomes of two simulation studies +of Poplawski and Binder [28] and Bi and Liu [30]. From the six power analysis tools mentioned in +Table 1, we first considered ‘RnaSeqSampleSize’, ‘ssizeRNA’, and ‘PROPER’ based on their +FDR-targeting nature and focus on a single DEG analysis tool. However, depending on the +performance of the simulation studies, we decided to exclude ‘RnaSeqSampleSize’ from +consideration. Specifically, according to Poplawski and Binder [28], ‘RnaSeqSampleSize’ typically +recommends a very large sample size. ‘RnaSeqSampleSize’ performs well in Bi and Liu [30] when +the model is simple, and gene-specific parameters are absent. When the simulation model +became realistic, the sample size suggested by ‘RnaSeqSampleSize’ was either too large to +significantly exceed the desired power or too small to adequately regulate power. The subsequent +selection was based on speed. The simulation results presented in both papers indicate that both +the ‘PROPER’ and ‘ssizeRNA’ tools recommend sample sizes with target power levels. Due to +the conservative nature of the voom method, the ‘ssizeRNA’ tool typically recommends a few + +more samples. In terms of usability, however, we recommend the ‘ssizeRNA’ tool, which is faster +due to its analytical nature. + +3. Power analysis for single-cell RNA-seq (scRNA-seq) experiments + +scRNA-seq technologies have revolutionized the study of transcriptomics by profiling +genome-wide gene expression at the individual cell level. The cell-level information provides +unprecedented opportunities for studying cellular heterogeneity and expands our understanding +of developmental biology [3]. Even though the context of a single-cell transcriptomic study differs +from that of a bulk transcriptomic study, DEG detection remains a fascinating study area. In +addition, the cell information enables researchers to answer questions about cell subpopulations. +The relevant power analysis has been developed in response to the distinct research questions. +In general, the sample size in scRNA-seq experiments refers to the number of cells. Due to the +additional technical steps required to distinguish cells, scRNA-seq data contain more zeros than +bulk RNA-seq data [11], and a zero-inflated model is frequently employed when developing +statistical approaches [35]. Sections 3.1 and 3.2 discuss power analysis for identifying cell sub- +populations and detecting DEGs, respectively, for scRNA-seq experiments. The information +presented in Table 2 outlines a variety of power analysis tools applicable to single-cell +transcriptomic experiments with distinct research questions. + +3.1 Power analysis for cell subpopulation detection + +Unlike bulk RNA-seq experiments, scRNA-seq experiments frequently attempt to identify +the characteristics underlying cell subpopulations. A cell subpopulation refers to a group of cells +determined by various cell types, states, or subclones. Bulk RNA-seq data does not allow cell +subpopulation-level investigation, especially for rare cell subpopulations. In contrast, the scRNA- +seq data provides the cell subpopulation-level resolution [36]. The research questions and +associated power analysis can be further divided into two categories, depending on whether +scRNA-seq experiments examine the proportion of cell subpopulations within a single tissue +(Section 3.1.1) or the proportional differences across experimental conditions for a given cell +subpopulation (Section 3.1.2). + +Table 2: A table with information about different software tools for scRNA-seq power analysis with two distinct detection targets. +Experimental Factors: Number of cells (1), Number of individuals (2), Sequencing depth (3). + + +Detection +Target +# of +Samples +Tool Name +Experimental +Factor +Software +Model +Power +Assessment +Cell sub- +population +Single sample +‘SCOPIT’ [37] +(1) +R package & +Web application +Multinomial +Analytical +'howmanycells' +Web application +Negative Binomial +Multi sample +‘Sensei‘ [38] +(1) , (2) +Beta Binomial +‘scPOST’ [39] +R package +Linear mixed model +Simulation- +based +DEG +‘scPower’ [43] +(1), (2), (3) +R package & +Web server +Negative Binomial +Pseudobulk +‘hierarchicell’ [41] +R package +Simulation- +based +Single sample +‘powsimR’ [40] +(1) +‘POWSC’ [42] +(1), (3) +A mixture of zero-inflated +Poisson and log-normal +Poisson distributions +‘scDesign’ [44] +Gamma-Normal +mixture model + +3.1.1 Ascertaining cell subpopulation proportions in a single tissue + +Multiple cell types in varying proportions compose a biological tissue. In the experimental +design phase, power analysis is indispensable for ensuring that enough cells are sampled to +adequately represent both normal and rare cell types. The following sections discuss the power +analysis for sufficient cell numbers (sample size) in a single tissue. +Two software tools, 'howmanycells' (https://satijalab.org/howmanycells) and ‘SCOPIT’ +[37], were developed specifically for cell number calculation. Using statistical models, they both +approached the problem by calculating the probability of sampling at least a predetermined +number of cells from each subpopulation. The 'howmanycells' function uses the NB distribution +to estimate the total number of cells required for adequate representation of a given cell +subpopulation under the assumption that the number of cells in each cell type is statistically +independent. This assumption may not hold in practice, but the results can be used to determine +the required minimum sample size. On the other hand, 'SCOPIT' employs the Dirichlet- +multinomial model for the distribution of the number of cells from each subpopulation, which more +accurately reflects the constraint on the proportion of cell subpopulation (i.e., proportions sum to +one). +Both 'howmanycells' and 'SCOPIT' are comparable in that they use analytical approaches, +identify the proportion of the rarest cell type as the most significant statistical factor affecting power, +and offer lightweight web applications to facilitate quick and intuitive power calculation. Above all, +their estimates of the required sample size are comparable in general. An important distinction is +that 'SCOPIT' permits retrospective analysis for hypothetical experiments, i.e., determining how +many cells would be required based on the number of sequenced cells, the number of +subpopulations detected, and their frequencies. In addition, 'SCOPIT' reports Bayesian credible +intervals for the estimated probability and number of cells to account for the uncertainty +associated with the observed empirical subpopulation frequencies. +The methods mentioned above only consider the effects of cell subpopulation proportions +and total cell number but do not account for technical factors such as sequencing depth. This is +partially due to the difficulties in obtaining an analytical solution when other factors are considered. +Note that these methods are intended to estimate the total number of cells in a single biological +sample to identify subpopulations. When dealing with multiple samples in scRNA-seq +experiments, the detection target may change, and different approaches are needed. The +following section describes the problem. + + +3.1.2 Ascertaining differential cell subpopulation proportions between distinct +experimental conditions + +In a multi-sample scRNA-seq experiment, researchers are primarily interested in +determining whether a specific cell subpopulation has differential abundances between +experimental conditions (e.g., diseased vs. healthy). In this case, the difference in cell +subpopulation proportion represents the effect size, and the number of biological samples (such +as patients and mice) represents the sample size. The proportion of cell subpopulation, the +number of cells, and the number of (biological) samples influence the power. Since cell +subpopulations are often identified by comparing marker gene expression levels, sequencing +depth may affect power since it influences technical variation. Moreover, since this is a multiple- +sample experiment, the batch effect may be significant, and the experimental design may be +unbalanced. Consequently, batch effect and experimental design (balanced or unbalanced, +paired or unpaired) may also impact the power. +Two approaches, ‘Sensei‘ [38] and ‘scPOST’ [39], have been developed for the power +analysis of distinguishing proportional differences within a cell subpopulation. The former provides +an analytical solution after a reasonable approximation, whereas the latter relies on simulation. +They both consider the potential impact of the proportion of cell subpopulation (biological factor), +the number of cells, and the number of samples (experimental factors), but only 'scPOST' +considers the effect of gene expression variation. Both works attempt to explain how to balance +the number of biological samples and the number of cells within a limited budget, and both +suggest that increasing the sample size yields greater power than increasing the number of cells +per sample. In addition, ‘scPOST’ indicates that modest reductions in sequencing depth have +negligible effects on power. +Specifically, ‘Sensei’ integrates the impacts of the number of cells and the number of +biological replicates in a mathematical framework. It models the abundance of cell types using a +beta-binomial distribution and estimates the sample size based on Welch's t-test. Under this +framework, beta distribution captures the biological difference in cell type abundance between +groups, as well as variance among samples within a group, while binomial distribution models the +technical variation caused by a limited number of cells. ‘Sensei’ provides a closed-form +representation for the statistical power upon reasonable approximation, which makes a +lightweight web application possible. As an output, ‘Sensei’ shows a table of false negative rates +for each feasible sample size combination. +Although 'Sensei' attempted to account for some biological and technical variations, the +pursuit of an analytical representation of power necessitates the adoption of assumptions and + +simplifications that may not apply to real data (e.g., assume no batch effect). In contrast, ‘scPOST’ +employs a simulation-based method to account for the effects of more factors. It begins by +estimating key parameters based on the prototype or pilot data supplied by the user. Specifically, +it assumes gene expression variation in principal components (PCs) space that arises from three +sources (batch, sample, and residual), and employs linear mixed effects models to decompose +the total variance for each PC and each cluster. Both fixed and random effects are extracted from +the fitted models, and cluster frequency mean and covariance is estimated from the prototype +dataset. In the second step, the previously estimated parameters and user-specified batch and +sample effect scale parameters are used in linear mixed effects models to simulate PC +coordinates for cells. In the final step, ‘scPOST’ employs a test based on logistic mixed effects +models to determine whether the mean frequency of a cluster differs significantly between two +conditions. The power is computed as the proportion of simulation runs in which at least one +cluster represented differential abundance. + +3.2 Power analysis for DEG detection + +Identifying DEGs is another important goal of scRNA-seq data analysis. DEG analysis can +also be divided into two categories, depending on whether the goal is to identify (i) DEGs across +different conditions (e.g., treatment vs. control) for a specific cell type or (ii) DEGs that are +differentially expressed across cell types for a given biological sample. Numerous factors can +influence power, such as effect size, number of cells, number of biological replicates, sequencing +depth, dropout rates, cell subpopulation proportion, and multiple testing methods. Given that so +many factors may affect power, it is hard to provide an analytical framework to assess power. +Therefore, most of the existing work employs simulation-based approaches, which consist of +three key steps: parameter estimation, data simulation, and power evaluation. In the parameter +estimation step, important parameters like gene-wise mean and standard deviation are estimated +from user-provided data or representative example data based on a data model. In the simulation +step, gene expression values are simulated based on the estimated parameters. Finally, in the +power evaluation step, existing DEG analysis or detection methods are applied to the simulated +data to assess power. The subsequent sections discuss the approaches in detail. + +3.2.1 DEGs across different conditions for a cell type + +Similar to bulk RNA-seq experiments, a DEG analysis can be performed to identify genes +whose expression levels vary significantly between experimental conditions. In scRNA-seq + +experiments, such DEG analysis is often performed for a specific cell type. Four software tools +are available for this type of power analysis: ‘powsimR’ [40], ‘hierarchicell’ [41], ‘POWSC’ [42], +and ‘scPower’ [43]. ‘powsimR’ and ‘POWSC’ are more suitable for single-sample experiments, +while ‘hierarchicell’ and ‘scPower’ are designed for multi-sample experiments. ‘powsimR’ +assumes an NB distribution for the count data and emphasizes the mean-dispersion relationship +during simulation. The existing package is used for DEG detection, and power-related statistics +including FDR and true positive rate (TPR) are calculated to evaluate power based on estimated +and simulated expression differences. The ‘hierarchicell’ also assumes an NB distribution for gene +expression value, and it highlights the hierarchical structure of scRNA-seq data from multiple +individuals. For power evaluation, it implements a two-part hurdle model. +‘scPower’ uses an analytical-based approach for this task. The fundamental idea behind +‘scPower’ is that a gene needs to be expressed and exceed a significance cutoff to be identified +as DEG. Therefore, it decomposes the power as the product of the expression probability +(probability of detecting an expressed gene) and the DE power (probability of significantly +expressed). For the expression probability, a pseudobulk approach is adopted. Specifically, it +sums the expression of a gene over all cells of the cell type of interest within an individual to get +the pseudobulk count for that gene. Then it calculates the probability of this pseudobulk count +greater than a threshold based on an NB distribution. Based on this probability, the probability +that the gene is expressed is obtained from a cumulative binomial distribution. The DE power is +calculated analytically based on an NB model using existing tools. + +3.2.2 DEGs across different cell types + +Identifying genes that are differentially expressed across different cell types under the +same experimental condition is another common DEG analysis, aiming to identify genes that +could distinguish from one cell type to another. ‘scDesign’ [44] and ‘POWSC’ [42] were developed +for the power analysis, and both are simulation-based approaches designed for studies involving +a single biological sample. ‘scDesign’ assumes gamma-normal distribution for log-transformed +count data. ‘POWSC’ assumes a mixture of zero-inflated Poisson and lognormal-Poisson +distributions for the count data. ‘scDesign’ and ‘POWSC’ allow user-supplied data for parameter +estimation, while ‘POWSC’ also provides precalculated parameter estimates from various tissue +types. The parameters to be estimated for ‘scDesign’ include the cell library size and cell-wise +dropout rate, as well as the gene-wise mean, standard deviation, and dropout rate. The +parameters to be estimated for ‘POWSC’ include the cell-wise zero inflation point mass and +Poisson rate, as well as gene-wise mixture proportion, mean, and variance. In the data simulation + +step, both approaches consider the constraint on total reads and allow users to choose the +number of cells, and sequencing depths under the constraint. Therefore, they can provide insights +regarding how to optimize the tradeoffs between these two experimental factors. ‘scDesign’ +performs DEG analysis using a two-sample t-test and reports five power-related measures. On +the other hand, ‘POWSC’ utilizes existing DEG analysis tools and reports both stratified and +marginal power. + +3.3 scRNA-seq power analysis tool recommendations + +As illustrated in Table 2, for the scRNA-seq experiments, a unique set of software tools +for power analysis has been developed for a specific research objective. Specifically, the tools' +distinctive features include the factors considered and the data models. Therefore, users should +consider the previously stated distinctive features when selecting an appropriate power analysis +tool. Here, we make recommendations based on these considerations. +First, the 'SCOPIT' tool is recommended when detecting cell subpopulations is the +purpose of the research. In this case, one can choose between the 'howmanycells' and 'SCOPIT'. +Both offer lightweight web applications to facilitate fast and intuitive power calculations, and their +estimates for the required number of cells are nearly identical. However, we recommend 'SCOPIT' +for this research purpose given its more comprehensive and kinder documentation. +Second, when the differential proportion of cell subpopulations is the main goal of the +research, one can choose between 'Sensei' and 'scPOST'. 'Sensei' provides a lightweight web +application that is quick and intuitive. However, 'scPOST' allows considering more factors because +it is a simulation-based method. If users desire a quick and approximate estimate of the number +of cells, 'Sensei' is a suitable option. On the other hand, 'scPOST' may be preferred if users wish +to consider various experimental and biological factors, such as the batch effect and gene +expression variation, in the statistical power analysis. +Third, 'scPower' and 'hierarchicell' are available tools for power analysis if researchers +wish to identify the genes whose expression levels differ under different experimental conditions +within a particular cell type, and multiple biological samples are involved. Between these two tools, +we recommend 'scPower' over 'hierarchicell' due to its user-friendly web application. Likewise, +'POWSC' and 'powsimR' can accomplish the task with a single sample. Between these two tools, +we recommend 'POWSC' over 'powsimR' because of the richer documentation for 'POWSC'. +Finally, if the genes characterizing one cell type from another are the primary objective, then +'scDesign' and 'POWSC' can assist. They address the restriction on total sequencing depth and +the zero-inflation issue, although they employ different data models. Between these two tools, we + +recommend 'POWSC' over 'scDesign' because 'POWSC' also reports the stratified power, i.e., +stratified based on gene expression level or zero fractions, which makes more sense given that +power depends on these two factors. + +4. Power analysis for spatial transcriptomic experiments + +4.1 Introduction of high-throughput spatial transcriptomics (HST) technology + +The lack of spatial information has limited the scope of scRNA-seq data analysis. +Technological advancements in HST have made it possible to collect gene expression data along +with spatial coordinates. HST technology enables gene expression profiling while preserving the +spatial location (coordinate) of each observational unit, depicted in Figure 1. The observational +unit can be a cell or a group of cells (spot). There are two main categories of technological +variations of HST technology: imaging-based and sequencing-based. seqFISH+ [45] and +MERFISH [46] are representative technologies for generating imaging-based HST data with a cell +as the observational unit. Due to its probe hybridization-based gene detection, imaging-based +HST data can only observe a limited number of genes. 10X Visium [47] is a standard technology +for generating sequencing-based HST data with a spot as the observational unit. Since +sequencing-based technology employs NGS technology, there are fewer restrictions on the +number of genes compared to imaging-based technology. Accordingly, there is currently a +technological trade-off between cell resolution and the number (dimension) of genes. For +instance, imaging-based HST data can be described as high-resolution and low-dimensional data, +while sequencing-based HST data can be considered as low-resolution and high-dimensional +data. Note that the spatial information from various HST data types is derived from distinct +observational units (cells and spots), which affects the type of inferences we can make. For +example, image-based HST data would be more suitable for statistical inferences requiring cell- +level resolution. +As illustrated in Figure 2, researchers can answer multiple research questions using the +HST data, including spatially variable gene (SVG) detection, tissue architecture identification, and +cell-cell communication prediction. Answering these research questions requires understanding +how to incorporate spatial information into a model to define the SVGs, tissue architecture, and +cellular phenotype. First, the SVG detection method determines which genes exhibit spatial +patterns within the target tissue, where examples include spatialDE [48], SPARK [49], and +Trendsceek [50]. spatialDE and SPARK utilize the Gaussian random effect model and the +Poisson log-normal model, respectively, with distinct normalization strategies. On the other hand, + +the Trendsceek approach detects spatial variation using a nonparametric approach. Second, the +main goal of tissue architecture identification is to group the observational units (i.e., cells or spots) +into biologically distinct clusters. Before the advent of HST technologies, previous studies +employed clustering based only on the gene expression data [51,52]. Now, additional spatial +information available in the HST data allows one to also consider the proximity between cells to +improve such clustering. Gitto [53], BayesSpace [54], and SPRUCE [55] are examples of models +employing spatial associations between observational units to identify clustering patterns. Third, +cell-cell communication analysis is to predict interactions between cells. The spatial closeness or +adjacency can provide important information to improve this type of analysis because spatially +closer cells are more likely to interact with each other. Previously, with the absence of spatial +information, interactions between ligands and receptors were predicted only based on their gene +expression patterns [56,57]. For example, CellChat [58] estimates the interaction between ligands +and receptors based on the latent distance between cells, which is calculated solely based on +gene expression data. This does not reflect the fact that cells located nearby are more likely to +interact with each other; incorporating such information can lead to higher accuracy. + +Figure 2: The figure depicts three representative research questions for the analysis of HST data. SVG +denotes the identification of a gene with a spatial pattern of gene expression. Tissue architecture refers to +the identification of a tissue's structure through the clustering of similar gene expression patterns. Cell-cell +communication, on the other hand, detects the interaction between cells using their spatial information and +gene expression data. + +SpatiallyVariableGene(SVG) +Single-cell/spot +20 +Expression +Gene +10 +TissueArchitecture +5 +A +BC +D +Cell/Spot +sub-population2 +Cell/SpotCoordinates +uojiendod-qns +Cell-CellCommunication +Given the coordinates from each observation in HST data, the spatial patterns are +modeled through the distances among observations. We note that the optimal approach to +calculate the distances among observations can be different for different data type. Figure 3 +illustrates how the imaging-based and sequence-based HST data can be regarded as different +types of spatial data. First, one can consider the imaging-based HST data as geostatistical data +or spatial point process data. Here, geostatistical data follows a spatial process that varies +continuously, but observed only at discrete points (coordinates). By using the coordinate +information, we can define the distance (e.g., Euclidean distance) among cells. The existing +models, including spatialDE and SPARK, define the spatial closeness by calculating the distances +among cell coordinates. On the other hand, the sequencing-based HST data can be thought of +as lattice or areal data observed at the discrete points or spots on a regular or irregular grid. In +the lattice data structure, the neighborhood is defined by the adjacency on the grid and the +distance between two spots is measured by the least number of spots that need to be visited +while moving from one spot to the other on the lattice. + +Figure 3: Depending on the type of HST data, it can be considered as either point process data or areal +data. First, imaging-based HST data can be regarded as point process data. For example, cell locations +are analogous to the spatial coordinates of birds’ habitats in the US. Its spatial information is modeled +through the distance among habitats. Sequencing-based HST data, on the other hand, can be regarded as +areal data on a regular grid. Here the spot, which is a group of cells, can be compared to the states' +aggregated bird habitats. Its spatial information is modeled through the adjacency or neighborhood +structure. + + +Imaging-basedHST +Sequencing-based HsT +PointProcessData +ArealDataAs shown in Figure 4, there are several key experimental factors that can affect the +generation of spatial features in HST data, including the choice of tissue area, size of the fields of +view (FoVs), the number of FoVs, and the number of cells or spots, where FoVs are defined as +the region on a tissue captured by an HST experiment. Note that such selection of FoVs and +tissue area is needed as it is often not possible to capture the whole tissue using the HST +experiment. These experimental factors can affect capturing transcripts at a specific location on +a tissue [59] or lead to a different context for capturing the region of interest, e.g., building a +neighborhood network [60]. Hence, the power analysis for HST data needs to take these +experimental factors into account to estimate the minimum number of samples to achieve a +specific analysis goal using HST data. First, the size of FoVs determines how large we measure +spatial features and gene expression locally (i.e., local capture efficiency). On the other hand, the +number of FoVs affects how many different regions on a tissue we check on a tissue (i.e., global +capture efficiency). Second, because these FoVs are not qualitatively and biologically identical, it +also matters where we capture on the tissue. For example, for the tissue architecture identification, +one might want to include the regions that contain interesting and/or rare cell sub-populations. +Likewise, for the cell-cell communication prediction, one might hope that the regions with active +cell-cell interactions are included in our HST data. Third, because the number of cells and spots +can affect signal-to-noise ratios of the generated HST data, one needs to make sure that sufficient +cells and spots are captured to avoid potential analytical and computational issues. In summary, +a rigorous experimental design that systematically considers these experimental factors will +facilitate the effective use of resources (e.g., experimental cost) by improving efficiency in +capturing the spatial features with gene expression data. + + + +Figure 4: Key experimental factors in designing HST experiments include: (1) the choice of tissue area, (2) +the number and sizes of fields of view (FoVs), and (3) the number of cells and spots. These experimental +factors can affect the statistical power to achieve the research goals, e.g., those mentioned in Figure 2. +For example, the choice of tissue area, along with the number and sizes of FoVs, can determine the degree +that biological aspects of our interest (e.g., interesting cell sub-populations, or cell-cell communications) +are captured in the generated HST data. Likewise, the number of cells and spots can affect the signal-to- +noise ratios (effect sizes) of the generated HST data. + +4.2 Literature reviews of power analysis for HST data + +Recently, Bost et al. [61] implemented several experiments to figure out how the number +of FoVs and their widths affect the coverage of the true clusters in a tissue. By changing the +number and the size of FoVs, they examined the ratio of the number of covered clusters to the +true number of clusters. It was the first attempt to investigate how the experimental design affects +the HST data analysis. For example, they calculated the required number of FoVs to discover the +true clusters in the cell phenotype and compared it between tumor samples and healthy samples. +The result showed that a larger number of FoVs are needed to capture the true clusters in tumor +samples compared to healthy samples, likely because of the complex and heterogeneous tissue + +TissueArea +ExperimentalFactor +NumberofFoVsandSizeofFoVs +NumberofCellsandSpotsstructure generated through tumorigenesis. They also applied this experiment to real data on +heart disease and breast cancer. They concluded that different types of data, such as human +body and animal tissue, have different required numbers and sizes of FoVs to recover the true +clusters. Moreover, the technologies of generating the HST data also affect the relationship +between the identification of cell clustering and the number and size of FoVs. However, the +investigation of Bost et al. [61] is limited in the sense that it was based on an empirical equation +that was not justified by any statistical model or machine learning model. Moreover, its ratio of +discovering the true cluster is not the power to discover the true clusters, whose computation +requires a large number of iterations. +In contrast to Bost et al. [61], which used an empirical equation to calculate the ratio of +covering true clusters, Baker et al. [62] employed a simulated HST approach to investigate the +design of HST experiments. Here, they performed a spatial power analysis experiment with their +devised HST data generation, called "in silico” approach. Using the in silico approach, they +generated various types of HST data as spatial profiling data such as cells in random states or +cells in self-preference states to proceed with an exploratory computational framework. They +pointed out three experimental factors to be considered in calculating the power: the number of +cells, the number of FoVs, and the size of FoVs. They applied their approach to two analytical +tasks, including cell type discovery (tissue architecture identification) and cell-cell communication. +Based on these simulation strategies, they used statistical models such as the Gamma-Poisson +model to predict how many FoVs are required to discover the cell types or cell interactions. +Through their simulation studies, they discovered that the size of FoVs and the number of FoVs +impacted the statistical power. First, in cell type discovery, they concluded that the nature of tissue +structure affects the required number of cells and FoVs to discover the true cell types. They +demonstrated this by applying the power analysis model to unstructured data of human breast +cancer, highly ordered and heterogeneous data from the mouse brain, and complex and +recurrently structured data from the mouse spleen. Second, for the cell-cell communication task, +they argued that the interactions among the cells might not be captured with the insufficient FoV +size. However, the investigation of Baker et al. [62] also has multiple limitations. First, it is hard to +directly apply their approach to point-referenced data (point process data). Specifically, the +simulation data generation model ("in silico”) is based on the blank tissue scaffold where the +random circle packing forms a planar graph, which requires strong prior knowledge for cluster +labels. This cannot capture all the variations in point reference data whose spatial locations are +randomly distributed, and the resulting pattern often exhibits non-trivial microscale variation. +Second, their investigation was limited to the number and sizes of FoVs while they ignored other + +important experimental factors that can affect the statistical power, e.g., the choice of tissue area +and the number of cells/spots mentioned in Figure 4. In summary, at this point, the optimal +strategies for statistical power analysis for HST experiments remain to be explored. + +5. Conclusions + +The advancement of transcriptomic technology has allowed researchers to expand their +scope of questioning. In order to guarantee biologically meaningful findings, rigorous experimental +design is critical, including statistical power analysis that carefully considers research questions +and data characteristics. In this review paper, we investigated the power analysis for three distinct +types of transcriptomic technologies from a practical standpoint. First, in the case of the bulk RNA- +seq experiment, the primary objective is to identify DEGs and we recommend the R package +‘ssizeRNA’ as a tool for power analysis. Second, in the case of the scRNA-seq experiment, two +main analytical goals are cell subpopulation identification and DEG detection. Specifically, +regarding cell subpopulation detection, we recommend ‘SCOPIT’ for detecting cell +subpopulations and ‘scPOST’ for inferring proportional differences across cell subpopulations. +Regarding DEG detection, we recommend ‘scPower’ for DEG detection across multiple cell sub- +populations using multiple samples, and ‘POWSC’ for DEG detection across cell sub-populations +with a single sample and within a cell subpopulation under varying experimental conditions. Third, +in the case of the HST experiment, its power analysis framework is still under-developed and we +highlight key aspects that need to be considered for the power analysis framework of HST +experiments, including research questions (SVG, tissue architecture, cell-cell communications), +technological variations (imaging- and sequencing-based HST), and experimental factors (tissue +area, the number and size of FoVs, and the number of cells or spots). We believe that this review +paper can be a useful guideline for the future design and statistical power analysis of +transcriptomic experiments. + +References +1. +Morozova, O.; Hirst, M.; Marra, M.A. Applications of new sequencing technologies for transcriptome +analysis. Annu Rev Genomics Hum Genet 2009, 10, 135-151, doi:10.1146/annurev-genom-082908-145957. +2. +Liang, K.-H.; ScienceDirect. Bioinformatics for biomedical science and clinical applications, 1st edition ed.; +Woodhead Pub: Philadelphia, NY, 2013. +3. +Haque, A.; Engel, J.; Teichmann, S.A.; Lonnberg, T. A practical guide to single-cell RNA-sequencing +for biomedical research and clinical applications. Genome Med 2017, 9, 75, doi:10.1186/s13073-017-0467- +4. + +4. +Hong, M.; Tao, S.; Zhang, L.; Diao, L.T.; Huang, X.; Huang, S.; Xie, S.J.; Xiao, Z.D.; Zhang, H. RNA +sequencing: new technologies and applications in cancer research. J Hematol Oncol 2020, 13, 166, +doi:10.1186/s13045-020-01005-x. +5. +Rao, M.S.; Van Vleet, T.R.; Ciurlionis, R.; Buck, W.R.; Mittelstadt, S.W.; Blomme, E.A.G.; Liguori, M.J. +Comparison of RNA-Seq and Microarray Gene Expression Platforms for the Toxicogenomic +Evaluation of Liver From Short-Term Rat Toxicity Studies. Front Genet 2018, 9, 636, +doi:10.3389/fgene.2018.00636. +6. +Burgess, D.J. Spatial transcriptomics coming of age. Nat Rev Genet 2019, 20, 317, doi:10.1038/s41576- +019-0129-z. +7. +Marioni, J.C.; Mason, C.E.; Mane, S.M.; Stephens, M.; Gilad, Y. RNA-seq: an assessment of technical +reproducibility and comparison with gene expression arrays. Genome Res 2008, 18, 1509-1517, +doi:10.1101/gr.079558.108. +8. +Bacher, R.; Kendziorski, C. Design and computational analysis of single-cell RNA-sequencing +experiments. Genome Biol 2016, 17, 63, doi:10.1186/s13059-016-0927-y. +9. +Schurch, N.J.; Schofield, P.; Gierliński, M.; Cole, C.; Sherstnev, A.; Singh, V.; Wrobel, N.; Gharbi, K.; +Simpson, G.G.; Owen-Hughes, T.; et al. How many biological replicates are needed in an RNA-seq +experiment and which differential expression tool should you use? Rna 2016, 22, 839-851, +doi:10.1261/rna.053959.115. +10. Liu, Y.; Zhou, J.; White, K.P. RNA-seq differential expression studies: more sequence or more +replication? Bioinformatics 2014, 30, 301-304, doi:10.1093/bioinformatics/btt688. +11. Pollen, A.A.; Nowakowski, T.J.; Shuga, J.; Wang, X.; Leyrat, A.A.; Lui, J.H.; Li, N.; Szpankowski, L.; +Fowler, B.; Chen, P.; et al. Low-coverage single-cell mRNA sequencing reveals cellular heterogeneity +and activated signaling pathways in developing cerebral cortex. Nat Biotechnol 2014, 32, 1053-1058, +doi:10.1038/nbt.2967. +12. Cohen, J. The statistical power of abnormal-social psychological research: a review. The Journal of +Abnormal and Social Psychology 1962, 65, 145. +13. Cohen, J. Statistical power analysis. Current directions in psychological science 1992, 1, 98-101. +14. Thomas, L. Retrospective power analysis. Conservation Biology 1997, 11, 276-280. +15. Chuan, C.L.; Penyelidikan, J. Sample size estimation using Krejcie and Morgan and Cohen statistical +power analysis: A comparison. Jurnal Penyelidikan IPBL 2006, 7, 78-86. +16. Sanger, F.; Nicklen, S.; Coulson, A.R. DNA sequencing with chain-terminating inhibitors. Proc Natl +Acad Sci U S A 1977, 74, 5463-5467, doi:10.1073/pnas.74.12.5463. +17. Stark, R.; Grzelak, M.; Hadfield, J. RNA sequencing: the teenage years. Nat Rev Genet 2019, 20, 631-656, +doi:10.1038/s41576-019-0150-2. +18. Van den Berge, K.; Hembach, K.M.; Soneson, C.; Tiberi, S.; Clement, L.; Love, M.I.; Patro, R.; Robinson, +M.D. RNA sequencing data: Hitchhiker's guide to expression analysis. Annual Review of Biomedical +Data Science 2019, 2, 139-173. +19. Hart, S.N.; Therneau, T.M.; Zhang, Y.; Poland, G.A.; Kocher, J.P. Calculating sample size estimates for +RNA sequencing data. J Comput Biol 2013, 20, 970-978, doi:10.1089/cmb.2012.0283. +20. Robinson, M.D.; McCarthy, D.J.; Smyth, G.K. edgeR: a Bioconductor package for differential +expression analysis of digital gene expression +data. +Bioinformatics +2010, 26, 139-140, +doi:10.1093/bioinformatics/btp616. +21. Anders, S.; Huber, W. Differential expression analysis for sequence count data. Genome Biol 2010, 11, +R106, doi:10.1186/gb-2010-11-10-r106. +22. Love, M.I.; Huber, W.; Anders, S. Moderated estimation of fold change and dispersion for RNA-seq +data with DESeq2. Genome Biol 2014, 15, 550, doi:10.1186/s13059-014-0550-8. +23. Hardcastle, T.J.; Kelly, K.A. baySeq: empirical Bayesian methods for identifying differential expression +in sequence count data. BMC Bioinformatics 2010, 11, 422, doi:10.1186/1471-2105-11-422. +24. Kvam, V.M.; Liu, P.; Si, Y. A comparison of statistical methods for detecting differentially expressed +genes from RNA-seq data. Am J Bot 2012, 99, 248-256, doi:10.3732/ajb.1100340. + +25. Li, D.; Zand, M.S.; Dye, T.D.; Goniewicz, M.L.; Rahman, I.; Xie, Z. An evaluation of RNA-seq +differential analysis methods. PLoS One 2022, 17, e0264246, doi:10.1371/journal.pone.0264246. +26. Lund, S.P.; Nettleton, D.; McCarthy, D.J.; Smyth, G.K. Detecting differential expression in RNA- +sequence data using quasi-likelihood with shrunken dispersion estimates. Stat Appl Genet Mol Biol +2012, 11, doi:10.1515/1544-6115.1826. +27. Law, C.W.; Chen, Y.; Shi, W.; Smyth, G.K. voom: Precision weights unlock linear model analysis tools +for RNA-seq read counts. Genome Biol 2014, 15, R29, doi:10.1186/gb-2014-15-2-r29. +28. Poplawski, A.; Binder, H. Feasibility of sample size calculation for RNA-seq studies. Brief Bioinform +2018, 19, 713-720, doi:10.1093/bib/bbw144. +29. Li, C.I.; Su, P.F.; Shyr, Y. Sample size calculation based on exact test for assessing differential +expression analysis in RNA-seq data. BMC Bioinformatics 2013, 14, 357, doi:10.1186/1471-2105-14-357. +30. Bi, R.; Liu, P. Sample size calculation while controlling false discovery rate for differential expression +analysis with RNA-sequencing experiments. BMC Bioinformatics 2016, 17, 146, doi:10.1186/s12859-016- +0994-9. +31. Wu, H.; Wang, C.; Wu, Z. PROPER: comprehensive power evaluation for differential expression using +RNA-seq. Bioinformatics 2015, 31, 233-241, doi:10.1093/bioinformatics/btu640. +32. Busby, M.A.; Stewart, C.; Miller, C.A.; Grzeda, K.R.; Marth, G.T. Scotty: a web tool for designing RNA- +Seq experiments to measure differential gene expression. Bioinformatics 2013, 29, 656-657, +doi:10.1093/bioinformatics/btt015. +33. Zhao, S.; Li, C.I.; Guo, Y.; Sheng, Q.; Shyr, Y. RnaSeqSampleSize: real data based sample size estimation +for RNA sequencing. BMC Bioinformatics 2018, 19, 191, doi:10.1186/s12859-018-2191-5. +34. Ching, T.; Huang, S.; Garmire, L.X. Power analysis and sample size estimation for RNA-Seq +differential expression. Rna 2014, 20, 1684-1696, doi:10.1261/rna.046011.114. +35. Pierson, E.; Yau, C. ZIFA: Dimensionality reduction for zero-inflated single-cell gene expression +analysis. Genome Biol 2015, 16, 241, doi:10.1186/s13059-015-0805-z. +36. Qi, R.; Ma, A.; Ma, Q.; Zou, Q. Clustering and classification methods for single-cell RNA-sequencing +data. Brief Bioinform 2020, 21, 1196-1208, doi:10.1093/bib/bbz062. +37. Davis, A.; Gao, R.; Navin, N.E. SCOPIT: sample size calculations for single-cell sequencing +experiments. BMC Bioinformatics 2019, 20, 566, doi:10.1186/s12859-019-3167-9. +38. Liang, S.; Willis, J.; Dou, J.; Mohanty, V.; Huang, Y.; Vilar, E.; Chen, K. Sensei: how many samples to +tell a change in cell type abundance? BMC Bioinformatics 2022, 23, 2, doi:10.1186/s12859-021-04526-5. +39. Millard, N.; Korsunsky, I.; Weinand, K.; Fonseka, C.Y.; Nathan, A.; Kang, J.B.; Raychaudhuri, S. +Maximizing statistical power to detect differentially abundant cell states with scPOST. Cell Rep Methods +2021, 1, doi:10.1016/j.crmeth.2021.100120. +40. Vieth, B.; Ziegenhain, C.; Parekh, S.; Enard, W.; Hellmann, I. powsimR: power analysis for bulk and +single cell RNA-seq experiments. Bioinformatics 2017, 33, 3486-3488, doi:10.1093/bioinformatics/btx435. +41. Zimmerman, K.D.; Langefeld, C.D. Hierarchicell: an R-package for estimating power for tests of +differential expression with single-cell data. BMC Genomics 2021, 22, 319, doi:10.1186/s12864-021- +07635-w. +42. Su, K.; Wu, Z.; Wu, H. Simulation, power evaluation and sample size recommendation for single-cell +RNA-seq. Bioinformatics 2020, 36, 4860-4868, doi:10.1093/bioinformatics/btaa607. +43. Schmid, K.T.; Höllbacher, B.; Cruceanu, C.; Böttcher, A.; Lickert, H.; Binder, E.B.; Theis, F.J.; Heinig, +M. scPower accelerates and optimizes the design of multi-sample single cell transcriptomic studies. +Nat Commun 2021, 12, 6625, doi:10.1038/s41467-021-26779-7. +44. Li, W.V.; Li, J.J. A statistical simulator scDesign for rational scRNA-seq experimental design. +Bioinformatics 2019, 35, i41-i50, doi:10.1093/bioinformatics/btz321. +45. Maynard, K.R.; Collado-Torres, L.; Weber, L.M.; Uytingco, C.; Barry, B.K.; Williams, S.R.; Catallini, +J.L., 2nd; Tran, M.N.; Besich, Z.; Tippani, M.; et al. Transcriptome-scale spatial gene expression in the +human dorsolateral prefrontal cortex. Nat Neurosci 2021, 24, 425-436, doi:10.1038/s41593-020-00787-0. + +46. Chen, K.H.; Boettiger, A.N.; Moffitt, J.R.; Wang, S.; Zhuang, X. RNA imaging. Spatially resolved, +highly +multiplexed +RNA +profiling +in +single +cells. +Science +2015, +348, +aaa6090, +doi:10.1126/science.aaa6090. +47. Ståhl, P.L.; Salmén, F.; Vickovic, S.; Lundmark, A.; Navarro, J.F.; Magnusson, J.; Giacomello, S.; Asp, +M.; Westholm, J.O.; Huss, M.; et al. Visualization and analysis of gene expression in tissue sections by +spatial transcriptomics. Science 2016, 353, 78-82, doi:10.1126/science.aaf2403. +48. Svensson, V.; Teichmann, S.A.; Stegle, O. SpatialDE: identification of spatially variable genes. Nat +Methods 2018, 15, 343-346, doi:10.1038/nmeth.4636. +49. Sun, S.; Zhu, J.; Zhou, X. Statistical analysis of spatial expression patterns for spatially resolved +transcriptomic studies. Nat Methods 2020, 17, 193-200, doi:10.1038/s41592-019-0701-7. +50. Edsgärd, D.; Johnsson, P.; Sandberg, R. Identification of spatial expression trends in single-cell gene +expression data. Nat Methods 2018, 15, 339-342, doi:10.1038/nmeth.4634. +51. Shi, J.; Luo, Z. Nonlinear dimensionality reduction of gene expression data for visualization and +clustering +analysis +of +cancer +tissue +samples. +Comput +Biol +Med +2010, +40, +723-732, +doi:10.1016/j.compbiomed.2010.06.007. +52. Ben-Dor, A.; Bruhn, L.; Friedman, N.; Nachman, I.; Schummer, M.; Yakhini, Z. Tissue classification +with gene expression profiles. J Comput Biol 2000, 7, 559-583, doi:10.1089/106652700750050943. +53. Dries, R.; Zhu, Q.; Dong, R.; Eng, C.L.; Li, H.; Liu, K.; Fu, Y.; Zhao, T.; Sarkar, A.; Bao, F.; et al. Giotto: +a toolbox for integrative analysis and visualization of spatial expression data. Genome Biol 2021, 22, 78, +doi:10.1186/s13059-021-02286-2. +54. Zhao, E.; Stone, M.R.; Ren, X.; Guenthoer, J.; Smythe, K.S.; Pulliam, T.; Williams, S.R.; Uytingco, C.R.; +Taylor, S.E.B.; Nghiem, P.; et al. Spatial transcriptomics at subspot resolution with BayesSpace. Nat +Biotechnol 2021, 39, 1375-1384, doi:10.1038/s41587-021-00935-2. +55. Allen, C.; Chang, Y.; Neelon, B.; Chang, W.; Kim, H.J.; Li, Z.; Ma, Q.; Chung, D. A Bayesian +multivariate mixture model for high throughput spatial transcriptomics. Biometrics 2022, +doi:10.1111/biom.13727. +56. Browaeys, R.; Saelens, W.; Saeys, Y. NicheNet: modeling intercellular communication by linking +ligands to target genes. Nat Methods 2020, 17, 159-162, doi:10.1038/s41592-019-0667-5. +57. Chen, Z.; Yang, X.; Bi, G.; Liang, J.; Hu, Z.; Zhao, M.; Li, M.; Lu, T.; Zheng, Y.; Sui, Q.; et al. Ligand- +receptor interaction atlas within and between tumor cells and T cells in lung adenocarcinoma. Int J +Biol Sci 2020, 16, 2205-2219, doi:10.7150/ijbs.42080. +58. Jin, S.; Guerrero-Juarez, C.F.; Zhang, L.; Chang, I.; Ramos, R.; Kuan, C.H.; Myung, P.; Plikus, M.V.; +Nie, Q. Inference and analysis of cell-cell communication using CellChat. Nat Commun 2021, 12, 1088, +doi:10.1038/s41467-021-21246-9. +59. Williams, C.G.; Lee, H.J.; Asatsuma, T.; Vento-Tormo, R.; Haque, A. An introduction to spatial +transcriptomics for biomedical research. Genome Med 2022, 14, 68, doi:10.1186/s13073-022-01075-1. +60. Lohoff, T.; Ghazanfar, S.; Missarova, A.; Koulena, N.; Pierson, N.; Griffiths, J.A.; Bardot, E.S.; Eng, C.L.; +Tyser, R.C.V.; Argelaguet, R.; et al. Integration of spatial and single-cell transcriptomic data elucidates +mouse organogenesis. Nat Biotechnol 2022, 40, 74-85, doi:10.1038/s41587-021-01006-2. +61. Bost, P.; Schulz, D.; Engler, S.; Wasserfall, C.; Bodenmiller, B. Optimizing multiplexed imaging +experimental design through tissue spatial segregation estimation. bioRxiv 2021. +62. Baker, E.A.G.; Schapiro, D.; Dumitrascu, B.; Vickovic, S.; Regev, A. Power analysis for spatial omics. +bioRxiv 2022. + + + + diff --git a/LdE1T4oBgHgl3EQfGwON/content/tmp_files/load_file.txt b/LdE1T4oBgHgl3EQfGwON/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df9d94f36fe19d1e78eb1dda1c32e67021723569 --- /dev/null +++ b/LdE1T4oBgHgl3EQfGwON/content/tmp_files/load_file.txt @@ -0,0 +1,1348 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf,len=1347 +page_content='Statistical Power Analysis for Designing Bulk, Single-Cell, and Spatial Transcriptomics Experiments: Review, Tutorial, and Perspectives Hyeongseon Jeon1,2,*, Juan Xie1,2,3,*, Yeseul Jeon1,4,5,*, Kyeong Joo Jung6, Arkobrato Gupta1,2,3, Won Chang7, Dongjun Chung1,2,# 1: Department of Biomedical Informatics, The Ohio State University, Columbus, OH, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 2: Pelotonia Institute for Immuno-Oncology, The James Comprehensive Cancer Center, The Ohio State University, Columbus, OH 43210, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3: The Interdisciplinary PhD program in Biostatistics, The Ohio State University, Columbus, Ohio, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 4: Department of Statistics and Data Science, Yonsei University, Seoul, South Korea 5: Department of Applied Statistics, Yonsei University, Seoul, South Korea 6: Department of Computer Science and Engineering, The Ohio State University, Columbus, Ohio, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 7: Division of Statistics and Data Science, University of Cincinnati, Cincinnati, Ohio, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' : Joint first authors #: Correspondence (chung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='911@osu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='edu) Abstract Gene expression profiling technologies have been used in various applications such as cancer biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The development of gene expression profiling has expanded the scope of target discovery in transcriptomic studies, and each technology produces data with distinct characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In order to guarantee biologically meaningful findings using transcriptomic experiments, it is important to consider various experimental factors in a systematic way through statistical power analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In this paper, we review and discuss the power analysis for three types of gene expression profiling technologies from a practical standpoint, including bulk RNA-seq, single-cell RNA-seq, and high-throughput spatial transcriptomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, we describe the existing power analysis tools for each research objective for each of the bulk RNA-seq and scRNA-seq experiments, along with recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' On the other hand, since there are no power analysis tools for high-throughput spatial transcriptomics at this point, we instead investigate the factors that can influence power analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Keywords Transcriptomics, gene expression analysis, power analysis, RNA-seq, scRNA-seq, high- throughput spatial transcriptomics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Introduction Transcriptomics refers to either gene expression profiling or the study of the transcriptome using gene expression profiling technologies, where transcriptome refers to the collection of all the ribonucleic acid (RNA) molecules expressed in a cell, cell type, or organism [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' According to the central dogma, RNA transcripts are generated by the cellular transcription process, play a role in protein-coding, and connect the genome, proteome, and cellular phenotype [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therefore, as a proxy for proteome analysis, numerous transcriptomic studies have analyzed messenger RNA (mRNA) molecules encoding proteins [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In addition, transcriptomic approaches have contributed to the advancement of various biological and medical studies, such as cancer biology by identifying possible prognostic biomarkers [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Transcriptomic studies can be categorized by underlying gene expression profiling technology, and technological advancements have increased the scope of target discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Figure 1 provides a summary of three types of gene expression profiling technologies in terms of their profiling resolution, data structure, and potential target discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [4] illustrate the evolution of RNA sequencing technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Unlike microarrays, which profile predefined transcript through hybridization, bulk RNA sequencing (bulk RNA-seq) allows genome-wide analysis across the whole transcriptome within a cell population by employing next-generation sequencing (NGS) technology [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In contrast to bulk RNA-seq, single-cell RNA sequencing (scRNA-seq) enables the comparison of the transcriptomes of individual cells and the analysis of heterogeneity within a cell population [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The high-throughput spatial transcriptomics (HST) technology permits gene expression profiles at the cell or close-to-cell level while also preserving spatial tissue context information [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' We note that the characteristics of the transcriptomic data are contingent on the underlying technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bulk RNA-seq data are highly reproducible, indicating that technical replicates display minimal systemic changes and are thus unnecessary [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bacher and Kendziorski [8] demonstrate that scRNA-seq data has a greater proportion of zeros, more variability, and a more complex distribution than bulk RNA-seq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Figure 1: Comparison of bulk RNA-seq, single-cell RNA-seq, and high-throughput spatial transcriptomics technologies in terms of the profiling resolution (level), data structure, and target discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' When designing a transcriptomic experiment, it is crucial to determine the experimental factors, such as the number of biological replicates, the number of cells and sequencing depth, to guarantee sufficient power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the statistical framework, power refers to the probability of detecting target discoveries, also known as sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In bulk RNA-seq analysis, Schurch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [9] provided an empirical guideline for the number of biological replicates to guarantee sufficient power, and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [10] demonstrated that the number of biological replicates has a greater influence on power than sequencing depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Pollen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [11] demonstrated that low-coverage scRNA-seq is sufficient for cell-type classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Despite the existence of basic guidelines, there exists no unifying rule due to the complexity of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For example, biological factors of the experimental unit, such as sex and breeding type, may impact power and should be considered when selecting experimental parameters more systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therefore, to determine experimental factors in transcriptomic experiments in a systematic way, a power analysis can be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Cohen [12] pioneered the concept of power analysis, which refers to the examination of the relationship between power and all parameters influencing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The parameters include desired error rate and size of the experimental effect of interest (effect size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In practice, power analysis aims to identify a parameter under the assumption that all other parameters remain constant, with power itself being considered a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In power analysis, sample size or power itself is a common target parameter [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In this review paper, the sample size refers to either the number of biological replicates or the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In an experimental study, power analysis provides crucial information at each stage of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Bulk RNA-Seq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Single-cell RNA-seq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='High-throughputSpatial Transcriptomics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Bulk Expression Profile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Single-cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Single-cell/spot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Cell/Spot Coordinates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Cell/Spot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Cell/SpotxGeneExpressionCountData ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='SubjectxGeneExpression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='CellxGeneExpressionCountData ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='CountData ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Cell/Spot2-dimensionalCoordinates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='SpatiallyVariableGenes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='DifferentiallyExpressedGenes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='DifferentiallyExpressedGenes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='TissueArchitecture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Cell Sub-populations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Cell-CellCommunicationBefore the study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' prospective power analysis helps determine the experimental factors that will provide sufficient power for detecting target discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Researchers can conduct a retrospective power analysis to evaluate the experiment, despite differing opinions regarding how to use the collected data for the power analysis, as discussed in Thomas [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Power analysis varies according to the underlying objectives of the study and how the data will be analyzed to achieve the research objective [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' As previously discussed, the employed technology affects the scope of target discoveries and transcriptomic data characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In this context, the power analysis for three distinct transcriptomic technologies will be examined, including bulk RNA-seq, scRNA-seq, and HST technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' From Sections 2 through 4, each transcriptomic technology is covered in a separate section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For a given technology, we examine the power analysis for transcriptomic experiments with respect to experimental factors, research objectives, and explanations of existing power analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' If there are power analysis tools for a particular technology and research objective, we provide recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Power analysis for bulk RNA-seq experiments 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1 Bulk RNA-seq experiment Sequencing technologies originate from Sanger sequencing, first introduced by Sanger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In 2005, the introduction of Next-Generation Sequencing (NGS), also known as massively parallel sequencing, improved sequencing in terms of high throughput, scalability, and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Especially, NGS technology enables the bulk RNA-seq profiling of gene expression levels in over ten thousand genes simultaneously in a specific tissue or cell population, where the gene expression is characterized by an abundance of messenger RNA (mRNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Typical bulk RNA-seq protocol includes sample preparation, mRNA fragmentation, reverse transcription to complementary DNA (cDNA), and mapping of cDNA fragments to a reference genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" A gene's expression level is ultimately determined by counting the cDNA fragments, called reads, that are mapped to the gene." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' See Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [17] and Van den Berge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [18] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Sequencing depth is defined as the total number of reads, influencing the sequencing's technical precision [19]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" The bulk RNA-seq profiling platforms include Illumina's HiSeq and MiSeq and ABI's SOLID." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [4] illustrate the RNA sequencing technological evolution over time and in- depth explanations of the related platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bulk RNA-seq transcriptomic experiments typically aim to identify differentially expressed genes (DEGs) across various experimental conditions, where multiple biological replicates are expected in each condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' DEGs are the bulk RNA-seq experiment’s detection target, with their detection probability determining the associated power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, the power of the bulk RNA- seq gene expression analysis is defined by the expected proportion of DEGs detected among all DEGs, following a prespecified statistical procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Unlike conventional microarray technology that generates continuous data, bulk RNA-seq generates count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Due to the discrete nature, the Poisson distribution was originally employed to model the bulk RNA-seq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' However, due to its one-parameter nature, the Poisson distribution cannot account for extra-biological variation in bulk RNA-seq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therefore, the negative binomial (NB) distribution, which can be viewed as a Poisson-gamma mixture, has gained popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Under a model assumption, a DEG is characterized as a gene whose mean expression ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', fold change) deviates from 1 for any pair of experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The difference or ratio can be understood as a measure of the effect size that characterizes DEGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioconductor packages of edgeR [20], DESeq [21], DESeq2 [22], and baySeq [23] employ the NB model to identify DEGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' While NB-based methods generally have a higher detection power, there are also reports indicating its FDR inflation [24,25] due to ignoring the uncertainty of the estimated dispersion parameters [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Alternatively, the voom method [27] can be used to detect DEGs by applying normal-based theory to the log-transformed count data, which is implemented in the limma Bioconductor package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Even though count data is not directly modeled, the voom method adjusts heterogeneous variances across all observations concurrently by utilizing an adequate mean and variance relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Additional software tools for DEG analysis are described in Schurch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [9] and Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the case of a bulk RNA-seq experiment, it is essential to determine the number of biological replicates that will provide sufficient DEG detection power, a type of power analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Consider the factors that may affect the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Note that the power depends on the assumed model's parameters and the software tools that provide the p-value for each gene under consideration." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Additionally, the power is affected by the considered error rate and the target level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bulk RNA-seq gene expression analysis typically considers multiple genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' When multiple genes are simultaneously inferred, it is common to control the false discovery rate (FDR) rather than the type 1 error rate, which is appropriate for inferring a single gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' By controlling FDR, it is possible to regulate the proportion of non-DEGs among genes declared to be DEGs on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Consequently, when inferring multiple genes and conducting power analysis, it is necessary to consider the target FDR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2 Bulk RNA-seq power analysis tools Numerous power analysis software tools calculating the number of biological replicates, alternatively sample size, for bulk RNA-seq experiments have been developed according to the factors affecting the power: model assumptions, the testing type employed for each gene, and desired error rates to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Model parameters are often estimated using pilot data, and some tools provide stored data for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' As demonstrated by data analysis in Poplawski and Binder [28], if the stored data are utilized carelessly, a highly inappropriate sample size can be suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In addition to sample size, some software tools consider sequencing depth to be an experimental factor that influences the power to be chosen during experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [10] demonstrated the tradeoff between biological replicates and sequencing depth in the context of statistical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [19] suggested a flexible power analysis approach that calculates the sample size for a single gene expression analysis using the NB model, which is implemented in the ‘RNASeqPower’ Bioconductor package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Due to the asymptotic normality of the score test statistic, a closed-form power function is obtained as a function of all possible parameters, including sample size, fold change, average sequencing depth, target type 1 error rate, and coefficient of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Due to the simplicity of the inference situation and the closed-form power function, it is possible to perceive the relationship between all parameters affecting the detection power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" [19] also suggested a sequencing depth motivated by the parameters' relationship and demonstrated that although the method does not assume FDR control, it can be extended to multiple gene inference by setting the p-value threshold α to a small value, such as 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [29] proposed a tool for calculating sample size based on the NB model and FDR control via a gene-specific power function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The approach is effectively implemented in the ‘RnaSeqSampleSize’ Bioconductor package, with an additional parameter estimation procedure supported by data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' However, the ‘RnaSeqSampleSize’ tool tends to overestimate sample size in the data analysis and data-based simulation study of Poplawski and Binder [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' To overcome this overestimation, Bi and Liu [30] suggested a method that assumes the NB model but uses the normal-based test statistic via the voom method to assess the power function partially analytically, implemented in the ‘ssizeRNA’ R package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' According to the data-driven simulation study of Poplawski and Binder [28], this approach is faster and provides the sample size closer to the actual number required to achieve the desired power, compared to other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Additionally, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [31] proposed a simulation-based FDR controlling approach, implemented in the ‘PROPER’ tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Table 1 provides a summary of the information from different power analysis tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The tools are chosen from the methods with relevant literature described in Poplawski and Binder [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Table 1: A table shows six software tools for statistical power analysis for bulk RNA-seq experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Each tool is presented along with the citation and the software environments that have been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Tool Name [Citation] (Implementation) Pilot Data Pilot Data with Stored Data Type 1 Error Poisson Lognormal ‘Scotty’ [32] (Web Interface) Negative Binomial ‘RNASeqPower’ [19] (R package) FDR ‘ssizeRNA’ [30] (R package) ‘RnaSeqSampleSize’ [33] (R package) ‘RNASeqPowerCalculator’ [34] (R package) ‘PROPER’ [31] (R package) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='3 Bulk RNA-seq power analysis tool recommendation The ‘ssizeRNA’ R package was chosen based on the outcomes of two simulation studies of Poplawski and Binder [28] and Bi and Liu [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' From the six power analysis tools mentioned in Table 1, we first considered ‘RnaSeqSampleSize’, ‘ssizeRNA’, and ‘PROPER’ based on their FDR-targeting nature and focus on a single DEG analysis tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' However, depending on the performance of the simulation studies, we decided to exclude ‘RnaSeqSampleSize’ from consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, according to Poplawski and Binder [28], ‘RnaSeqSampleSize’ typically recommends a very large sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘RnaSeqSampleSize’ performs well in Bi and Liu [30] when the model is simple, and gene-specific parameters are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' When the simulation model became realistic, the sample size suggested by ‘RnaSeqSampleSize’ was either too large to significantly exceed the desired power or too small to adequately regulate power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The subsequent selection was based on speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The simulation results presented in both papers indicate that both the ‘PROPER’ and ‘ssizeRNA’ tools recommend sample sizes with target power levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Due to the conservative nature of the voom method, the ‘ssizeRNA’ tool typically recommends a few more samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In terms of usability, however, we recommend the ‘ssizeRNA’ tool, which is faster due to its analytical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Power analysis for single-cell RNA-seq (scRNA-seq) experiments scRNA-seq technologies have revolutionized the study of transcriptomics by profiling genome-wide gene expression at the individual cell level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The cell-level information provides unprecedented opportunities for studying cellular heterogeneity and expands our understanding of developmental biology [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Even though the context of a single-cell transcriptomic study differs from that of a bulk transcriptomic study, DEG detection remains a fascinating study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In addition, the cell information enables researchers to answer questions about cell subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The relevant power analysis has been developed in response to the distinct research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In general, the sample size in scRNA-seq experiments refers to the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Due to the additional technical steps required to distinguish cells, scRNA-seq data contain more zeros than bulk RNA-seq data [11], and a zero-inflated model is frequently employed when developing statistical approaches [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2 discuss power analysis for identifying cell sub- populations and detecting DEGs, respectively, for scRNA-seq experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The information presented in Table 2 outlines a variety of power analysis tools applicable to single-cell transcriptomic experiments with distinct research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1 Power analysis for cell subpopulation detection Unlike bulk RNA-seq experiments, scRNA-seq experiments frequently attempt to identify the characteristics underlying cell subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' A cell subpopulation refers to a group of cells determined by various cell types, states, or subclones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bulk RNA-seq data does not allow cell subpopulation-level investigation, especially for rare cell subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In contrast, the scRNA- seq data provides the cell subpopulation-level resolution [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The research questions and associated power analysis can be further divided into two categories, depending on whether scRNA-seq experiments examine the proportion of cell subpopulations within a single tissue (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1) or the proportional differences across experimental conditions for a given cell subpopulation (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Table 2: A table with information about different software tools for scRNA-seq power analysis with two distinct detection targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Experimental Factors: Number of cells (1), Number of individuals (2), Sequencing depth (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Detection Target # of Samples Tool Name Experimental Factor Software Model Power Assessment Cell sub- population Single sample ‘SCOPIT’ [37] (1) R package & Web application Multinomial Analytical 'howmanycells' Web application Negative Binomial Multi sample ‘Sensei‘ [38] (1) ," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' (2) Beta Binomial ‘scPOST’ [39] R package Linear mixed model Simulation- based DEG ‘scPower’ [43] (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' (3) R package & Web server Negative Binomial Pseudobulk ‘hierarchicell’ [41] R package Simulation- based Single sample ‘powsimR’ [40] (1) ‘POWSC’ [42] (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' (3) A mixture of zero-inflated Poisson and log-normal Poisson distributions ‘scDesign’ [44] Gamma-Normal mixture model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1 Ascertaining cell subpopulation proportions in a single tissue Multiple cell types in varying proportions compose a biological tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the experimental design phase, power analysis is indispensable for ensuring that enough cells are sampled to adequately represent both normal and rare cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The following sections discuss the power analysis for sufficient cell numbers (sample size) in a single tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Two software tools, 'howmanycells' (https://satijalab." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='org/howmanycells) and ‘SCOPIT’ [37], were developed specifically for cell number calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Using statistical models, they both approached the problem by calculating the probability of sampling at least a predetermined number of cells from each subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" The 'howmanycells' function uses the NB distribution to estimate the total number of cells required for adequate representation of a given cell subpopulation under the assumption that the number of cells in each cell type is statistically independent." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' This assumption may not hold in practice, but the results can be used to determine the required minimum sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" On the other hand, 'SCOPIT' employs the Dirichlet- multinomial model for the distribution of the number of cells from each subpopulation, which more accurately reflects the constraint on the proportion of cell subpopulation (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', proportions sum to one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Both 'howmanycells' and 'SCOPIT' are comparable in that they use analytical approaches, identify the proportion of the rarest cell type as the most significant statistical factor affecting power, and offer lightweight web applications to facilitate quick and intuitive power calculation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Above all, their estimates of the required sample size are comparable in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" An important distinction is that 'SCOPIT' permits retrospective analysis for hypothetical experiments, i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', determining how many cells would be required based on the number of sequenced cells, the number of subpopulations detected, and their frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" In addition, 'SCOPIT' reports Bayesian credible intervals for the estimated probability and number of cells to account for the uncertainty associated with the observed empirical subpopulation frequencies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The methods mentioned above only consider the effects of cell subpopulation proportions and total cell number but do not account for technical factors such as sequencing depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' This is partially due to the difficulties in obtaining an analytical solution when other factors are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Note that these methods are intended to estimate the total number of cells in a single biological sample to identify subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' When dealing with multiple samples in scRNA-seq experiments, the detection target may change, and different approaches are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The following section describes the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2 Ascertaining differential cell subpopulation proportions between distinct experimental conditions In a multi-sample scRNA-seq experiment, researchers are primarily interested in determining whether a specific cell subpopulation has differential abundances between experimental conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', diseased vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' healthy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In this case, the difference in cell subpopulation proportion represents the effect size, and the number of biological samples (such as patients and mice) represents the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The proportion of cell subpopulation, the number of cells, and the number of (biological) samples influence the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Since cell subpopulations are often identified by comparing marker gene expression levels, sequencing depth may affect power since it influences technical variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Moreover, since this is a multiple- sample experiment, the batch effect may be significant, and the experimental design may be unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Consequently, batch effect and experimental design (balanced or unbalanced, paired or unpaired) may also impact the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Two approaches, ‘Sensei‘ [38] and ‘scPOST’ [39], have been developed for the power analysis of distinguishing proportional differences within a cell subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The former provides an analytical solution after a reasonable approximation, whereas the latter relies on simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" They both consider the potential impact of the proportion of cell subpopulation (biological factor), the number of cells, and the number of samples (experimental factors), but only 'scPOST' considers the effect of gene expression variation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Both works attempt to explain how to balance the number of biological samples and the number of cells within a limited budget, and both suggest that increasing the sample size yields greater power than increasing the number of cells per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In addition, ‘scPOST’ indicates that modest reductions in sequencing depth have negligible effects on power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, ‘Sensei’ integrates the impacts of the number of cells and the number of biological replicates in a mathematical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" It models the abundance of cell types using a beta-binomial distribution and estimates the sample size based on Welch's t-test." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Under this framework, beta distribution captures the biological difference in cell type abundance between groups, as well as variance among samples within a group, while binomial distribution models the technical variation caused by a limited number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘Sensei’ provides a closed-form representation for the statistical power upon reasonable approximation, which makes a lightweight web application possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' As an output, ‘Sensei’ shows a table of false negative rates for each feasible sample size combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Although 'Sensei' attempted to account for some biological and technical variations, the pursuit of an analytical representation of power necessitates the adoption of assumptions and simplifications that may not apply to real data (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', assume no batch effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In contrast, ‘scPOST’ employs a simulation-based method to account for the effects of more factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' It begins by estimating key parameters based on the prototype or pilot data supplied by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, it assumes gene expression variation in principal components (PCs) space that arises from three sources (batch, sample, and residual), and employs linear mixed effects models to decompose the total variance for each PC and each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Both fixed and random effects are extracted from the fitted models, and cluster frequency mean and covariance is estimated from the prototype dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the second step, the previously estimated parameters and user-specified batch and sample effect scale parameters are used in linear mixed effects models to simulate PC coordinates for cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the final step, ‘scPOST’ employs a test based on logistic mixed effects models to determine whether the mean frequency of a cluster differs significantly between two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The power is computed as the proportion of simulation runs in which at least one cluster represented differential abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2 Power analysis for DEG detection Identifying DEGs is another important goal of scRNA-seq data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' DEG analysis can also be divided into two categories, depending on whether the goal is to identify (i) DEGs across different conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', treatment vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' control) for a specific cell type or (ii) DEGs that are differentially expressed across cell types for a given biological sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Numerous factors can influence power, such as effect size, number of cells, number of biological replicates, sequencing depth, dropout rates, cell subpopulation proportion, and multiple testing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Given that so many factors may affect power, it is hard to provide an analytical framework to assess power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therefore, most of the existing work employs simulation-based approaches, which consist of three key steps: parameter estimation, data simulation, and power evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the parameter estimation step, important parameters like gene-wise mean and standard deviation are estimated from user-provided data or representative example data based on a data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the simulation step, gene expression values are simulated based on the estimated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Finally, in the power evaluation step, existing DEG analysis or detection methods are applied to the simulated data to assess power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The subsequent sections discuss the approaches in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1 DEGs across different conditions for a cell type Similar to bulk RNA-seq experiments, a DEG analysis can be performed to identify genes whose expression levels vary significantly between experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In scRNA-seq experiments, such DEG analysis is often performed for a specific cell type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Four software tools are available for this type of power analysis: ‘powsimR’ [40], ‘hierarchicell’ [41], ‘POWSC’ [42], and ‘scPower’ [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘powsimR’ and ‘POWSC’ are more suitable for single-sample experiments, while ‘hierarchicell’ and ‘scPower’ are designed for multi-sample experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘powsimR’ assumes an NB distribution for the count data and emphasizes the mean-dispersion relationship during simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The existing package is used for DEG detection, and power-related statistics including FDR and true positive rate (TPR) are calculated to evaluate power based on estimated and simulated expression differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The ‘hierarchicell’ also assumes an NB distribution for gene expression value, and it highlights the hierarchical structure of scRNA-seq data from multiple individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For power evaluation, it implements a two-part hurdle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘scPower’ uses an analytical-based approach for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The fundamental idea behind ‘scPower’ is that a gene needs to be expressed and exceed a significance cutoff to be identified as DEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therefore, it decomposes the power as the product of the expression probability (probability of detecting an expressed gene) and the DE power (probability of significantly expressed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For the expression probability, a pseudobulk approach is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, it sums the expression of a gene over all cells of the cell type of interest within an individual to get the pseudobulk count for that gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Then it calculates the probability of this pseudobulk count greater than a threshold based on an NB distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Based on this probability, the probability that the gene is expressed is obtained from a cumulative binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The DE power is calculated analytically based on an NB model using existing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2 DEGs across different cell types Identifying genes that are differentially expressed across different cell types under the same experimental condition is another common DEG analysis, aiming to identify genes that could distinguish from one cell type to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘scDesign’ [44] and ‘POWSC’ [42] were developed for the power analysis, and both are simulation-based approaches designed for studies involving a single biological sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘scDesign’ assumes gamma-normal distribution for log-transformed count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘POWSC’ assumes a mixture of zero-inflated Poisson and lognormal-Poisson distributions for the count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘scDesign’ and ‘POWSC’ allow user-supplied data for parameter estimation, while ‘POWSC’ also provides precalculated parameter estimates from various tissue types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The parameters to be estimated for ‘scDesign’ include the cell library size and cell-wise dropout rate, as well as the gene-wise mean, standard deviation, and dropout rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The parameters to be estimated for ‘POWSC’ include the cell-wise zero inflation point mass and Poisson rate, as well as gene-wise mixture proportion, mean, and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the data simulation step, both approaches consider the constraint on total reads and allow users to choose the number of cells, and sequencing depths under the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therefore, they can provide insights regarding how to optimize the tradeoffs between these two experimental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ‘scDesign’ performs DEG analysis using a two-sample t-test and reports five power-related measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' On the other hand, ‘POWSC’ utilizes existing DEG analysis tools and reports both stratified and marginal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='3 scRNA-seq power analysis tool recommendations As illustrated in Table 2, for the scRNA-seq experiments, a unique set of software tools for power analysis has been developed for a specific research objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Specifically, the tools' distinctive features include the factors considered and the data models." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therefore, users should consider the previously stated distinctive features when selecting an appropriate power analysis tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Here, we make recommendations based on these considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" First, the 'SCOPIT' tool is recommended when detecting cell subpopulations is the purpose of the research." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" In this case, one can choose between the 'howmanycells' and 'SCOPIT'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Both offer lightweight web applications to facilitate fast and intuitive power calculations, and their estimates for the required number of cells are nearly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" However, we recommend 'SCOPIT' for this research purpose given its more comprehensive and kinder documentation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Second, when the differential proportion of cell subpopulations is the main goal of the research, one can choose between 'Sensei' and 'scPOST'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" 'Sensei' provides a lightweight web application that is quick and intuitive." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" However, 'scPOST' allows considering more factors because it is a simulation-based method." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" If users desire a quick and approximate estimate of the number of cells, 'Sensei' is a suitable option." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" On the other hand, 'scPOST' may be preferred if users wish to consider various experimental and biological factors, such as the batch effect and gene expression variation, in the statistical power analysis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Third, 'scPower' and 'hierarchicell' are available tools for power analysis if researchers wish to identify the genes whose expression levels differ under different experimental conditions within a particular cell type, and multiple biological samples are involved." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Between these two tools, we recommend 'scPower' over 'hierarchicell' due to its user-friendly web application." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Likewise, 'POWSC' and 'powsimR' can accomplish the task with a single sample." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Between these two tools, we recommend 'POWSC' over 'powsimR' because of the richer documentation for 'POWSC'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Finally, if the genes characterizing one cell type from another are the primary objective, then 'scDesign' and 'POWSC' can assist." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' They address the restriction on total sequencing depth and the zero-inflation issue, although they employ different data models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Between these two tools, we recommend 'POWSC' over 'scDesign' because 'POWSC' also reports the stratified power, i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', stratified based on gene expression level or zero fractions, which makes more sense given that power depends on these two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Power analysis for spatial transcriptomic experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1 Introduction of high-throughput spatial transcriptomics (HST) technology The lack of spatial information has limited the scope of scRNA-seq data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Technological advancements in HST have made it possible to collect gene expression data along with spatial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' HST technology enables gene expression profiling while preserving the spatial location (coordinate) of each observational unit, depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The observational unit can be a cell or a group of cells (spot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' There are two main categories of technological variations of HST technology: imaging-based and sequencing-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' seqFISH+ [45] and MERFISH [46] are representative technologies for generating imaging-based HST data with a cell as the observational unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Due to its probe hybridization-based gene detection, imaging-based HST data can only observe a limited number of genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 10X Visium [47] is a standard technology for generating sequencing-based HST data with a spot as the observational unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Since sequencing-based technology employs NGS technology, there are fewer restrictions on the number of genes compared to imaging-based technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Accordingly, there is currently a technological trade-off between cell resolution and the number (dimension) of genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For instance, imaging-based HST data can be described as high-resolution and low-dimensional data, while sequencing-based HST data can be considered as low-resolution and high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Note that the spatial information from various HST data types is derived from distinct observational units (cells and spots), which affects the type of inferences we can make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For example, image-based HST data would be more suitable for statistical inferences requiring cell- level resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' As illustrated in Figure 2, researchers can answer multiple research questions using the HST data, including spatially variable gene (SVG) detection, tissue architecture identification, and cell-cell communication prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Answering these research questions requires understanding how to incorporate spatial information into a model to define the SVGs, tissue architecture, and cellular phenotype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' First, the SVG detection method determines which genes exhibit spatial patterns within the target tissue, where examples include spatialDE [48], SPARK [49], and Trendsceek [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' spatialDE and SPARK utilize the Gaussian random effect model and the Poisson log-normal model, respectively, with distinct normalization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' On the other hand, the Trendsceek approach detects spatial variation using a nonparametric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Second, the main goal of tissue architecture identification is to group the observational units (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', cells or spots) into biologically distinct clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Before the advent of HST technologies, previous studies employed clustering based only on the gene expression data [51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Now, additional spatial information available in the HST data allows one to also consider the proximity between cells to improve such clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Gitto [53], BayesSpace [54], and SPRUCE [55] are examples of models employing spatial associations between observational units to identify clustering patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Third, cell-cell communication analysis is to predict interactions between cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The spatial closeness or adjacency can provide important information to improve this type of analysis because spatially closer cells are more likely to interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Previously, with the absence of spatial information, interactions between ligands and receptors were predicted only based on their gene expression patterns [56,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For example, CellChat [58] estimates the interaction between ligands and receptors based on the latent distance between cells, which is calculated solely based on gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' This does not reflect the fact that cells located nearby are more likely to interact with each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' incorporating such information can lead to higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Figure 2: The figure depicts three representative research questions for the analysis of HST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' SVG denotes the identification of a gene with a spatial pattern of gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Tissue architecture refers to the identification of a tissue's structure through the clustering of similar gene expression patterns." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Cell-cell communication, on the other hand, detects the interaction between cells using their spatial information and gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' SpatiallyVariableGene(SVG) Single-cell/spot 20 Expression Gene 10 TissueArchitecture 5 A BC D Cell/Spot sub-population2 Cell/SpotCoordinates uojiendod-qns Cell-CellCommunication Given the coordinates from each observation in HST data, the spatial patterns are modeled through the distances among observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' We note that the optimal approach to calculate the distances among observations can be different for different data type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Figure 3 illustrates how the imaging-based and sequence-based HST data can be regarded as different types of spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' First, one can consider the imaging-based HST data as geostatistical data or spatial point process data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Here, geostatistical data follows a spatial process that varies continuously, but observed only at discrete points (coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' By using the coordinate information, we can define the distance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', Euclidean distance) among cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The existing models, including spatialDE and SPARK, define the spatial closeness by calculating the distances among cell coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' On the other hand, the sequencing-based HST data can be thought of as lattice or areal data observed at the discrete points or spots on a regular or irregular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In the lattice data structure, the neighborhood is defined by the adjacency on the grid and the distance between two spots is measured by the least number of spots that need to be visited while moving from one spot to the other on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Figure 3: Depending on the type of HST data, it can be considered as either point process data or areal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' First, imaging-based HST data can be regarded as point process data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For example, cell locations are analogous to the spatial coordinates of birds’ habitats in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Its spatial information is modeled through the distance among habitats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sequencing-based HST data, on the other hand, can be regarded as areal data on a regular grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" Here the spot, which is a group of cells, can be compared to the states' aggregated bird habitats." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Its spatial information is modeled through the adjacency or neighborhood structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Imaging-basedHST Sequencing-based HsT PointProcessData ArealDataAs shown in Figure 4, there are several key experimental factors that can affect the generation of spatial features in HST data, including the choice of tissue area, size of the fields of view (FoVs), the number of FoVs, and the number of cells or spots, where FoVs are defined as the region on a tissue captured by an HST experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Note that such selection of FoVs and tissue area is needed as it is often not possible to capture the whole tissue using the HST experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' These experimental factors can affect capturing transcripts at a specific location on a tissue [59] or lead to a different context for capturing the region of interest, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', building a neighborhood network [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hence, the power analysis for HST data needs to take these experimental factors into account to estimate the minimum number of samples to achieve a specific analysis goal using HST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' First, the size of FoVs determines how large we measure spatial features and gene expression locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', local capture efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' On the other hand, the number of FoVs affects how many different regions on a tissue we check on a tissue (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', global capture efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Second, because these FoVs are not qualitatively and biologically identical, it also matters where we capture on the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For example, for the tissue architecture identification, one might want to include the regions that contain interesting and/or rare cell sub-populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Likewise, for the cell-cell communication prediction, one might hope that the regions with active cell-cell interactions are included in our HST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Third, because the number of cells and spots can affect signal-to-noise ratios of the generated HST data, one needs to make sure that sufficient cells and spots are captured to avoid potential analytical and computational issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In summary, a rigorous experimental design that systematically considers these experimental factors will facilitate the effective use of resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', experimental cost) by improving efficiency in capturing the spatial features with gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Figure 4: Key experimental factors in designing HST experiments include: (1) the choice of tissue area, (2) the number and sizes of fields of view (FoVs), and (3) the number of cells and spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' These experimental factors can affect the statistical power to achieve the research goals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', those mentioned in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For example, the choice of tissue area, along with the number and sizes of FoVs, can determine the degree that biological aspects of our interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', interesting cell sub-populations, or cell-cell communications) are captured in the generated HST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Likewise, the number of cells and spots can affect the signal-to- noise ratios (effect sizes) of the generated HST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2 Literature reviews of power analysis for HST data Recently, Bost et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [61] implemented several experiments to figure out how the number of FoVs and their widths affect the coverage of the true clusters in a tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' By changing the number and the size of FoVs, they examined the ratio of the number of covered clusters to the true number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' It was the first attempt to investigate how the experimental design affects the HST data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' For example, they calculated the required number of FoVs to discover the true clusters in the cell phenotype and compared it between tumor samples and healthy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The result showed that a larger number of FoVs are needed to capture the true clusters in tumor samples compared to healthy samples, likely because of the complex and heterogeneous tissue TissueArea ExperimentalFactor NumberofFoVsandSizeofFoVs NumberofCellsandSpotsstructure generated through tumorigenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' They also applied this experiment to real data on heart disease and breast cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' They concluded that different types of data, such as human body and animal tissue, have different required numbers and sizes of FoVs to recover the true clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Moreover, the technologies of generating the HST data also affect the relationship between the identification of cell clustering and the number and size of FoVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' However, the investigation of Bost et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [61] is limited in the sense that it was based on an empirical equation that was not justified by any statistical model or machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Moreover, its ratio of discovering the true cluster is not the power to discover the true clusters, whose computation requires a large number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In contrast to Bost et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [61], which used an empirical equation to calculate the ratio of covering true clusters, Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [62] employed a simulated HST approach to investigate the design of HST experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Here, they performed a spatial power analysis experiment with their devised HST data generation, called "in silico” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Using the in silico approach, they generated various types of HST data as spatial profiling data such as cells in random states or cells in self-preference states to proceed with an exploratory computational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' They pointed out three experimental factors to be considered in calculating the power: the number of cells, the number of FoVs, and the size of FoVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' They applied their approach to two analytical tasks, including cell type discovery (tissue architecture identification) and cell-cell communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Based on these simulation strategies, they used statistical models such as the Gamma-Poisson model to predict how many FoVs are required to discover the cell types or cell interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Through their simulation studies, they discovered that the size of FoVs and the number of FoVs impacted the statistical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' First, in cell type discovery, they concluded that the nature of tissue structure affects the required number of cells and FoVs to discover the true cell types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' They demonstrated this by applying the power analysis model to unstructured data of human breast cancer, highly ordered and heterogeneous data from the mouse brain, and complex and recurrently structured data from the mouse spleen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Second, for the cell-cell communication task, they argued that the interactions among the cells might not be captured with the insufficient FoV size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' However, the investigation of Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' [62] also has multiple limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' First, it is hard to directly apply their approach to point-referenced data (point process data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, the simulation data generation model ("in silico”) is based on the blank tissue scaffold where the random circle packing forms a planar graph, which requires strong prior knowledge for cluster labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' This cannot capture all the variations in point reference data whose spatial locations are randomly distributed, and the resulting pattern often exhibits non-trivial microscale variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Second, their investigation was limited to the number and sizes of FoVs while they ignored other important experimental factors that can affect the statistical power, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', the choice of tissue area and the number of cells/spots mentioned in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In summary, at this point, the optimal strategies for statistical power analysis for HST experiments remain to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Conclusions The advancement of transcriptomic technology has allowed researchers to expand their scope of questioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In order to guarantee biologically meaningful findings, rigorous experimental design is critical, including statistical power analysis that carefully considers research questions and data characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' In this review paper, we investigated the power analysis for three distinct types of transcriptomic technologies from a practical standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' First, in the case of the bulk RNA- seq experiment, the primary objective is to identify DEGs and we recommend the R package ‘ssizeRNA’ as a tool for power analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Second, in the case of the scRNA-seq experiment, two main analytical goals are cell subpopulation identification and DEG detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Specifically, regarding cell subpopulation detection, we recommend ‘SCOPIT’ for detecting cell subpopulations and ‘scPOST’ for inferring proportional differences across cell subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Regarding DEG detection, we recommend ‘scPower’ for DEG detection across multiple cell sub- populations using multiple samples, and ‘POWSC’ for DEG detection across cell sub-populations with a single sample and within a cell subpopulation under varying experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Third, in the case of the HST experiment, its power analysis framework is still under-developed and we highlight key aspects that need to be considered for the power analysis framework of HST experiments, including research questions (SVG, tissue architecture, cell-cell communications), technological variations (imaging- and sequencing-based HST), and experimental factors (tissue area, the number and size of FoVs, and the number of cells or spots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' We believe that this review paper can be a useful guideline for the future design and statistical power analysis of transcriptomic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Morozova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hirst, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Marra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Applications of new sequencing technologies for transcriptome analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Annu Rev Genomics Hum Genet 2009, 10, 135-151, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1146/annurev-genom-082908-145957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ScienceDirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics for biomedical science and clinical applications, 1st edition ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Woodhead Pub: Philadelphia, NY, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Haque, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Engel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Teichmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lonnberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' A practical guide to single-cell RNA-sequencing for biomedical research and clinical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Med 2017, 9, 75, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s13073-017-0467- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Tao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Diao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Xiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' RNA sequencing: new technologies and applications in cancer research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' J Hematol Oncol 2020, 13, 166, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s13045-020-01005-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Rao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Van Vleet, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ciurlionis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Buck, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Mittelstadt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Blomme, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liguori, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Comparison of RNA-Seq and Microarray Gene Expression Platforms for the Toxicogenomic Evaluation of Liver From Short-Term Rat Toxicity Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Front Genet 2018, 9, 636, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='3389/fgene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='00636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Burgess, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Spatial transcriptomics coming of age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Rev Genet 2019, 20, 317, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41576- 019-0129-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Marioni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Mason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Mane, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Stephens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Gilad, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' RNA-seq: an assessment of technical reproducibility and comparison with gene expression arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Res 2008, 18, 1509-1517, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1101/gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='079558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bacher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Kendziorski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Design and computational analysis of single-cell RNA-sequencing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Biol 2016, 17, 63, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s13059-016-0927-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Schurch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Schofield, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Gierliński, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Cole, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sherstnev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Singh, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wrobel, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Gharbi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Simpson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Owen-Hughes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' How many biological replicates are needed in an RNA-seq experiment and which differential expression tool should you use?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Rna 2016, 22, 839-851, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1261/rna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='053959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' White, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' RNA-seq differential expression studies: more sequence or more replication?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics 2014, 30, 301-304, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bioinformatics/btt688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Pollen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nowakowski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Shuga, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Leyrat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Szpankowski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Fowler, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Low-coverage single-cell mRNA sequencing reveals cellular heterogeneity and activated signaling pathways in developing cerebral cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Biotechnol 2014, 32, 1053-1058, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/nbt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Cohen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The statistical power of abnormal-social psychological research: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' The Journal of Abnormal and Social Psychology 1962, 65, 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Cohen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Statistical power analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Current directions in psychological science 1992, 1, 98-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Thomas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Retrospective power analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Conservation Biology 1997, 11, 276-280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Penyelidikan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sample size estimation using Krejcie and Morgan and Cohen statistical power analysis: A comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Jurnal Penyelidikan IPBL 2006, 7, 78-86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sanger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nicklen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Coulson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' DNA sequencing with chain-terminating inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Proc Natl Acad Sci U S A 1977, 74, 5463-5467, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='5463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Stark, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Grzelak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hadfield, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' RNA sequencing: the teenage years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Rev Genet 2019, 20, 631-656, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41576-019-0150-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Van den Berge, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hembach, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Soneson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Tiberi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Clement, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Love, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Patro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Robinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=" RNA sequencing data: Hitchhiker's guide to expression analysis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Annual Review of Biomedical Data Science 2019, 2, 139-173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hart, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Therneau, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Poland, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Kocher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Calculating sample size estimates for RNA sequencing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' J Comput Biol 2013, 20, 970-978, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1089/cmb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='0283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Robinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' McCarthy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Smyth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' edgeR: a Bioconductor package for differential expression analysis of digital gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics 2010, 26, 139-140, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bioinformatics/btp616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Anders, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Huber, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Differential expression analysis for sequence count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Biol 2010, 11, R106, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/gb-2010-11-10-r106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Love, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Huber, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Anders, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Biol 2014, 15, 550, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s13059-014-0550-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hardcastle, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Kelly, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' baySeq: empirical Bayesian methods for identifying differential expression in sequence count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' BMC Bioinformatics 2010, 11, 422, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/1471-2105-11-422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Kvam, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Si, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' A comparison of statistical methods for detecting differentially expressed genes from RNA-seq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Am J Bot 2012, 99, 248-256, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='3732/ajb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1100340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Dye, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Goniewicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Rahman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Xie, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' An evaluation of RNA-seq differential analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' PLoS One 2022, 17, e0264246, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='0264246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lund, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nettleton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' McCarthy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Smyth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Detecting differential expression in RNA- sequence data using quasi-likelihood with shrunken dispersion estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Stat Appl Genet Mol Biol 2012, 11, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1515/1544-6115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Law, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Shi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Smyth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' voom: Precision weights unlock linear model analysis tools for RNA-seq read counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Biol 2014, 15, R29, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/gb-2014-15-2-r29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Poplawski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Binder, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Feasibility of sample size calculation for RNA-seq studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Brief Bioinform 2018, 19, 713-720, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bib/bbw144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Su, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Shyr, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sample size calculation based on exact test for assessing differential expression analysis in RNA-seq data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' BMC Bioinformatics 2013, 14, 357, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/1471-2105-14-357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sample size calculation while controlling false discovery rate for differential expression analysis with RNA-sequencing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' BMC Bioinformatics 2016, 17, 146, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s12859-016- 0994-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' PROPER: comprehensive power evaluation for differential expression using RNA-seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics 2015, 31, 233-241, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bioinformatics/btu640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Busby, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Stewart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Miller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Grzeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Marth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Scotty: a web tool for designing RNA- Seq experiments to measure differential gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics 2013, 29, 656-657, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bioinformatics/btt015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sheng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Shyr, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' RnaSeqSampleSize: real data based sample size estimation for RNA sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' BMC Bioinformatics 2018, 19, 191, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s12859-018-2191-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ching, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Garmire, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Power analysis and sample size estimation for RNA-Seq differential expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Rna 2014, 20, 1684-1696, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1261/rna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='046011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Pierson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Yau, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ZIFA: Dimensionality reduction for zero-inflated single-cell gene expression analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Biol 2015, 16, 241, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s13059-015-0805-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Qi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Clustering and classification methods for single-cell RNA-sequencing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Brief Bioinform 2020, 21, 1196-1208, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bib/bbz062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Davis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Gao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Navin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' SCOPIT: sample size calculations for single-cell sequencing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' BMC Bioinformatics 2019, 20, 566, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s12859-019-3167-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Willis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Dou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Mohanty, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Vilar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sensei: how many samples to tell a change in cell type abundance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' BMC Bioinformatics 2022, 23, 2, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s12859-021-04526-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Millard, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Korsunsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Weinand, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Fonseka, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nathan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Kang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Raychaudhuri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Maximizing statistical power to detect differentially abundant cell states with scPOST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Cell Rep Methods 2021, 1, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='crmeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='100120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Vieth, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ziegenhain, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Parekh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Enard, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hellmann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' powsimR: power analysis for bulk and single cell RNA-seq experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics 2017, 33, 3486-3488, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bioinformatics/btx435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zimmerman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Langefeld, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hierarchicell: an R-package for estimating power for tests of differential expression with single-cell data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' BMC Genomics 2021, 22, 319, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s12864-021- 07635-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Su, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Simulation, power evaluation and sample size recommendation for single-cell RNA-seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics 2020, 36, 4860-4868, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bioinformatics/btaa607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Schmid, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Höllbacher, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Cruceanu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Böttcher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lickert, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Binder, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Theis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Heinig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' scPower accelerates and optimizes the design of multi-sample single cell transcriptomic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Commun 2021, 12, 6625, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41467-021-26779-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' A statistical simulator scDesign for rational scRNA-seq experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bioinformatics 2019, 35, i41-i50, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1093/bioinformatics/btz321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Maynard, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Collado-Torres, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Weber, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Uytingco, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Barry, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Williams, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Catallini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=', 2nd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Tran, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Besich, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Tippani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Transcriptome-scale spatial gene expression in the human dorsolateral prefrontal cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Neurosci 2021, 24, 425-436, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41593-020-00787-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Boettiger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Moffitt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhuang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' RNA imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Spatially resolved, highly multiplexed RNA profiling in single cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Science 2015, 348, aaa6090, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='aaa6090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ståhl, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Salmén, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Vickovic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lundmark, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Navarro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Magnusson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Giacomello, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Asp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Westholm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Huss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Visualization and analysis of gene expression in tissue sections by spatial transcriptomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Science 2016, 353, 78-82, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='aaf2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Svensson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Teichmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Stegle, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' SpatialDE: identification of spatially variable genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Methods 2018, 15, 343-346, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/nmeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='4636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Statistical analysis of spatial expression patterns for spatially resolved transcriptomic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Methods 2020, 17, 193-200, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41592-019-0701-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Edsgärd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Johnsson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sandberg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Identification of spatial expression trends in single-cell gene expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Methods 2018, 15, 339-342, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/nmeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='4634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Luo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nonlinear dimensionality reduction of gene expression data for visualization and clustering analysis of cancer tissue samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Comput Biol Med 2010, 40, 723-732, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='compbiomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ben-Dor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bruhn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Friedman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nachman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Schummer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Yakhini, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Tissue classification with gene expression profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' J Comput Biol 2000, 7, 559-583, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1089/106652700750050943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Dries, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Dong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Eng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sarkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Giotto: a toolbox for integrative analysis and visualization of spatial expression data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Biol 2021, 22, 78, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s13059-021-02286-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhao, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Stone, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Guenthoer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Smythe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Pulliam, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Williams, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Uytingco, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Taylor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nghiem, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Spatial transcriptomics at subspot resolution with BayesSpace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Biotechnol 2021, 39, 1375-1384, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41587-021-00935-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Allen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Neelon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ma, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chung, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' A Bayesian multivariate mixture model for high throughput spatial transcriptomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Biometrics 2022, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1111/biom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='13727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Browaeys, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Saelens, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Saeys, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' NicheNet: modeling intercellular communication by linking ligands to target genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Methods 2020, 17, 159-162, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41592-019-0667-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Sui, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ligand- receptor interaction atlas within and between tumor cells and T cells in lung adenocarcinoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Int J Biol Sci 2020, 16, 2205-2219, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='7150/ijbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='42080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Jin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Guerrero-Juarez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Chang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ramos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Kuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Myung, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Plikus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nie, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Inference and analysis of cell-cell communication using CellChat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Commun 2021, 12, 1088, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41467-021-21246-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Williams, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Asatsuma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Vento-Tormo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Haque, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' An introduction to spatial transcriptomics for biomedical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Genome Med 2022, 14, 68, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1186/s13073-022-01075-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Lohoff, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Ghazanfar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Missarova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Koulena, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Pierson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Griffiths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bardot, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Eng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Tyser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Argelaguet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Integration of spatial and single-cell transcriptomic data elucidates mouse organogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Nat Biotechnol 2022, 40, 74-85, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='1038/s41587-021-01006-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bost, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Engler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Wasserfall, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Bodenmiller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Optimizing multiplexed imaging experimental design through tissue spatial segregation estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' bioRxiv 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Baker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Schapiro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Dumitrascu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Vickovic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Regev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' Power analysis for spatial omics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} +page_content=' bioRxiv 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf'} diff --git a/PNAyT4oBgHgl3EQf7fod/content/tmp_files/2301.00838v1.pdf.txt b/PNAyT4oBgHgl3EQf7fod/content/tmp_files/2301.00838v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..29e779551498317ed1a8b9aaa33ad01bcbcf8fb8 --- /dev/null +++ b/PNAyT4oBgHgl3EQf7fod/content/tmp_files/2301.00838v1.pdf.txt @@ -0,0 +1,1462 @@ + + + + +IAC-22- B3.7.5 + + + + + +1 +IAC-22-B3.7.5 + +Categorisation of future applications for Augmented Reality in human lunar exploration + +Paul Topf Aguiar de Medeirosa, Paul Njayoub, Flavie A. A. S. D. T. Rometschc, Dr. Tommy Nilssond, Leonie +Beckere, Dr. Aidan Cowleyf + +a European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany +hello@pauldemedeiros.nl +b European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany +paul.njayou@stud.ph-weingarten.de +c European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany +flavie.rometsch@ext.esa.int +d European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany +tommy.nilsson@esa.int +e European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany +leonie.becker1010@gmail.com +f European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany +aidan.cowley@esa.int + +Abstract +The European Space Agency (ESA) has a clear mission to go forward to the Moon in preparation of human +presence on Mars. One of the technologies looked at to increase safety and efficiency of astronauts in this context +is Augmented Reality (AR). This technology allows digital visual information to be overlaid onto the user's +environment through some type of display or projector. In recent years separate studies have been conducted to +test the potential value of AR for astronauts by implementing a few functionalities on an AR display followed by +testing in terrestrial analogue environments. One of the groups contributing to these investigations is Spaceship +EAC (SSEAC). SSEAC is a group of interns and trainees at the European Astronaut Centre (EAC) focusing on +emerging technologies for human space exploration. +This paper presents an outcome of SSEAC's activities related to AR for lunar extravehicular activities (EVAs), in +which an approach similar to design thinking was used to explore, identify, and structure the opportunities offered +by this technology. The resulting categorization of AR use cases can be used to identify new functionalities to test +through prototyping and usability tests and can also be used to relate individual studies to each other to gain insight +into the overall potential value AR has to offer to human lunar exploration. +The approach adopted in this paper is based on the Fuzzy Front End (FFE) model from the innovation management +domain. Utilising a user-driven instead of technology-driven method resulted in findings that are relevant +irrespective of the hardware and software implementation. Instead, the outcome is an overview of use cases in +which some type of AR system could provide value by contributing to increased astronaut safety, efficiency and/or +efficacy. + An initial overview of AR functions for lunar EVAs was created based on existing literature. These were +expanded on through a multidisciplinary brainstorm within SSEAC. A subsequent clustering activity resulted in +a categorisation of potential AR applications. +The following categories were defined: EVA navigation, Scientific measurements and observations, Sample +Collection, Maintenance, Repair, Overhaul (MRO) and Construction, Logistics and Inventory Management, +Medical Procedures, Biomedical and System Status Monitoring, Collaboration and Support. + +Keywords: Augmented Reality, use case classification, user centred design, Fuzzy Front End, lunar exploration, +astronaut systems + + + + + +IAC-22- B3.7.5 + + + + + +2 +Acronyms / abbreviations +AR + +Augmented Reality +COTS +Commercial Off The Shelf +ESA + +European Space Agency +EVA + +Extravehicular Activity +FFE + +Fuzzy Front end +SLS + +Space Launch System +HUD + +Heads Up Display +ISS + +International Space Station +MRO + +Maintenance, Repair, Overhaul +NASA +National Aeronautics and Space +Administration +xEMU +eXploration Extravehicular +Mobility Unit, + +1. Introduction +The international aerospace community is once again +preparing for the exploration of the lunar surface by +astronauts. Leading up to the anticipated crewed +Artemis missions, scientists and engineers are +working to define what lunar exploration will look like +in the 21st century. Humanity has come a long way +since the Apollo era, and one should expect higher +standards of safety, increased science return and +hopefully missions with a longer duration leading to +the establishment of a sustainable human presence on +the Moon. The new technological paradigm affects +every single aspect of future missions, from the suits +used during Extravehicular Activities (EVAs) to the +communication infrastructure and the tools used for +in-situ science and sample return. +This paper presents the results of a project +which aimed to create an overview and classification +of potential use cases of Augmented Reality (AR) in +the context of Lunar EVAs. Through a review of +literature, a list of applications which have been +investigated was made. Subsequently a guided +brainstorm served to generate new ideas and concepts +for novel use cases. Through a clustering activity, all +the use cases were grouped together, and a +classification +was +made +to +describe +distinct +application areas. +The +aim +was +not +to +make +a +fully +comprehensive categorization, but rather to lay the +groundwork for a user-centred design approach which +can take these and other application areas into account +in the design and development of the entire AR +system. Secondarily, the overview made in this project +can be helpful to others wishing to evaluate the +potential benefits of AR for lunar EVAs across use +cases. This more complete view of the benefits which +could be derived from such a technology development +could aid in decision-making regarding the allocation +of funds for a lunar EVA AR interface. +This paper is the result of an investigation into +the potential of AR applications for Lunar exploration +which was performed by interns at the European +Astronaut Center (EAC) and more specifically within +the Spaceship EAC group. This group consists of +interns and trainees and aims to investigate low +Technology Readiness Level technologies for space +exploration. + +1.1 Lunar exploration context +Although there have been fluctuations in the level of +interest in and funding for human space exploration +since the end of the Apollo program, there are +indications that the current upwards trend will +continue. There is international support for a strategy +in which human exploration of the Moon will be used +as a steppingstone towards human exploration of Mars +[1]. This year NASA’s Space Launch System (SLS) +and the Orion spacecraft, a collaborative achievement +between NASA and ESA, are scheduled to launch as +part of the Artemis I mission. This inaugural uncrewed +mission will prove the system’s capability to bring +humans into Lunar orbit. Meanwhile, an international +collaboration of space agencies has started working on +the next long-term human orbital outpost called +‘Lunar Gateway’, for the first time in history to be +built in Lunar orbit. NASA’s next-generation EVA +spacesuit is also being developed with Lunar surface +operations in mind [2]. The Human Landing System +is the last piece of the puzzle which will allow +astronauts to access the Lunar surface, and its +development is being funded by NASA [3]. +Later phases of the Artemis program aim to +establish longer-duration crewed lunar missions. ESA +has also envisioned the establishment of an +international lunar village, an outpost for long +duration manned planetary missions. This would be an +ideal platform not only for detailed science, but also +to prepare for the first manned Mars missions [4]. +With the prospect of increasing human deep +space exploration, the return of planetary EVAs and +all the challenges related to long-term astronaut +presence on lunar and planetary surfaces, we must +consolidate efforts to develop optimized state-of-the- +art technologies and tools to enable astronauts to work +safely and efficiently. + +1.2 Augmented reality +One such technology which has gained some interest +in the context of EVAs is AR. Augmented reality +involves the overlay of digital information onto the +user’s physical environment. There are three main +types of AR technologies currently on the market [5]: +Optical See-through AR consists of a transparent +display which allows the user to see their physical +environment behind digital projections. Video See- +through AR, commonly used in Mobile Augmented +Reality found on smartphones, consists of a display +which shows a real-time video feed from a camera +with overlayed digital information. Finally, Spatial +AR does not make use of a display, but rather projects +digital information directly onto the physical +environment. + + + + + +IAC-22- B3.7.5 + + + + + +3 +Augmented reality emerged several decades +ago and has since then been developed in a multitude +of technologies for various applications. Some of the +earliest examples were found in military cockpits to +aid pilots. Other use cases have been found in +education, training, industry and more. The use of +Augmented Reality for astronauts is also not a new +concept. As far back as the 1980s and 90s, concepts +were made for Heads up displays (HUDs) to be +integrated in EVA suit helmets [6][7]. Practical +experience has since been gained in microgravity +through experimentation with both bespoke and +Commercial Off the Shelf (COTS) AR interfaces on +board the International Space Station (ISS)[8][9]. + +1.3 AR for lunar exploration +The integration of a HUD system has been +documented as being one of the design goals for +NASA’s next generation EVA suit, called xEMU [10]. +Although there have been some published tests with +AR in the xEMU helmet [11], based on the lack of +publicly available information it appears that this +functionality is currently not on the critical design path +for the system. +Numerous studies have been performed in +which some specific functionality was implemented +as a prototype on either bespoke or COTS hardware, +to enable testing of AR functionalities in use cases +analogous to astronaut operations in space [7] [12] +[13] [11] [14] [15] [16] [17] [18] [19] [20] [21] [22] +[23] [24] [25]. With some exceptions, the studies do +not tend to adopt user-centred design processes, +instead opting to work with available technology to +demonstrate the benefits of AR in a specific use case. +In the setup of these studies, it is rarely +mentioned why the hardware used for the study was +chosen. If it is mentioned, it tends to be in the form of +an evaluation of a few available options, comparing +the suitability of these technologies to the specific use +case intended for the study. There seems to be a +knowledge gap concerning the wider context of +potential applications for AR. This makes it difficult +to +select +optimal +technologies +and +system +architectures for development, since one cannot +predict the suitability of any given technology for all +use cases if no overview of use cases exists. +The practical studies listed above choose a few +highly specific use cases or applications, but do not +tend to elaborate on how the choice for a specific use +case was made, beyond establishing that they are +relevant to the human space exploration context. This +presents a limitation in the state of the art, since one +must assume that a complex and presumably +expensive system such as an AR interface rated for use +inside an EVA suit, should be used for as broad a +range of applications as is possible and useful. +Although individual studies have contributed +significantly +to +showing +applications +of +AR +technology for human space exploration and the +benefits which can be derived from them, there seems +to be a need for a more comprehensive study of +potential applications of this technology [26]. Such an +overview would allow for a better understanding of +the full benefits which can be derived from an AR +system across use cases, which could form a stronger +basis for the allocation of the necessary funding to +develop such a system. Additionally, understanding +potential use cases of AR irrespective of the +technology used for implementation allows for a user- +centred instead of a technology-driven design +approach. + +2. Approach +The aim of this project was to create an overview and +classification of potential use cases of AR in the +context of Lunar EVAs. The adopted approach finds +similarities in the ‘Fuzzy Front End’ (FFE) phase of +the product development process from the innovation +management domain. +Defined as “the period between when an +opportunity is first considered and when an idea is +judged ready for development” [27], the FFE +approach assumes that significant value can be +derived from properly understanding the contexts, +stakeholder needs and problem definitions of a new +product before investing heavily into its development. +This is reflected in the first half of the British Design +Council’s Double Diamond model for a structured +design approach (Figure 1) [28], a widely utilized +model in the Industrial Design Engineering industry. +FFE also shares common attributes with the widely +known +‘Design +Thinking’ +approach +which +emphasizes a human-centred, iterative approach +including analysis and synthesis phases which +employ, amongst other things, brainstorms, and +clustering activities [29]. +FFE aims to develop more optimized products +by spending time to properly understand what is being +developed and why. This should result in a higher +return on investment and can prevent costly late-stage +design changes which might incur significant delays +in the delivery of a product or system [30]. +Additionally, integrating relevant data in new ways +during a well-structured FFE phase can lead to novel +and innovative solutions. [31] + + + + + +IAC-22- B3.7.5 + + + + + +4 + +Figure 1, the Double Design approach as described by the British Design Council [28] + +Characteristics of a well-structured FFE phase +tend to be multi-disciplinary, collaborative, and +iterative. The process should consist of multiple +rounds of convergent and divergent activities and can +include guided brainstorm sessions with experts, users +and/or stakeholders. This allows a learning process to +take place in which the problem is further defined, the +user is better understood, and the context is further +mapped out. Breuer et al. describe a classic FFE +approach in which certain inputs are given to an expert +brainstorm, which triggers a wide range of ideas +(divergent) +which +are +subsequently +clustered +(convergent) to form search areas. These search areas +can then form the basis for further investigation, +definition, ideation, and design (Figure 2) [32] +The classification generated in this project can +be seen as analogous to the search areas in FFE, in that +they do not specify a design or technology but rather +represent insights into user needs and context factors +such as science goals, and form demarcated areas +which aid further ideation and concept development, +breaking +free +from +convention +and +existing +assumptions about the applications of AR to develop +user-centred solutions. +The approach to forming the classification also +reflects processes commonly employed in FFE. +Starting with contextual research, existing literature +was studied to create an overview of applications +which have previously been described and/or + +Figure 2. The iterative divergent and convergent +process as described by Breuer et al. [32] + + +investigated. Subsequently, a guided brainstorm with +a multi-disciplinary team of SSEAC interns and staff +served to generate a large quantity of ideas for +potential use cases. These were then clustered to +create a categorization of AR use cases for lunar +surface exploration. Finally, the categorization was +tested against the applications described in literature +to ensure they were representative of the existing body +of work. +During the project it was decided to limit the +scope to applications and use cases of AR during lunar +EVAs. Although an even wider evaluation of +applications for all elements of a human lunar +exploration mission could be valuable, the more +limited scope helped to gather useful insights within +the limited timeframe of the project. +Publications related to among others NASA’s +IDEAS system, Holo-SEXTANT, SUITS program + +★ +Stuetions- +Twodustering +Specificalionaf +impulsesbrainstoming +contentafeach +&oorcept- +searchfelds +searchfieid. +brainstomingENGAGEMENT +DESIGN +PRINCIPLES +OUTCOME +METHODS +BANK +LEADERSHIP + + + +IAC-22- B3.7.5 + + + + + +5 +were included in the review of existing literature. Due +to the scope of the project, publications related to real- +world experiments with AR in terrestrial industry and +on the ISS such as ESA’s MobiPV4Hololens were +purposefully omitted. Inclusion of a wider selection of +studies could be beneficial to find more potential +applications, however the limitation of the scope was +instrumental to complete the project within its limited +timeframe. +The guided brainstorm was organized on +August 3, 2020. Due to restrictions related to the +COVID-19 pandemic, the brainstorm was organized +remotely, and an online whiteboard tool was used in +conjunction with video conference software. This +allowed a group of interns, trainees, and staff from +SSEAC with a wide variety of backgrounds from +computer science to aerospace engineering and +industrial design to join the session and contribute to +the ideation of potential use cases of AR for human +lunar exploration. +The first step in the brainstorm was not to +directly talk about AR applications for lunar +exploration. Instead, the ‘principle of detour’ [32] was +applied and participants were asked to write down +abstracted +potential +values +offered +by +AR +technologies regardless of their application area. +Additionally, participants were asked to write down as +many activities as they could think of that could +possibly be a part of future human lunar exploration, +without thinking about AR at all. +Subsequently, participants were asked to +combine these two inputs and generate a large number +of use cases. They were also instructed that not all use +cases had to be linked to inputs which were defined in +the previous step. To the contrary, the synthesis of use +cases from insights should ideally trigger new ideas +and insights, thereby leading to the identification of +more use cases. The brainstorm lasted 2.5 hours, and +the resulting use cases are described in section 3. +After the divergent phase, the seemingly +random and chaotic collection of ideas needs to be +ordered in some way. More than just an organization +of ideas, the process of clustering also adds value to +the creative process. By linking ideas together and +choosing specific words to describe idea-spaces, new +search areas are created which can form the basis for +whole new concepts to be developed [33], indirectly +triggered by the earlier discovery and definition steps. +The clustering activity was performed by two +authors, in an iterative process that included feedback +from other co-authors. The resulting classification can +be found in the section 3. +Finally, the classification was tested against the +applications found in existing literature. Through this +process, it was realized that there was no category +accurately representing the display of telemetry from +various external sensors and that science operations +outside of geological sampling had not been discussed +during the brainstorm. To address this, a category was +added to represent these use cases. + + + + + +IAC-22- B3.7.5 + + + + + +6 +3. Results +16 publications were included in the review of +applications mentioned and/or investigated by +existing literature. Table 1 shows an overview of the +applications per publication, worded as they are in the +original text. + + + + + + +Reference +AR applications which are investigated or suggested + +Griffin, B. (1990)[7] +Map-type graphics for navigation, pre-recorded video instructions, remote +live-streamed video from cameras, gauge readings for consumables +Hogson, E. et al. (2003) [12] +Life support and comfort control, communications, mission and task +planning, localization and situational awareness, navigation, task execution +Di Capua, M. (2008) [13] +Life support and comfort control, mission and task planning, localization +and situational awareness, navigation, task execution and human-robot +interfaces +Stolen, M. et al. (2008) [11] +Monitor the status of their own and other’s biometrics, monitor the status +of their and other’s spacesuit systems, monitor the status of robotic systems +Jacobs, S. et al. (2009) [14] +Navigation package, remaining consumables, crewmember health, suit +status +Villorin, A. (2016) [15] +Procedure lists and task instructions, consumables status, camera tools, +video communications, sensor telemetry views +Morrison, M. et al. (2017) [16] +Procedure checklists, navigational aids, display of biomedical data +Anandapadmanaban, E. et al. +(2018) [17] +Traverse plans +Gibson, A. et al. (2018) [18] +Obstacle avoidance and wayfinding +Mitra, P. (2018) [19] +Cuff checklist, suit data display, camera control, communications, caution +and warning system +Valencio D’souza, G. (2019) +[20] +Maintenance task, navigation and rocks sample collection task +Fox, K. (2020) [21] +Task instructions +McHenry, N. et al. (2020) [22] +Visual display of suit vitals, telemetry, waypoints and checklist items +Radway, S. et al. (2020) [23] +Task instruction, sampling assistance, note taking, telemetry monitoring +and display +Rometsch, F. (2020) [24] +Geological site inspection, data logging, photo documentation, taking site +coordinates, verbal field notebook, waypoints, display of suit diagnostics +Miller, L. et al. (2021) [25] +Livestream of biometric values, procedure overview, reference resources +to support activities with detailed information +Table 1: Applications described and investigated in existing literature. + +To generate a list which is more workable than the +information in table 1, the list in table 2 was made, +somewhat generalizing, and grouping specific +applications together. + + + + + + + + + + +IAC-22- B3.7.5 + + + + + +7 +Application +References +Navigation +[7][12][13][14], [16][17][18][20][22][24] +Procedure information +[7][12][13][15][16][19][20][21][22][23] +Camera live feed +[7][15][19] +Consumables monitoring +[7][11][14][15] +Life support control +[12][13] +Communications +[12][13][15][19] +Procedure planning +[12][13] +Situational awareness +[12][13] +Human-robot +and +Human-machine +interfaces +[13] [11][15] +Biometrics monitoring +[11] [14][16] +Suit system status monitoring +[11][14][19][22][23] +Note taking and data logging +[23][24] +Table 2: Generalized overview of applications described and investigated in existing literature. + + +As described in the approach section, a brainstorm +was organized in which participants were asked to +document ideas for potential values derived from AR +irrespective of application type, and to document +potential activities which might be a part of future +human lunar exploration missions. + +The following types of value which could be derived +from a lunar AR system were identified: + +For astronauts +- +Reduce cognitive load +- +More agency in accessing data +- +Increase amount of information crew can +access +- +Enhance capabilities to control vessel in +flight +- +Easier crew to crew communication +- +Free hands +- +Increased situational awareness +- +Ground can send information directly to +crew’s feed +- +Enhanced +communication +between +astronauts and ground +- +Faster assembly / maintenance +- +Decrease time needed to perform a task +- +Live adaptable instructions +- +Visual text-based communication messages +- +Sharing target of attention +- +Extend visual senses +- +Ability to reconfigure the multipurpose +interface +- +Adaptable setting +- +Integration into existing hardware + +Programmatic value +- +Enhanced PR content +- +Lower risk for accidents +- +More collaborative possibilities +- +Increased general well-being of astronauts +- +Avoid distractions for astronauts +- +Increase astronauts’ focus +- +Less need for training +- +Improved emergency response + +The following lunar activities were described: +Gateway +- +Communication, planning and preparation of +day-to-day tasks +- +Crop cultivation +- +Hardware troubleshooting +- +Tele-medicine +- +Payload deployment +- +Retrieving regolith samples +- +Hardware status observations +- +Construction of infrastructure +- +Post- and pre- EVA activities +- +Performing experiments +- +Leak detection +- +In-Situ medical care +- +Post- and pre- flight activities +- +Spare part manufacture +- +Payload maintenance +- +Resting/ sleeping +- +Cargo and stowage logistics +- +Payload upgrading + + + +Human Landing System +- +Dust mitigation in habitat +- +Collaboration between Gateway and lunar +surface + + + + + +IAC-22- B3.7.5 + + + + + +8 +- +Terrain awareness +- +Live flight data +- +Hardware troubleshooting +- +Retrieving/handling regolith samples +- +Tele-medicine +- +Preparing samples for return to Earth +- +System integrity checks +- +Leak detection +- +Communication, planning and preparation of +day-to-day tasks +- +Resting/sleeping +- +Post- and pre- EVA activities +- +Proof of concepts for fuel and oxygen storage +and transportation +- +In-situ medical care +- +Synthetic landing site markers + +Lunar surface +- +Harvesting lunar volatiles +- +Terrain awareness +- +In-situ analysis of geological samples +- +Crop cultivation +- +Tele-geology +- +Exploration of Permanently Shadowed +Regions +- +Retrieving regolith samples +- +Hardware troubleshooting +- +Dust mitigation on equipment +- +Tele-medicine +- +Traverse over rough terrain +- +proof-of-concepts for fuel and oxygen +storage and transportation +- +Co-bot operations +- +Performing experiments +- +Mapping and characterization of macro +geological features +- +Construction of infrastructure +- +Leak detection +- +Construction of roads or landing pads +- +Live checklists +- +Communication, planning and preparation of +day-to-day tasks +- +Spare part manufacture +- +Construction of infrastructure +- +In-situ medical care + +Subsequently, participants were asked to write down +as many use cases of AR for lunar exploration as they +could come up with. Each use case should have a title, +and one or two sentences detailing the function and +added value of AR in this use case (Table 3). + + + +Use Case Title +Function +Value +1 +Rover / +instrument +maintenance +Display procedures, schematics to do +maintenance work on an instrument +Less training required as procedures are +automatic and updated accordingly, easy +to follow and highlights and displays +overlays on the +2 +Construction of +roads / landing +pads +Helps astronauts in selecting areas to +construct basic infrastructure and helps +them in finding level ground to build on. +Support for construction tasks that would +require additional hardware, integrated +into a HUD. +3 +Instructions +Overlay +overlay visual assembly or maintenance +cues (highlight next screw holes, +insertion path/orientation of parts etc.) +Faster assembly / maintenance, less +training required, fewer errors. +4 +Sample +selection HUD +HUD provides overlay of information +from an IR camera to provide more +information about potential sample +composition +Increased science return from samples +more efficient use of astronaut time +5 +Communication +between +astronauts +during EVA +HUD allows astronauts to communicate +by highlighting physical objects, and by +transferring data from one to another +(e.g., location, health monitoring). +Reduces the likelihood of +misunderstandings, increases the ability +of astronauts to assist each other (e.g., +rescue), makes communication more +effective, decreases the amount of verbal +communication needed. +6 +Sample +retrieval +Display the location of a sample and +protocols to follow for retrieval +Minimize sample retrieval time +7 +Classic flight / +landing HUD +Will display flight data, landing data and +environmental data on a classic HUD +allowing astronauts to observe the Lunar +environment during critical phases. + less accidents, better situational +awareness + + + + + +IAC-22- B3.7.5 + + + + + +9 +8 +Non-vocal +one-way +communication +Messages by ground control or Gateway +can be sent to the astronaut’s HUD and +displayed there. +no need for vocal communication +9 +Medical +information in +HUD +Displaying personal vitals and vitals of +crew members. Basic vitals (e.g., blood +pressure, heart rate, O2sat). Can also +display energy expenditure and give +warnings if overexerting oneself. +Reduces the need to request medical +information. Can increase safety, increase +emergency response +10 +Checklists in +HUD +Checklists of items (i.e., deployment of +stuff, or procedures). Collaborative +checklists could possibly be +synchronized in real time. +No need for an additional device for +checklists +11 +Construction +enhancer +Simulate beams and loads and payloads +to calculate the optimal structure or +deployment + +12 + Mission +markers +Visual representation of items to be +interacted with +Good overview of where to go for the +next objective +13 +Remote support +during medical +operations +Enables an expert on the ground (i.e., +medical doctor) to provide relevant +visual information to an astronaut +performing a minor surgery. This +information can be: checklists in text +format, pre-recorded visual instructions, +virtual pointer/highlighting to guide +astronaut, live video feed from +instructor. +Reduces the amount of training needed, +increases the odds of success of surgery, +increases the flexibility in terms of +performable operations (instructor can +adapt to exact situation) +14 +Telepresence of +expert / +instructor +Overlay of video-feed of expert or +instructor enabling additional +communication channels (gestures, +demonstration of movements etc.) +Higher quality communication, easier +interaction with instructor or expert +15 +EVA mini map +Display current position around ISS, or +on lunar and/or planetary surface relative +to base camp (including surface features +etc. from satellite imagery) as well as +teammate [Gä1] ’s positions. +Increased situational or locational +awareness of self and crew. This is good +for safety, efficiency, and cooperation. +Table 3: Use cases resulting from the brainstorm + +After the brainstorm, the resulting use cases were +clustered in a collaborative and iterative process +amongst the co-authors of this publication. The +following classification (Table 4.) was deemed to be +representative of all use cases, while maintaining +sufficient differentiation between each class. It should +be noted that each class of use cases can contain +multiple specific use cases and each use case can +involve a combination of AR applications (e.g., +waypoints, procedure list) and UI elements (e.g., +video feed, overlaid data on the physical terrain). +Table 4, ‘related use cases from literature’ only refers +to use cases found in literature listed in Table 1 + + + + + +IAC-22- B3.7.5 + + + + + +10 +Use case +classification +Description +Related use cases from literature +EVA navigation +Navigation on the surface with or without +vehicle. Positioning, situational awareness and +interpretation of terrain features. +Navigation, Procedure planning, +Situational +awareness, +Human- +Robot +and +human-machine +interfaces +Scientific +measurements and +observations +Observation and interpretation of data from +science +instruments, +control +of +science +instruments, annotation and tagging of data. +Camera live feed +Sample collection +Sample collection process, sample and site +documentation and data logging. +Procedure information, procedure +planning, Camera live feed +MRO and +construction +Maintenance, Repair and Overhaul (MRO) and +construction +procedures, +instructions, +annotation, simulation, compliance testing and +data logging. +Procedure information, procedure +planning, Human-Robot and human- +machine interfaces +Logistics and +inventory +management +Inventory +tracking, +equipment +and +consumables management, process and storage +optimization. + +Medical procedures +Diagnostic, +emergency, +and +scientific +procedures. +Procedure information, procedure +planning, Huma-Robot and human- +machine interfaces +Biomedical and +system status +monitoring +Monitoring of crew member’s vitals and +critical system telemetry. +Consumables +monitoring, +Life +support control, Human-machine +interfaces, biometric monitoring, +suit system status monitoring. +Collaboration and +support +Collaboration between crew members, crew +and ground, EVA crew and crew inside a +habitat, lunar surface crew and Gateway crew +or crew and (semi)-autonomous robotic +systems. +Camera live feed, Communications, +Human-robot and Human-machine +interfaces +Table 4, classification of use cases of a lunar EVA AR +4. Discussion +The results of this project encompass a wide variety of +applications, and the classification should be useful in +the generation of new concepts and the development +of a user-centred system design. +Although efforts were made to include a wide +variety of activities and use cases, the overview of use +cases cannot be seen as comprehensive, even within +the limited scope of lunar EVAs. This is evidenced by +the fact that a significant group of activities was not +found during the brainstorm and was instead added +later, which indicates that there are likely to be other +use cases which have not been found during this +project. Ostensibly, making a complete overview of +activities might not be possible until the actual mission +profiles have been decided on. Until that time, one can +however assume a certain value to be inherent in +insights which aim to be diverse if not complete. +A certain transition is evident between the +‘applications’ +of +technology-driven +design +developments and evaluations - which constitute most +of the existing literature - and the ‘use cases’ which +are more relevant for the user-centred approach. The +difference +can +be +described +as +applications +representing +technical +functions +(i.e., +placing +waypoints, displaying a list of procedures, controlling +the Life Support System, see ‘Table 1’) whereas use +cases +represent +activities +with +more +clear +stakeholders, contexts and goals (i.e., ‘guiding non- +geologists during geological inspection tasks’ [34]). +The latter feeds directly into user-centred concept +development and could allow designs to let go of +conventions informed by the paradigm of outdated +technologies. Any realistic system should however +keep in mind the proven processes and designs which +have been in use for decades. Future designs should +incorporate these to benefit from their reliability and +compatibility with existing systems. +Although a user-centred approach can lead to +novel and optimized designs, one could argue that +technical limitations should be given as much +importance as design considerations as user needs. +Especially for a technology which should work inside +an EVA suit in use, extreme technical challenges need +to be overcome to create a functioning system. For +example, the electronics must be safe to use in the +oxygen-rich environment inside a suit, integration of +multiple systems such as GPS and IoT networks can +rapidly increase complexity and cost, and redundancy +must be built into systems which are critical for +mission success and astronaut safety. All this +considered, the technology-driven approach does not + + + + + +IAC-22- B3.7.5 + + + + + +11 +guarantee that these limitations are considered, since +many studies are based on terrestrial COTS systems +and would not fulfil these requirements. And a user- +centred approach would include considerations for +technical limitations in the design embodiment and +detailing phases, as represented for example in the +iterative ‘develop and deliver’ diamond shown in +Figure 1. +This project has proven that there are relevant +methodologies from the innovation management +domain that could be applied to the development of +complex systems for human space exploration. Future +studies could potentially identify more opportunities +for the development of user-centred systems for +astronauts when applying methodologies from the +innovation management and design engineering +domains, as also evidenced by Rometsch et al. [35]. + +The main subject of this project was the +classification of potential AR use cases for human +lunar exploration. Although the outcome should be +useful in its current form, one can imagine an even +more comprehensive classification process which +would not limit the scope to EVAs but to all activities +related to human lunar and planetary exploration. +Furthermore, the approach which was used to +create the classification could be formalized further, +ensuring +that +the +resulting +categorization +is +comprehensive and individual classes are sufficiently +differentiated from each other. An example of an +excellent formalized classification of AR use cases +was performed by Röltgen and Dumitrescu and could +serve as an inspiration for further work in the subject +area of this publication [36]. +By focusing specifically on visual AR systems, +the potential value of multi-modal AR systems might +have been overlooked. Multi-modal AR systems use a +mix of stimuli to provide data to the user instead of +solely using visual displays. For example, Gibson et +al. studied the use of haptic feedback in astronaut +boots for obstacle avoidance +[18]. Although +challenging, it is likely worthwhile to include multi- +modal interfaces as a consideration in the further +development and evaluation of AR systems for lunar +exploration. + +5. Conclusion +This project has fulfilled its aim of generating a +classification of potential use cases of AR for human +lunar surface exploration. Although the scope had to +be narrowed down to AR for EVAs, the hope is that +future work can identify use cases for every potential +context of use for an astronaut AR system . A more +formalized process for classification might yield +results which are more comprehensive with more +precisely defined categories. However, it is expected +that the results from this project already in their +current form can help to evaluate potential AR +technologies, support concept development of novel +AR functions and provide a framework to bring +together results from individual studies and start to +form a picture of the full potential value which might +be gained from the development of an AR system for +human space exploration. +The following categories were defined: EVA +navigation, +Scientific +measurements +and +observations, +Sample +Collection, +Maintenance, +Repair, Overhaul (MRO) and Construction, Logistics +and Inventory Management, Medical Procedures, +Biomedical +and +System +Status +Monitoring, +Collaboration and Support. + + + + + + +IAC-22- B3.7.5 + + + + + +12 +References + +[1] +International Space Exploration Coordination Group, “The Global Exploration Roadmap,” Jan. 2018. +Accessed: Aug. 31, 2022. [Online]. Available: www.globalspaceexploration.org. +[2] +B. K. Alpert and B. J. Johnson, “Extravehicular activity framework for exploration - 2019,” 2019. +Accessed: Aug. 26, 2022. [Online]. Available: https://ntrs.nasa.gov/citations/20190028714 +[3] +R. C. Weber et al., “The Artemis III Science Definition Report,” in 52nd Lunar and Planetary Science +Conference, 2021, p. 1261. +[4] +J. Woerner, “ESA - Moon Village,” 2016. +https://www.esa.int/About_Us/Ministerial_Council_2016/Moon_Village (accessed Aug. 31, 2022). +[5] +A. Samini, K. L. Palmerius, and P. Ljung, “A Review of Current, Complete Augmented Reality +Solutions,” in 2021 International Conference on Cyberworlds (CW), Sep. 2021, pp. 49–56. doi: +10.1109/CW52790.2021.00015. +[6] +C. C. Gernux, R. W. Blaser, and J. Marmolejo, “A Helmet Mounted Display Demonstration unit for a +Space Station Application,” Jul. 1989, p. 891583. doi: 10.4271/891583. +[7] +B. Griffin, “A space suit for lunar construction and exploration,” Sep. 1990. doi: 10.2514/6.1990-3885. +[8] +V. Byrne, J. Mauldin, and B. Munson, “Treadmill 2 Augmented Reality (T2 AR) ISS Flight +Demonstration,” System Problem Resolution Team Meeting. Jul. 2019. +[9] +“NASA, Microsoft Collaborate to Bring Science Fiction to Science Fact | NASA.” +https://www.nasa.gov/press-release/nasa-microsoft-collaborate-to-bring-science-fiction-to-science-fact +(accessed Aug. 31, 2022). +[10] +D. Coan, “Relevant Environments for Analysis and Development (READy): Enabling Human Space +Exploration Through Integrated Operational Testing Enabling human space exploration through the +integration of mission planning, EVA, science, engineering, and operations,” Jul. 2019. Accessed: Sep. +01, 2022. [Online]. Available: +https://ntrs.nasa.gov/api/citations/20190027335/downloads/20190027335.pdf +[11] +M. F. Stolen, B. Dillow, S. E. Jacobs, and D. L. Akin, “Interface for EVA Human-Machine Interaction,” +Jun. 2008, pp. 2008–01–1986. doi: 10.4271/2008-01-1986. +[12] +E. Hogson et al., “Requirements and Potential for Enhanced EVA Information Interfaces,” Jul. 2003. +[13] +M. di Capua, “Augmented reality for space applications,” University of Maryland, 2008. +[14] +S. E. Jacobs, M. di Capua, S.-A. A. Husain, A. Mirvis, and D. L. Akin, “Incorporating Advanced +Controls, Displays and other Smart Elements into Space Suit Design,” SAE International Journal of +Aerospace, vol. 4, no. 1, pp. 374–384, Jul. 2009, doi: 10.4271/2009-01-2472. +[15] +A. Villorin, “IDEAS: Integrated Display and Environmental Awareness System.” Jun. 2016. +[16] +M. Morrison et al., “Research Gaps and Opportunities in Sensor-Based Medical Exploration +Capabilities in Extravehicular Astronaut Suits,” J Biosens Bioelectron, vol. 08, no. 04, 2017, doi: +10.4172/2155-6210.1000248. +[17] +E. Anandapadmanaban, J. Tannady, J. Norheim, D. J. Newman, and J. A. Hoffman, “Holo-SEXTANT: +an Augmented Reality Planetary EVA Navigation Interface,” Dec. 2018. [Online]. Available: https://ttu- +ir.tdl.org/bitstream/handle/2346/74212/ICES_2018_264.pdf?sequence=1 +[18] +A. Gibson, A. Webb, and L. Stirling, “Evaluation of a Visual–Tactile Multimodal Display for Surface +Obstacle Avoidance During Walking,” IEEE Trans Hum Mach Syst, vol. 48, no. 6, pp. 604–613, Dec. +2018, doi: 10.1109/THMS.2018.2849018. +[19] +P. Mitra, “Human systems integration of an extravehicular activity space suit augmented reality display +system,” Mississippi State University, 2018. +[20] +G. Valencio D’souza, “The Effectiveness of Augmented Reality for Astronauts on Lunar Missions: An +Analog Study,” Embry-Riddle Aeronautical University, 2019. [Online]. Available: +https://commons.erau.edu/edt/483/ +[21] +K. Fox et al., “Developing an Integrated Heads-Up Display for Astronauts,” 2020, doi: 10.13016/TTT1- +Q3XJ. +[22] +N. McHenry et al., “Design of an AR Visor Display System for Extravehicular Activity Operations,” in +2020 IEEE Aerospace Conference, Mar. 2020, pp. 1–11. doi: 10.1109/AERO47225.2020.9172268. +[23] +S. Radway et al., “Beyond LunAR: An augmented reality UI for deep-space exploration missions,” +arXiv:2011.14535 [cs], Nov. 2020, [Online]. Available: http://arxiv.org/abs/2011.14535 +[24] +F. Rometsch, “Design of an AR-IoT Tool for Future Human Space Exploration,” Delft University of +Technology, Delft, 2020. + + + + + +IAC-22- B3.7.5 + + + + + +13 +[25] +L. S. Miller, M. J. Fornito, R. Flanagan, and R. L. Kobrick, “Development of an Augmented Reality +Interface to Aid Astronauts in Extravehicular Activities,” in 2021 IEEE Aerospace Conference (50100), +Mar. 2021, pp. 1–12. doi: 10.1109/AERO50100.2021.9438430. +[26] +K. H. Beaton et al., “Mission enhancing capabilities for science-driven exploration extravehicular +activity derived from the NASA BASALT research program,” Planet Space Sci, vol. 193, p. 105003, +Nov. 2020, doi: 10.1016/j.pss.2020.105003. +[27] +J. Kim and D. Wilemon, “Focusing the fuzzy front–end in new product development,” R&D +Management, vol. 32, no. 4, pp. 269–279, Sep. 2002, doi: 10.1111/1467-9310.00259. +[28] +“Framework for Innovation: Design Council’s evolved Double Diamond - Design Council.” +https://www.designcouncil.org.uk/our-work/skills-learning/tools-frameworks/framework-for- +innovation-design-councils-evolved-double-diamond/ (accessed Aug. 29, 2022). +[29] +J. Auernhammer and B. Roth, “The origin and evolution of Stanford University’s design thinking: From +product design to design thinking in innovation management,” Journal of Product Innovation +Management, vol. 38, no. 6, pp. 623–644, Nov. 2021, doi: 10.1111/JPIM.12594. +[30] +U. Brentani and S. E. Reid, “The Fuzzy Front-End of Discontinuous Innovation: Insights for Research +and Management: The Front-End of Discontinuous Innovation: Insights,” Journal of Product Innovation +Management, vol. 29, no. 1, pp. 70–87, Jan. 2012, doi: 10.1111/j.1540-5885.2011.00879.x. +[31] +D. J. Flint, “Compressing new product success-to-success cycle time,” Industrial Marketing +Management, vol. 31, no. 4, pp. 305–315, Jul. 2002, doi: 10.1016/S0019-8501(01)00165-1. +[32] +H. Breuer, M. Hewing, and F. Steinhoff, “Divergent innovation: Fostering and managing the fuzzy front +end of innovation,” in PICMET ’09 - 2009 Portland International Conference on Management of +Engineering & Technology, Aug. 2009, pp. 754–761. doi: 10.1109/PICMET.2009.5262066. +[33] +M. Tassoul and J. Buijs, “Clustering: An Essential Step from Diverging to Converging,” Creativity and +Innovation Management, vol. 16, no. 1, pp. 16–26, Mar. 2007, doi: 10.1111/j.1467-8691.2007.00413.x. +[34] +F. A. A. S. D. T. Rometsch, A. E. M. Casini, A. Drepper, A. Cowley, J. C. F. de Winter, and J. Guo, +“Design and evaluation of an Augmented Reality tool for future human space exploration aided by an +Internet of Things architecture,” Advances in Space Research, Jul. 2022, doi: +10.1016/J.ASR.2022.07.045. +[35] +F. A. A. S. D. T. Rometsch et al., “Towards a human-centred framework for conceptualization of lunar +surface solutions,” Sep. 2022. +[36] +D. Röltgen and R. Dumitrescu, “Classification of industrial Augmented Reality use cases,” Procedia +CIRP, vol. 91, pp. 93–100, Jan. 2020, doi: 10.1016/J.PROCIR.2020.01.137. + + diff --git a/PNAyT4oBgHgl3EQf7fod/content/tmp_files/load_file.txt b/PNAyT4oBgHgl3EQf7fod/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..afddd71aa64f2cb8269bc04249ac62760494c14c --- /dev/null +++ b/PNAyT4oBgHgl3EQf7fod/content/tmp_files/load_file.txt @@ -0,0 +1,851 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf,len=850 +page_content='IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 1 IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 Categorisation of future applications for Augmented Reality in human lunar exploration Paul Topf Aguiar de Medeirosa, Paul Njayoub, Flavie A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Rometschc, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Tommy Nilssond, Leonie Beckere, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Aidan Cowleyf a European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany hello@pauldemedeiros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='nl b European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='njayou@stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='ph-weingarten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='de c European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany flavie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='rometsch@ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='int d European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany tommy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='nilsson@esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='int e European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany leonie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='becker1010@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='com f European Space Agency (ESA), European Astronaut Centre (EAC), Linder Höhe, 51147 Cologne, Germany aidan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='cowley@esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='int Abstract The European Space Agency (ESA) has a clear mission to go forward to the Moon in preparation of human presence on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' One of the technologies looked at to increase safety and efficiency of astronauts in this context is Augmented Reality (AR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=" This technology allows digital visual information to be overlaid onto the user's environment through some type of display or projector." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' In recent years separate studies have been conducted to test the potential value of AR for astronauts by implementing a few functionalities on an AR display followed by testing in terrestrial analogue environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' One of the groups contributing to these investigations is Spaceship EAC (SSEAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' SSEAC is a group of interns and trainees at the European Astronaut Centre (EAC) focusing on emerging technologies for human space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=" This paper presents an outcome of SSEAC's activities related to AR for lunar extravehicular activities (EVAs), in which an approach similar to design thinking was used to explore, identify, and structure the opportunities offered by this technology." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The resulting categorization of AR use cases can be used to identify new functionalities to test through prototyping and usability tests and can also be used to relate individual studies to each other to gain insight into the overall potential value AR has to offer to human lunar exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The approach adopted in this paper is based on the Fuzzy Front End (FFE) model from the innovation management domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Utilising a user-driven instead of technology-driven method resulted in findings that are relevant irrespective of the hardware and software implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Instead, the outcome is an overview of use cases in which some type of AR system could provide value by contributing to increased astronaut safety, efficiency and/or efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' An initial overview of AR functions for lunar EVAs was created based on existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' These were expanded on through a multidisciplinary brainstorm within SSEAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A subsequent clustering activity resulted in a categorisation of potential AR applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The following categories were defined: EVA navigation, Scientific measurements and observations, Sample Collection, Maintenance, Repair, Overhaul (MRO) and Construction, Logistics and Inventory Management, Medical Procedures, Biomedical and System Status Monitoring, Collaboration and Support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Keywords: Augmented Reality, use case classification, user centred design, Fuzzy Front End, lunar exploration, astronaut systems IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 2 Acronyms / abbreviations AR Augmented Reality COTS Commercial Off The Shelf ESA European Space Agency EVA Extravehicular Activity FFE Fuzzy Front end SLS Space Launch System HUD Heads Up Display ISS International Space Station MRO Maintenance, Repair, Overhaul NASA National Aeronautics and Space Administration xEMU eXploration Extravehicular Mobility Unit, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Introduction The international aerospace community is once again preparing for the exploration of the lunar surface by astronauts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Leading up to the anticipated crewed Artemis missions, scientists and engineers are working to define what lunar exploration will look like in the 21st century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Humanity has come a long way since the Apollo era, and one should expect higher standards of safety, increased science return and hopefully missions with a longer duration leading to the establishment of a sustainable human presence on the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The new technological paradigm affects every single aspect of future missions, from the suits used during Extravehicular Activities (EVAs) to the communication infrastructure and the tools used for in-situ science and sample return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This paper presents the results of a project which aimed to create an overview and classification of potential use cases of Augmented Reality (AR) in the context of Lunar EVAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Through a review of literature, a list of applications which have been investigated was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Subsequently a guided brainstorm served to generate new ideas and concepts for novel use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Through a clustering activity, all the use cases were grouped together, and a classification was made to describe distinct application areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The aim was not to make a fully comprehensive categorization, but rather to lay the groundwork for a user-centred design approach which can take these and other application areas into account in the design and development of the entire AR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Secondarily, the overview made in this project can be helpful to others wishing to evaluate the potential benefits of AR for lunar EVAs across use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This more complete view of the benefits which could be derived from such a technology development could aid in decision-making regarding the allocation of funds for a lunar EVA AR interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This paper is the result of an investigation into the potential of AR applications for Lunar exploration which was performed by interns at the European Astronaut Center (EAC) and more specifically within the Spaceship EAC group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This group consists of interns and trainees and aims to investigate low Technology Readiness Level technologies for space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1 Lunar exploration context Although there have been fluctuations in the level of interest in and funding for human space exploration since the end of the Apollo program, there are indications that the current upwards trend will continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' There is international support for a strategy in which human exploration of the Moon will be used as a steppingstone towards human exploration of Mars [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This year NASA’s Space Launch System (SLS) and the Orion spacecraft, a collaborative achievement between NASA and ESA, are scheduled to launch as part of the Artemis I mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This inaugural uncrewed mission will prove the system’s capability to bring humans into Lunar orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Meanwhile, an international collaboration of space agencies has started working on the next long-term human orbital outpost called ‘Lunar Gateway’, for the first time in history to be built in Lunar orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' NASA’s next-generation EVA spacesuit is also being developed with Lunar surface operations in mind [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The Human Landing System is the last piece of the puzzle which will allow astronauts to access the Lunar surface, and its development is being funded by NASA [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Later phases of the Artemis program aim to establish longer-duration crewed lunar missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' ESA has also envisioned the establishment of an international lunar village, an outpost for long duration manned planetary missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This would be an ideal platform not only for detailed science, but also to prepare for the first manned Mars missions [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' With the prospect of increasing human deep space exploration, the return of planetary EVAs and all the challenges related to long-term astronaut presence on lunar and planetary surfaces, we must consolidate efforts to develop optimized state-of-the- art technologies and tools to enable astronauts to work safely and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2 Augmented reality One such technology which has gained some interest in the context of EVAs is AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Augmented reality involves the overlay of digital information onto the user’s physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' There are three main types of AR technologies currently on the market [5]: Optical See-through AR consists of a transparent display which allows the user to see their physical environment behind digital projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Video See- through AR, commonly used in Mobile Augmented Reality found on smartphones, consists of a display which shows a real-time video feed from a camera with overlayed digital information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Finally, Spatial AR does not make use of a display, but rather projects digital information directly onto the physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 3 Augmented reality emerged several decades ago and has since then been developed in a multitude of technologies for various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Some of the earliest examples were found in military cockpits to aid pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Other use cases have been found in education, training, industry and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The use of Augmented Reality for astronauts is also not a new concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' As far back as the 1980s and 90s, concepts were made for Heads up displays (HUDs) to be integrated in EVA suit helmets [6][7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Practical experience has since been gained in microgravity through experimentation with both bespoke and Commercial Off the Shelf (COTS) AR interfaces on board the International Space Station (ISS)[8][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='3 AR for lunar exploration The integration of a HUD system has been documented as being one of the design goals for NASA’s next generation EVA suit, called xEMU [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although there have been some published tests with AR in the xEMU helmet [11], based on the lack of publicly available information it appears that this functionality is currently not on the critical design path for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Numerous studies have been performed in which some specific functionality was implemented as a prototype on either bespoke or COTS hardware, to enable testing of AR functionalities in use cases analogous to astronaut operations in space [7] [12] [13] [11] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' With some exceptions, the studies do not tend to adopt user-centred design processes, instead opting to work with available technology to demonstrate the benefits of AR in a specific use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' In the setup of these studies, it is rarely mentioned why the hardware used for the study was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' If it is mentioned, it tends to be in the form of an evaluation of a few available options, comparing the suitability of these technologies to the specific use case intended for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' There seems to be a knowledge gap concerning the wider context of potential applications for AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This makes it difficult to select optimal technologies and system architectures for development, since one cannot predict the suitability of any given technology for all use cases if no overview of use cases exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The practical studies listed above choose a few highly specific use cases or applications, but do not tend to elaborate on how the choice for a specific use case was made, beyond establishing that they are relevant to the human space exploration context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This presents a limitation in the state of the art, since one must assume that a complex and presumably expensive system such as an AR interface rated for use inside an EVA suit, should be used for as broad a range of applications as is possible and useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although individual studies have contributed significantly to showing applications of AR technology for human space exploration and the benefits which can be derived from them, there seems to be a need for a more comprehensive study of potential applications of this technology [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Such an overview would allow for a better understanding of the full benefits which can be derived from an AR system across use cases, which could form a stronger basis for the allocation of the necessary funding to develop such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Additionally, understanding potential use cases of AR irrespective of the technology used for implementation allows for a user- centred instead of a technology-driven design approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Approach The aim of this project was to create an overview and classification of potential use cases of AR in the context of Lunar EVAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The adopted approach finds similarities in the ‘Fuzzy Front End’ (FFE) phase of the product development process from the innovation management domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Defined as “the period between when an opportunity is first considered and when an idea is judged ready for development” [27], the FFE approach assumes that significant value can be derived from properly understanding the contexts, stakeholder needs and problem definitions of a new product before investing heavily into its development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This is reflected in the first half of the British Design Council’s Double Diamond model for a structured design approach (Figure 1) [28], a widely utilized model in the Industrial Design Engineering industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' FFE also shares common attributes with the widely known ‘Design Thinking’ approach which emphasizes a human-centred, iterative approach including analysis and synthesis phases which employ, amongst other things, brainstorms, and clustering activities [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' FFE aims to develop more optimized products by spending time to properly understand what is being developed and why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This should result in a higher return on investment and can prevent costly late-stage design changes which might incur significant delays in the delivery of a product or system [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Additionally, integrating relevant data in new ways during a well-structured FFE phase can lead to novel and innovative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [31] IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 4 Figure 1, the Double Design approach as described by the British Design Council [28] Characteristics of a well-structured FFE phase tend to be multi-disciplinary, collaborative, and iterative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The process should consist of multiple rounds of convergent and divergent activities and can include guided brainstorm sessions with experts, users and/or stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This allows a learning process to take place in which the problem is further defined, the user is better understood, and the context is further mapped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Breuer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' describe a classic FFE approach in which certain inputs are given to an expert brainstorm, which triggers a wide range of ideas (divergent) which are subsequently clustered (convergent) to form search areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' These search areas can then form the basis for further investigation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' ideation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' and design (Figure 2) [32] The classification generated in this project can be seen as analogous to the search areas in FFE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' in that they do not specify a design or technology but rather represent insights into user needs and context factors such as science goals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' and form demarcated areas which aid further ideation and concept development,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' breaking free from convention and existing assumptions about the applications of AR to develop user-centred solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The approach to forming the classification also reflects processes commonly employed in FFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Starting with contextual research, existing literature was studied to create an overview of applications which have previously been described and/or Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The iterative divergent and convergent process as described by Breuer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [32] investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Subsequently, a guided brainstorm with a multi-disciplinary team of SSEAC interns and staff served to generate a large quantity of ideas for potential use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' These were then clustered to create a categorization of AR use cases for lunar surface exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Finally, the categorization was tested against the applications described in literature to ensure they were representative of the existing body of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' During the project it was decided to limit the scope to applications and use cases of AR during lunar EVAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although an even wider evaluation of applications for all elements of a human lunar exploration mission could be valuable, the more limited scope helped to gather useful insights within the limited timeframe of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Publications related to among others NASA’s IDEAS system, Holo-SEXTANT, SUITS program ★ Stuetions Twodustering Specificalionaf impulsesbrainstoming contentafeach &oorcept searchfelds searchfieid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' brainstomingENGAGEMENT DESIGN PRINCIPLES OUTCOME METHODS BANK LEADERSHIP IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 5 were included in the review of existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Due to the scope of the project, publications related to real- world experiments with AR in terrestrial industry and on the ISS such as ESA’s MobiPV4Hololens were purposefully omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Inclusion of a wider selection of studies could be beneficial to find more potential applications, however the limitation of the scope was instrumental to complete the project within its limited timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The guided brainstorm was organized on August 3, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Due to restrictions related to the COVID-19 pandemic, the brainstorm was organized remotely, and an online whiteboard tool was used in conjunction with video conference software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This allowed a group of interns, trainees, and staff from SSEAC with a wide variety of backgrounds from computer science to aerospace engineering and industrial design to join the session and contribute to the ideation of potential use cases of AR for human lunar exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The first step in the brainstorm was not to directly talk about AR applications for lunar exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Instead, the ‘principle of detour’ [32] was applied and participants were asked to write down abstracted potential values offered by AR technologies regardless of their application area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Additionally, participants were asked to write down as many activities as they could think of that could possibly be a part of future human lunar exploration, without thinking about AR at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Subsequently, participants were asked to combine these two inputs and generate a large number of use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' They were also instructed that not all use cases had to be linked to inputs which were defined in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' To the contrary, the synthesis of use cases from insights should ideally trigger new ideas and insights, thereby leading to the identification of more use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The brainstorm lasted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 hours, and the resulting use cases are described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' After the divergent phase, the seemingly random and chaotic collection of ideas needs to be ordered in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' More than just an organization of ideas, the process of clustering also adds value to the creative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' By linking ideas together and choosing specific words to describe idea-spaces, new search areas are created which can form the basis for whole new concepts to be developed [33], indirectly triggered by the earlier discovery and definition steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The clustering activity was performed by two authors, in an iterative process that included feedback from other co-authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The resulting classification can be found in the section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Finally, the classification was tested against the applications found in existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Through this process, it was realized that there was no category accurately representing the display of telemetry from various external sensors and that science operations outside of geological sampling had not been discussed during the brainstorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' To address this, a category was added to represent these use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Results 16 publications were included in the review of applications mentioned and/or investigated by existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Table 1 shows an overview of the applications per publication, worded as they are in the original text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Reference AR applications which are investigated or suggested Griffin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (1990)[7] Map-type graphics for navigation, pre-recorded video instructions, remote live-streamed video from cameras, gauge readings for consumables Hogson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2003) [12] Life support and comfort control, communications, mission and task planning, localization and situational awareness, navigation, task execution Di Capua, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2008) [13] Life support and comfort control, mission and task planning, localization and situational awareness, navigation, task execution and human-robot interfaces Stolen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2008) [11] Monitor the status of their own and other’s biometrics, monitor the status of their and other’s spacesuit systems, monitor the status of robotic systems Jacobs, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2009) [14] Navigation package, remaining consumables, crewmember health, suit status Villorin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2016) [15] Procedure lists and task instructions, consumables status, camera tools, video communications, sensor telemetry views Morrison, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2017) [16] Procedure checklists, navigational aids, display of biomedical data Anandapadmanaban, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2018) [17] Traverse plans Gibson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2018) [18] Obstacle avoidance and wayfinding Mitra, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2018) [19] Cuff checklist, suit data display, camera control, communications, caution and warning system Valencio D’souza, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2019) [20] Maintenance task, navigation and rocks sample collection task Fox, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2020) [21] Task instructions McHenry, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2020) [22] Visual display of suit vitals, telemetry, waypoints and checklist items Radway, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2020) [23] Task instruction, sampling assistance, note taking, telemetry monitoring and display Rometsch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2020) [24] Geological site inspection, data logging, photo documentation, taking site coordinates, verbal field notebook, waypoints, display of suit diagnostics Miller, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' (2021) [25] Livestream of biometric values, procedure overview, reference resources to support activities with detailed information Table 1: Applications described and investigated in existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' To generate a list which is more workable than the information in table 1, the list in table 2 was made, somewhat generalizing, and grouping specific applications together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 7 Application References Navigation [7][12][13][14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[16][17][18][20][22][24] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Procedure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[7][12][13][15][16][19][20][21][22][23] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='live ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='feed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[7][15][19] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Consumables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[7][11][14][15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Life ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='support ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[12][13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Communications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[12][13][15][19] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Procedure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='planning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[12][13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Situational ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='awareness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[12][13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Human-robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Human-machine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='interfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[11][15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Biometrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[14][16] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Suit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='status ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[11][14][19][22][23] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Note ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='taking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='logging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='[23][24] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Generalized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='overview ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='described ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='investigated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='existing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' As described in the approach section, a brainstorm was organized in which participants were asked to document ideas for potential values derived from AR irrespective of application type, and to document potential activities which might be a part of future human lunar exploration missions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='The following types of value which could be derived from a lunar AR system were identified: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='For ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='astronauts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Reduce ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='cognitive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='load ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='More ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='agency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='accessing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Increase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='amount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='crew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='can ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='access ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Enhance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='capabilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='vessel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='flight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Easier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='crew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='crew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Free ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='hands ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Increased ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='situational ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='awareness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Ground ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='can ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='send ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='directly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='crew’s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='feed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Enhanced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='between ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='astronauts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='ground ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Faster ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='assembly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='maintenance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Decrease ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='needed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='perform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Live ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='adaptable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='instructions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Visual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='text-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='messages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Sharing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Extend ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='visual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='senses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Ability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='reconfigure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='multipurpose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Adaptable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='setting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Integration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='existing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='hardware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Programmatic value Enhanced PR content Lower risk for accidents More collaborative possibilities Increased general well being of astronauts Avoid distractions for astronauts Increase astronauts’ focus Less need for training Improved emergency response ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='The following lunar activities were described: Gateway - Communication,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' planning and preparation of day-to-day tasks - Crop cultivation - Hardware troubleshooting - Tele-medicine - Payload deployment - Retrieving regolith samples - Hardware status observations - Construction of infrastructure - Post- and pre- EVA activities - Performing experiments - Leak detection - In-Situ medical care - Post- and pre- flight activities - Spare part manufacture - Payload maintenance - Resting/ sleeping - Cargo and stowage logistics - Payload upgrading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Human Landing System - Dust mitigation in habitat - Collaboration between Gateway and lunar surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 8 - Terrain awareness - Live flight data - Hardware troubleshooting - Retrieving/handling regolith samples - Tele-medicine - Preparing samples for return to Earth - System integrity checks - Leak detection - Communication,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' planning and preparation of day-to-day tasks - Resting/sleeping - Post- and pre- EVA activities - Proof of concepts for fuel and oxygen storage and transportation - In-situ medical care - Synthetic landing site markers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Lunar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Harvesting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='lunar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='volatiles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Terrain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='awareness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='In-situ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='geological ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Crop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='cultivation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Tele-geology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Exploration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Permanently ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Shadowed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Regions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Retrieving ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='regolith ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Hardware ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='troubleshooting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Dust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='mitigation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Tele-medicine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Traverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='over ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='rough ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='terrain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='proof-of-concepts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='fuel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='oxygen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='storage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='transportation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Co-bot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='operations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Performing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='experiments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='characterization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='macro ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='geological ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Construction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='infrastructure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Leak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Construction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='roads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='landing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='pads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Live ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='checklists ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='Communication,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' planning and preparation of day-to-day tasks - Spare part manufacture - Construction of infrastructure - In-situ medical care Subsequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' participants were asked to write down as many use cases of AR for lunar exploration as they could come up with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Each use case should have a title, and one or two sentences detailing the function and added value of AR in this use case (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Use Case Title Function Value 1 Rover / instrument maintenance Display procedures, schematics to do maintenance work on an instrument Less training required as procedures are automatic and updated accordingly, easy to follow and highlights and displays overlays on the 2 Construction of roads / landing pads Helps astronauts in selecting areas to construct basic infrastructure and helps them in finding level ground to build on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Support for construction tasks that would require additional hardware, integrated into a HUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 3 Instructions Overlay overlay visual assembly or maintenance cues (highlight next screw holes, insertion path/orientation of parts etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=') Faster assembly / maintenance, less training required, fewer errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 4 Sample selection HUD HUD provides overlay of information from an IR camera to provide more information about potential sample composition Increased science return from samples more efficient use of astronaut time 5 Communication between astronauts during EVA HUD allows astronauts to communicate by highlighting physical objects, and by transferring data from one to another (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', location, health monitoring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Reduces the likelihood of misunderstandings, increases the ability of astronauts to assist each other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', rescue), makes communication more effective, decreases the amount of verbal communication needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 6 Sample retrieval Display the location of a sample and protocols to follow for retrieval Minimize sample retrieval time 7 Classic flight / landing HUD Will display flight data, landing data and environmental data on a classic HUD allowing astronauts to observe the Lunar environment during critical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' less accidents, better situational awareness IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 9 8 Non-vocal one-way communication Messages by ground control or Gateway can be sent to the astronaut’s HUD and displayed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' no need for vocal communication 9 Medical information in HUD Displaying personal vitals and vitals of crew members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Basic vitals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', blood pressure, heart rate, O2sat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Can also display energy expenditure and give warnings if overexerting oneself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Reduces the need to request medical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Can increase safety, increase emergency response 10 Checklists in HUD Checklists of items (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', deployment of stuff, or procedures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Collaborative checklists could possibly be synchronized in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' No need for an additional device for checklists 11 Construction enhancer Simulate beams and loads and payloads to calculate the optimal structure or deployment 12 Mission markers Visual representation of items to be interacted with Good overview of where to go for the next objective 13 Remote support during medical operations Enables an expert on the ground (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', medical doctor) to provide relevant visual information to an astronaut performing a minor surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This information can be: checklists in text format, pre-recorded visual instructions, virtual pointer/highlighting to guide astronaut, live video feed from instructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Reduces the amount of training needed, increases the odds of success of surgery, increases the flexibility in terms of performable operations (instructor can adapt to exact situation) 14 Telepresence of expert / instructor Overlay of video-feed of expert or instructor enabling additional communication channels (gestures, demonstration of movements etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=') Higher quality communication, easier interaction with instructor or expert 15 EVA mini map Display current position around ISS, or on lunar and/or planetary surface relative to base camp (including surface features etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' from satellite imagery) as well as teammate [Gä1] ’s positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Increased situational or locational awareness of self and crew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This is good for safety, efficiency, and cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Table 3: Use cases resulting from the brainstorm After the brainstorm, the resulting use cases were clustered in a collaborative and iterative process amongst the co-authors of this publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The following classification (Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=') was deemed to be representative of all use cases, while maintaining sufficient differentiation between each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' It should be noted that each class of use cases can contain multiple specific use cases and each use case can involve a combination of AR applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', waypoints, procedure list) and UI elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', video feed, overlaid data on the physical terrain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Table 4, ‘related use cases from literature’ only refers to use cases found in literature listed in Table 1 IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 10 Use case classification Description Related use cases from literature EVA navigation Navigation on the surface with or without vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Positioning, situational awareness and interpretation of terrain features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Navigation, Procedure planning, Situational awareness, Human- Robot and human-machine interfaces Scientific measurements and observations Observation and interpretation of data from science instruments, control of science instruments, annotation and tagging of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Camera live feed Sample collection Sample collection process, sample and site documentation and data logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Procedure information, procedure planning, Camera live feed MRO and construction Maintenance, Repair and Overhaul (MRO) and construction procedures, instructions, annotation, simulation, compliance testing and data logging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Procedure information, procedure planning, Human-Robot and human- machine interfaces Logistics and inventory management Inventory tracking, equipment and consumables management, process and storage optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Medical procedures Diagnostic, emergency, and scientific procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Procedure information, procedure planning, Huma-Robot and human- machine interfaces Biomedical and system status monitoring Monitoring of crew member’s vitals and critical system telemetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Consumables monitoring, Life support control, Human-machine interfaces, biometric monitoring, suit system status monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Collaboration and support Collaboration between crew members, crew and ground, EVA crew and crew inside a habitat, lunar surface crew and Gateway crew or crew and (semi)-autonomous robotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Camera live feed, Communications, Human-robot and Human-machine interfaces Table 4, classification of use cases of a lunar EVA AR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Discussion The results of this project encompass a wide variety of applications, and the classification should be useful in the generation of new concepts and the development of a user-centred system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although efforts were made to include a wide variety of activities and use cases, the overview of use cases cannot be seen as comprehensive, even within the limited scope of lunar EVAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This is evidenced by the fact that a significant group of activities was not found during the brainstorm and was instead added later, which indicates that there are likely to be other use cases which have not been found during this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Ostensibly, making a complete overview of activities might not be possible until the actual mission profiles have been decided on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Until that time, one can however assume a certain value to be inherent in insights which aim to be diverse if not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A certain transition is evident between the ‘applications’ of technology-driven design developments and evaluations - which constitute most of the existing literature - and the ‘use cases’ which are more relevant for the user-centred approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The difference can be described as applications representing technical functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', placing waypoints, displaying a list of procedures, controlling the Life Support System, see ‘Table 1’) whereas use cases represent activities with more clear stakeholders, contexts and goals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', ‘guiding non- geologists during geological inspection tasks’ [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The latter feeds directly into user-centred concept development and could allow designs to let go of conventions informed by the paradigm of outdated technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Any realistic system should however keep in mind the proven processes and designs which have been in use for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Future designs should incorporate these to benefit from their reliability and compatibility with existing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although a user-centred approach can lead to novel and optimized designs, one could argue that technical limitations should be given as much importance as design considerations as user needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Especially for a technology which should work inside an EVA suit in use, extreme technical challenges need to be overcome to create a functioning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' For example, the electronics must be safe to use in the oxygen-rich environment inside a suit, integration of multiple systems such as GPS and IoT networks can rapidly increase complexity and cost, and redundancy must be built into systems which are critical for mission success and astronaut safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' All this considered, the technology-driven approach does not IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 11 guarantee that these limitations are considered, since many studies are based on terrestrial COTS systems and would not fulfil these requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' And a user- centred approach would include considerations for technical limitations in the design embodiment and detailing phases, as represented for example in the iterative ‘develop and deliver’ diamond shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' This project has proven that there are relevant methodologies from the innovation management domain that could be applied to the development of complex systems for human space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Future studies could potentially identify more opportunities for the development of user-centred systems for astronauts when applying methodologies from the innovation management and design engineering domains, as also evidenced by Rometsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The main subject of this project was the classification of potential AR use cases for human lunar exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although the outcome should be useful in its current form, one can imagine an even more comprehensive classification process which would not limit the scope to EVAs but to all activities related to human lunar and planetary exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Furthermore, the approach which was used to create the classification could be formalized further, ensuring that the resulting categorization is comprehensive and individual classes are sufficiently differentiated from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' An example of an excellent formalized classification of AR use cases was performed by Röltgen and Dumitrescu and could serve as an inspiration for further work in the subject area of this publication [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' By focusing specifically on visual AR systems, the potential value of multi-modal AR systems might have been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Multi-modal AR systems use a mix of stimuli to provide data to the user instead of solely using visual displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' For example, Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' studied the use of haptic feedback in astronaut boots for obstacle avoidance [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although challenging, it is likely worthwhile to include multi- modal interfaces as a consideration in the further development and evaluation of AR systems for lunar exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Conclusion This project has fulfilled its aim of generating a classification of potential use cases of AR for human lunar surface exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Although the scope had to be narrowed down to AR for EVAs, the hope is that future work can identify use cases for every potential context of use for an astronaut AR system .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A more formalized process for classification might yield results which are more comprehensive with more precisely defined categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' However, it is expected that the results from this project already in their current form can help to evaluate potential AR technologies, support concept development of novel AR functions and provide a framework to bring together results from individual studies and start to form a picture of the full potential value which might be gained from the development of an AR system for human space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' The following categories were defined: EVA navigation, Scientific measurements and observations, Sample Collection, Maintenance, Repair, Overhaul (MRO) and Construction, Logistics and Inventory Management, Medical Procedures, Biomedical and System Status Monitoring, Collaboration and Support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 12 References [1] International Space Exploration Coordination Group, “The Global Exploration Roadmap,” Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Accessed: Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 31, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Available: www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='globalspaceexploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Alpert and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Johnson, “Extravehicular activity framework for exploration - 2019,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Accessed: Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 26, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Available: https://ntrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='gov/citations/20190028714 [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “The Artemis III Science Definition Report,” in 52nd Lunar and Planetary Science Conference, 2021, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Woerner, “ESA - Moon Village,” 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='int/About_Us/Ministerial_Council_2016/Moon_Village (accessed Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 31, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Samini, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Palmerius, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Ljung, “A Review of Current, Complete Augmented Reality Solutions,” in 2021 International Conference on Cyberworlds (CW), Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 49–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1109/CW52790.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='00015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Gernux, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Blaser, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Marmolejo, “A Helmet Mounted Display Demonstration unit for a Space Station Application,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1989, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 891583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='4271/891583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Griffin, “A space suit for lunar construction and exploration,” Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2514/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1990-3885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [8] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Byrne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Mauldin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Munson, “Treadmill 2 Augmented Reality (T2 AR) ISS Flight Demonstration,” System Problem Resolution Team Meeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [9] “NASA, Microsoft Collaborate to Bring Science Fiction to Science Fact | NASA.” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='gov/press-release/nasa-microsoft-collaborate-to-bring-science-fiction-to-science-fact (accessed Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 31, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Coan, “Relevant Environments for Analysis and Development (READy): Enabling Human Space Exploration Through Integrated Operational Testing Enabling human space exploration through the integration of mission planning, EVA, science, engineering, and operations,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Accessed: Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 01, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Available: https://ntrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='gov/api/citations/20190027335/downloads/20190027335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='pdf [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Stolen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Dillow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Jacobs, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Akin, “Interface for EVA Human-Machine Interaction,” Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2008–01–1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='4271/2008-01-1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Hogson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “Requirements and Potential for Enhanced EVA Information Interfaces,” Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' di Capua, “Augmented reality for space applications,” University of Maryland, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Jacobs, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' di Capua, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Husain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Mirvis, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Akin, “Incorporating Advanced Controls, Displays and other Smart Elements into Space Suit Design,” SAE International Journal of Aerospace, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 374–384, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2009, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='4271/2009-01-2472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Villorin, “IDEAS: Integrated Display and Environmental Awareness System.” Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Morrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “Research Gaps and Opportunities in Sensor-Based Medical Exploration Capabilities in Extravehicular Astronaut Suits,” J Biosens Bioelectron, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 08, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 04, 2017, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='4172/2155-6210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1000248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Anandapadmanaban, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Tannady, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Norheim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Newman, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Hoffman, “Holo-SEXTANT: an Augmented Reality Planetary EVA Navigation Interface,” Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Available: https://ttu- ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='tdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='org/bitstream/handle/2346/74212/ICES_2018_264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='pdf?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='sequence=1 [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Gibson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Webb, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Stirling, “Evaluation of a Visual–Tactile Multimodal Display for Surface Obstacle Avoidance During Walking,” IEEE Trans Hum Mach Syst, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 48, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 604–613, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2018, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1109/THMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2849018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Mitra, “Human systems integration of an extravehicular activity space suit augmented reality display system,” Mississippi State University, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Valencio D’souza, “The Effectiveness of Augmented Reality for Astronauts on Lunar Missions: An Analog Study,” Embry-Riddle Aeronautical University, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Available: https://commons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='erau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='edu/edt/483/ [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “Developing an Integrated Heads-Up Display for Astronauts,” 2020, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='13016/TTT1- Q3XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [22] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' McHenry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “Design of an AR Visor Display System for Extravehicular Activity Operations,” in 2020 IEEE Aerospace Conference, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1109/AERO47225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='9172268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Radway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “Beyond LunAR: An augmented reality UI for deep-space exploration missions,” arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='14535 [cs], Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2020, [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='org/abs/2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='14535 [24] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Rometsch, “Design of an AR-IoT Tool for Future Human Space Exploration,” Delft University of Technology, Delft, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' IAC 22 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5 13 [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Miller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Fornito, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Flanagan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Kobrick, “Development of an Augmented Reality Interface to Aid Astronauts in Extravehicular Activities,” in 2021 IEEE Aerospace Conference (50100), Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1109/AERO50100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='9438430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Beaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “Mission enhancing capabilities for science-driven exploration extravehicular activity derived from the NASA BASALT research program,” Planet Space Sci, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 193, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 105003, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2020, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='pss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='105003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Kim and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Wilemon, “Focusing the fuzzy front–end in new product development,” R&D Management, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 269–279, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2002, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1111/1467-9310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='00259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [28] “Framework for Innovation: Design Council’s evolved Double Diamond - Design Council.” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='designcouncil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='uk/our-work/skills-learning/tools-frameworks/framework-for- innovation-design-councils-evolved-double-diamond/ (accessed Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 29, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Auernhammer and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Roth, “The origin and evolution of Stanford University’s design thinking: From product design to design thinking in innovation management,” Journal of Product Innovation Management, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 623–644, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2021, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1111/JPIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='12594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [30] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Brentani and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Reid, “The Fuzzy Front-End of Discontinuous Innovation: Insights for Research and Management: The Front-End of Discontinuous Innovation: Insights,” Journal of Product Innovation Management, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 70–87, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2012, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1540-5885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='00879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Flint, “Compressing new product success-to-success cycle time,” Industrial Marketing Management, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 305–315, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2002, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1016/S0019-8501(01)00165-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Breuer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Hewing, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Steinhoff, “Divergent innovation: Fostering and managing the fuzzy front end of innovation,” in PICMET ’09 - 2009 Portland International Conference on Management of Engineering & Technology, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 754–761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1109/PICMET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='5262066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Tassoul and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Buijs, “Clustering: An Essential Step from Diverging to Converging,” Creativity and Innovation Management, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 16–26, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2007, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1467-8691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='00413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Rometsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Casini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Drepper, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Cowley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' de Winter, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Guo, “Design and evaluation of an Augmented Reality tool for future human space exploration aided by an Internet of Things architecture,” Advances in Space Research, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1016/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='ASR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [35] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Rometsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=', “Towards a human-centred framework for conceptualization of lunar surface solutions,” Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' [36] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Röltgen and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' Dumitrescu, “Classification of industrial Augmented Reality use cases,” Procedia CIRP, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 91, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 93–100, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content=' 2020, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='1016/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='PROCIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} +page_content='137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQf7fod/content/2301.00838v1.pdf'} diff --git a/QdA0T4oBgHgl3EQfDf80/content/tmp_files/2301.02003v1.pdf.txt b/QdA0T4oBgHgl3EQfDf80/content/tmp_files/2301.02003v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f790235a8b046ee0447410be4a6749831b74d53 --- /dev/null +++ b/QdA0T4oBgHgl3EQfDf80/content/tmp_files/2301.02003v1.pdf.txt @@ -0,0 +1,3513 @@ +arXiv:2301.02003v1 [quant-ph] 5 Jan 2023 +One-Way Ticket to Las Vegas and the Quantum Adversary +Aleksandrs Belovs∗ +Duyal Yolcu† +Abstract +We propose a new definition of quantum Las Vegas query complexity. We show that +it is exactly equal to the quantum adversary bound. This is achieved by a new and very +simple way of transforming a feasible solution to the adversary optimisation problem into a +quantum query algorithm. This allows us to generalise the bound to include unidirectional +access, multiple input oracles, and input oracles that are not unitary. As an application, we +demonstrate a separation between unidirectional and bidirectional access to an input oracle +for a rather natural unitary permutation inversion problem. +1 +Introduction +This paper combines two topics: Las Vegas query complexity and the quantum adversary bound. +1.1 +Las Vegas Complexity +There are two main types of randomised query algorithms, with different complexity measures: +• A Monte Carlo query algorithm, also known as bounded-error, is allowed to output an +incorrect answer with some small probability ε, usually 1{3. The algorithm is allowed to +make a certain number of queries, which cannot be exceeded. This number is the query +complexity of the algorithm. +• A Las Vegas query algorithm, also known as zero-error, always has to give the correct +answer. On the other hand, it has no strict limit on the number of queries it can make. +Sometimes it can make few queries, sometimes a lot. Its query complexity is defined as +the expected number of queries it makes on a certain input. +Las Vegas algorithms have a number of nice properties. First, complexity is independent of +the choice of the error parameter ε. Hence, one can talk about the exact value of complexity +for a particular problem on a particular input, which can even not be an integer. Second, Las +Vegas algorithms can be nicely composed as there is no need for error reduction. For Monte +Carlo algorithms, one usually gets extra logarithmic factors due to the necessity to reduce the +error of the inner subroutines. +One can terminate a Las Vegas algorithm after a certain number of queries, turning it into +a Monte Carlo algorithm. By Markov’s inequality, a Las Vegas algorithm with complexity L +can be turned into a Monte Carlo algorithm with error parameter ε and complexity OpL{εq. +On the other hand, there exist functions whose Las Vegas complexity is much larger than their +Monte Carlo complexity. Ref. [5] features a quadratic separation for a total Boolean function. +For partial functions, the separation can be even larger. +∗Faculty of Computing, University of Latvia +†https://github.com/qudent +1 + +Let us now turn to quantum query complexity. +In the overwhelming majority of cases, +the complexity under consideration is Monte Carlo: the number of queries is fixed, and the +algorithm can output an incorrect output with small probability. +Zero-error quantum algorithms have also been defined and studied [10, 28, 23]. A zero- +error quantum algorithm is not allowed to give an incorrect output, but it can output ’?’ with +probability at most 1{2. The answer ’?’ means that the algorithm has not figured out what the +answer is. This model is indeed a quantum counterpart of one way of defining randomised Las +Vegas complexity. However, it lacks the nice features of the randomised Las Vegas complexity +outlined above. The definition depends on the value with which ’?’ can be outputted, and it +also does not compose nicely [29]. +Another related notion is variable-time model introduced by Ambainis. +In this case, a +quantum subroutine can run for an unpredicted number of steps, and the average running time +is the corresponding quadratic mean +ař +t ptt2, where pt is the probability the subroutine runs +for t steps. Ambainis showed how to perform search [3] and amplitude amplification [4] on +such subroutines. A very recent paper by Jeffery [38] also considers quantum walks with such +subroutines. Up to our knowledge, this notion has not been studied as a complexity measure +per se. Also, these results are mostly concerned with time complexity, while we study query +complexity in this paper. +1.2 +Adversary Bound +The quantum adversary bound was first developed as a powerful tool for proving quantum query +lower bounds, but it has been later extended to include upper bounds as well. +The adversary bound originates from the hybrid method by Bennett, Bernstein, Brassard, +and Vazirani [24], which was further refined by Ambainis in the first version of the adver- +sary bound [1]. Due to its attractive combinatorial formulation, it fostered a large number of +applications [34, 25, 30, 33] to name just a few. +The bound was strengthened by Høyer, Lee, and ˇSpalek [37]. Using the semidefinite for- +mulation of the adversary bound by Barnum, Saks, and Szegedy [9], they showed that the +same expression still yields a lower bound if one replaces non-negative entries by arbitrary +real numbers. This negative-weighted version of the bound is strictly more powerful than the +positive-weighted one, but it is also harder to apply. In a series of papers [55, 52, 53], Reichardt +et al. surprisingly proved that the negative-weighted version of the bound is tight: The dual +formulation of the bound (which is equal to the primal formulation due to strong duality) can +be transformed into a quantum query algorithm with the same complexity up to a constant +factor. +The negative-weighted adversary bound has been used to prove lower bounds [22, 19, 20], but +more frequently to prove upper bounds, in particular using the learning graph approach [12]. For +instance, the adversary bound (sometimes in the equivalent form of span programs) was used to +construct quantum algorithms for formula evaluation [55, 54, 61], finding subgraphs [43, 18, 42], +k-distinctness problem [11], and in learning and property testing [14, 16]. +The next steps came when the adversary bound was extended to state generation by Ambai- +nis, Magnin, R¨otteler, and Roland [7]; and state conversion by Lee, Mittal, Reichardt, ˇSpalek, +and Szegedy [44]. Belovs [15] extended the bound for various types of input oracles, including +the case when the input oracle can be an arbitrary unitary. These generalisations came with a +twist, as the bound became semi-tight: a lower bound for the exact version of the problem and +an upper bound for the approximate version of the bound. +Let us briefly touch on techniques used in the above papers. Ambainis [1] and Høyer et +al. [37] only proved lower bounds, which they did considering a so-called progress function of +2 + +the algorithm. The upper bounds by Reichardt [55, 52, 53] used a rather complicated quantum +walk, which was inspired by previous work on evaluating NAND-trees [36, 6]. The (discrete) +quantum walk comprises two reflections, one simple and input-dependent, and the other one +complicated and input-independent. The analysis of the algorithm required technically involved +spectral analysis. +The paper by Lee et al. [44] featured many important technical innovations. +First, the +problem was generalised to state conversion, where the task of the algorithm is to transform +one vector ξx into another τx on every input x in the domain D. This turned out to be a +very fruitful approach, as the algorithm can be broken into smaller steps, which can be then +analysed independently. Second, a very simple proof of the lower bound was presented, which +worked by a direct conversion of the algorithm into the bound. This essentially established the +adversary bound as a semi-definite relaxation of the algorithm. Third, the bound was formu- +lated as an instance of filtered γ2-norm, which is a generalisation of γ2-norm used previously in +other context, see Section 6 for more detail. Finally, the proof of the upper bound was signif- +icantly simplified by introducing easy and powerful Effective Spectral Gap Lemma to analyse +the resulting quantum walk. The lemma can be also used independently [13, 17]. +Lee et al. [44] assumed the standard input oracle that encodes a string x P rqsn for some +alphabet rqs. +The problem of choice by Belovs [15] was still state conversion ξx ÞÑ τx but +this time with general input oracles Ox, which is just an arbitrary unitary transformation. This +removed the oracle-specific details from the proof, thus making it more transparent. (Barnum [8] +already considered the problem of function evaluation with unitary input oracles, but that paper +went unnoticed at the time.) The bound was formulated as an instance of relative γ2-norm, +which further generalises filtered γ2-norm, and which, in our opinion, is more natural than the +latter. Belovs used the adversary bound for this problem to construct various adversaries for +function and relation evaluation. +Finally, let us note that all the versions considered above are that of the so-called additive +adversary bound. We leave out of consideration the multiplicative version by ˇSpalek [57] based +on the earlier work of Ambainis [2]. +1.3 +Our Results and Techniques +Las Vegas Complexity. +We propose a different definition of quantum Las Vegas query +complexity, which is very natural and more quantum in spirit than the previous notion of zero- +error quantum algorithm. We define it as the total sum of the squared norms of all the states +processed by the input oracle during the execution of the algorithm. Since the square of the +norm means probability in the quantum world, this quantity can be interpreted as the expected +number of queries performed by the algorithm on input x. +Our variant of quantum Las Vegas complexity possesses all the nice properties mentioned +above. +It does not feature any artificial constants. +It composes nicely as we will show in +Sections 5 and 8.3. Finally, as we will show in Section 7.4, every quantum Las Vegas algorithm +with complexity L can be turned into a Monte Carlo algorithm with error ε and complexity +OpL{εq. Since the term ‘zero-error’ is standard for the previous definition, we call our version +‘Las Vegas’. +Contrary to Monte Carlo complexity, Las Vegas complexity is input-dependent: different +inputs can have different complexity. Thus, we can study not only the worst-case complexity, +but also track complexity on each input. We capture this by introducing complexity profile, +which is the vector recording the complexity of the algorithm on all inputs. +Very recently, and independently of our paper, Jeffery [38] came up with essentially the same +notion, which was combined with variable-time quantum algorithm to get a composition result. +3 + +The results of Section 8.3 can be seen as a query analogue of the latter composition result. +New Simplified Upper Bound Construction. +As mentioned in Section 1.2, our paper +continues the line of work relating quantum query algorithms and the adversary bound. Fol- +lowing Lee et al. [44], the relation between the two can be depicted as in Figure 1. +Figure 1 +Computational problem +ÐÑ +Adversary optimisation problem +Input of the problem +ÐÑ +Variable-vector in the problem +Query algorithm solving the problem +ÐÑ +Feasible solution to the problem +Complexity of the algorithm +ÐÑ +Objective value of the feasible solution +Correspondence between quantum query algorithms and the adversary bound +After formulating the adversary optimisation problem in Row 1 of Figure 1, the main issue +is to prove the corresponding lower and upper bounds. A lower bound is a transformation of +an algorithm into a feasible solution in Row 3 of Figure 1, which respects the fourth row of the +same diagram. An upper bound is a transformation in the opposite direction, which turns a +feasible solution into an algorithm. +For the lower bound, we follow the same approach that was developed in [44] and used in [15]. +The variable-vector is the direct sum of all the queries made to the oracle on the corresponding +input. Therefore, its squared norm lower bounds query complexity, as the state processed on +one query has norm at most 1. +The crucial novel ingredient in our paper is a new construction of the upper bound. The +idea behind it is as follows. What we would like to do is to reverse the above process and to +give the variable-vector from the feasible solution as a query to the input oracle. At the first +sight, it is not clear how to achieve this. Indeed, the algorithm does not know the input, hence, +does not know which vector to give. And even if it knew, the latter vector generally has norm +much larger than 1, making it impossible to use it as a query. +We have found a very simple way around these complications. +Assume we add a small +“catalyst” to the state of algorithm, which is just a scaled down variable-vector from the feasible +solution. We process the catalyst by the input oracle, as we wanted, and use the result to change +a tiny part of the state in the required direction. The constraints of the adversary optimisation +problem ensure that the latter step can be implemented by an input-independent unitary. What +is remarkable, however, is that we get the catalyst back after this unitary! So we can use it again, +and again, until we perform the required transformation on all of the state. Thus, it suffices for +the algorithm to “guess” the catalyst just once to perform the transformation described above. +The “guessing” ability is folklore in quantum algorithms. What is meant here is that if the +catalysis is small, the distance between the original state and the state with the catalyst is also +small, and it gets preserved during the execution of the algorithm. Therefore, we end up close +to the target state even if we started without the catalyst. The smaller the catalyst, the larger +the number of queries needed, but the smaller the error induced by guessing it. +Let us compare our algorithm with two previous approaches. They are the aforementioned +quantum-walk-based algorithm by Lee et al. [44] and an adiabatic algorithm by Brandeho and +Roland [27]. Both of them use the guessing ability to extend the initial state with a small +state incorporating the feasible solution. After that the algorithm uses a quantum walk or an +adiabatic transformation, respectively. +What we demonstrated is that the same effect can be obtained by a very simple unitary +4 + +transformation. This substantially simplifies the analysis and makes the algorithm more trans- +parent. In particular, we see what queries are being made by the algorithm: it repeatedly calls +the input oracle on the scaled down variable-vector from the feasible solution. +This allows us to make various improvements. First, we see that the Las Vegas complexity of +the algorithm is exactly the objective value of the optimisation problem. Second, we can easily +incorporate multiple input oracles. +Third, the upper bound works assuming unidirectional +access to the input oracle, while the previous algorithm in [15] required bidirectional access (the +algorithm can query both the input oracle Ox and its inverse O˚ +x). Fourth, we do not even need +the input oracle to be unitary. Finally, we get a slightly better dependence of the number of +queries (in the traditional sense) on the error parameter ε, which is now tight up to constant +factors. Let us further discuss these improvements. +Relation to Las Vegas Complexity. +With the new upper bound, it becomes easy to cal- +culate the Las Vegas complexity of the algorithm, which leads to the main result of this paper: +The quantum adversary bound is equal to the Las Vegas complexity of state conversion. +We note that this is a threefold tighter connection between the adversary bound and the usual +(Monte Carlo) query complexity. First, the bound is tight, while connection to Monte Carlo +complexity is semi-tight. Second, the bound is exactly equal to Las Vegas complexity, and not +just up to a constant factor. Third, the bound holds for all inputs simultaneously, and not just +worst-case. +This result automatically carries over to all special cases of state conversion, including state +generation and function evaluation. +Multiple Input Oracles. +Considering multiple input oracles is often useful. For instance, +even the standard input oracle Ox : |i⟩|b⟩ ÞÑ |i⟩|b ‘ xi⟩ is a direct sum of multiple input oracles +Oxi : |b⟩ ÞÑ |b ‘ xi⟩, which encode individual symbols of the input string x. +These settings were investigated previously, most notably in the context of compositional re- +sults. Reichardt and ˇSpalek [55] considered span programs with costs, where costs were assigned +to individual symbols, and which were meant to capture the complexity of the corresponding +subproblems, and Ref. [52] similarly consider the adversary bound with costs. +Multiple oracles are also necessary in the study of trade-offs between different input re- +sources. +Kimmel, Lin, and Lin [39] used an adversary-based approach to show a trade-off +between two input oracles. +Again, the adversary featured costs. +Belovs and Rosmanis [21] +used a similar approach with general input oracles. Actually, the whole notion of Las Vegas +complexity, including the multiple oracle case, is greatly inspired by the latter paper. +In the case of Las Vegas complexity, dealing with several input oracles is easy. We can use +the same definition (the sum of the squared norms of the states processed by the oracle) for +each oracle independently. The complexity profile becomes a matrix, where, for each input, the +complexity of each of the input oracles is listed. +In the adversary bound, the variable-vector is similarly broken down into parts corresponding +to different input oracles, and all the results carry over with minimal changes. Having a vector +of complexities of all the input oracles, it is easy to get compositional results as well as trade-offs. +Unidirectionality. +The upper bound in [15] used quantum walk, which imposed bidirectional +access to the input oracle to implement the required reflection. In the new upper bound, there +are no reflections, hence, there is no need for this assumption. +Allowing bidirectional access has a lot of rationale. First, oracles are usually thought as +quantum subroutines, and each quantum subroutine can be easily inverted. Also, many basic +5 + +quantum algorithms like Grover’s search and amplitude amplification often require bidirectional +access to work. +On the other hand, unidirectional access also naturally comes up in some cases. For instance, +if we send the state to some other party to apply the input oracle, we may trust the recipient +to perform the required query, but we might not be able to ask them to perform it in reverse. +Finally, the assumption that we have unidirectional access to the input oracle simplifies +the upper and the lower bounds. The bidirectional case can be obtained as a special case, see +Section 9. +General Input Oracles. +As there is no bidirectionality assumption, we can replace the +unitary oracle assumed in [15] by an arbitrary linear transformation. It turns out that many of +results hold still hold even under such assumptions. It seems, however, that contraction oracles, +which are linear transformations of norm not exceeding 1, might be a good choice to consider. +Contraction oracles seem to contradict the unitarity condition usually imposed on quantum +algorithms. Nonetheless, such oracles naturally come up in practise. For example, the input +oracle can perform some measurement and continue only if the outcome is positive. Similar +settings appear in interaction-free measurements by Elitzur and Vaidman [35] and subsequent +bomb query algorithm by Lin and Lin [46], measure-many quantum finite automata [40], and +faulty oracles [51]. +Subspace Conversion Problem. +Finally, we define and study the subspace conversion prob- +lem, which is in between state conversion ξx ÞÑ τx, which we assume for the action of the +algorithm, and unitary (or contraction) Ox, which we use for the input oracle. +In this problem, the task is to implement a linear transformation Tx : Kx Ñ K defined on +a linear subspace Kx of the workspace K. If Kx is one-dimensional, this is state conversion; if +Kx “ K, this is unitary (or contraction). +We introduce a complexity notion for this problem, which is the largest Las Vegas complexity +of the algorithm achieved when executed on a unit vector in Kx. The definition turns out to be +natural for composition, and it is still exactly characterised by the corresponding version of the +adversary bound. +Unitary Permutation Inversion. +Finally, we use this occasion to demonstrate a separation +between unidirectional and bidirectional access to the input oracle. We consider the unitary +permutation inversion problem, where the oracle is a unitary that implements a permutation +|i⟩ ÞÑ |πpiq⟩, and the task is to find π´1p1q. We prove an Ωp?nq lower bound, where n is the +size of the domain of π, whereas the problem is trivially solvable with 1 query to the inverse +oracle. Up to our knowledge, these types of questions have not been addressed before. +1.4 +Overview of the Paper +In this subsection, we give a very brief overview of the paper, highlighting the most important +points. +The main part of the paper starts with Section 3. In Section 3.1, we define a quantum query +algorithm as a sequence of linear transformations +UT rO UT´1 rO ¨ ¨ ¨ U1 rO U0, +(1.1) +where Ui are some unitaries. The operator rO “ pO b I‚q ‘ I˝ is a query, where O is the input +oracle, and I‚ and I˝ are some identity transformations. Thus, the algorithm implements a +transformation O ÞÑ ApOq: from the input oracle to the linear operator (1.1). In Section 3.2, +6 + +we describe problems solved by the algorithm. We first give a general definition, capable of +describing a wide range of problems, but for the purposes of this paper, the most important +problem is state conversion. Given a family of input oracles Ox and pairs ξx ÞÑ τx, where x +ranges overs some set D, the task is to develop an algorithm A such that ApOxq maps ξx into +τx for all x P D. +The remaining part of the paper follows a similar division. Sections 4 and 5 are devoted to +the study of Las Vegas complexity of algorithms without connection to any particular problem. +In Section 7, we consider the state conversion problem, and in Section 8, subspace conversion. In +particular, the adversary bound makes its first appearance in Section 7 as it is tied to a particular +problem being solved. Other problems can be studied as well, for instance, in Section 7.7, we +consider the problem of Boolean function evaluation, and Ref. [15] considers a wide range of +other problems, which we leave outside the confines of this paper. Section 6 is an intermission, +and Sections 9 and 10 contain complementary results. Let us describe these sections in more +detail. +In Section 4, we define Las Vegas complexity a quantum query algorithm A. Let QtpA, Oqξ +be the state processed by the input oracle (the O bI‚ part of the query operator rO) on the t-th +query when executed on the input oracle O and the initial state ξ. We define the Las Vegas +complexity LpA, O, ξq as the sum of }QtpA, Oqξ}2 over all t. Note that it depends both on the +input oracle O and the initial state ξ. In Section 4.2, we define the same notion for multiple +input oracles. In essence, the complexity LpA, O, ξq becomes a tuple which accounts for the +total squared norm of the states processed by each of the input oracles. The difference between +the single-oracle and the multiple-oracles variants is mostly cosmetic. The reader may choose +to assume the single-oracle variant throughout the paper. +In Section 5, we study various properties of Las Vegas complexity without relation to any +particular task. We consider various ways algorithms can be composed: inversion, direct sum, +tensor product, sequential and functional composition, and show that our definition of Las +Vegas complexity encompasses these composition variants naturally. +The results are pretty +straightforward, but there is one subtlety involving functional composition, where one algorithm +B is used as an input oracle for another algorithm A. The thing is that the algorithm A executes +the input oracle as O b I‚, while we assume the algorithm B implements O. This means that +the complexity of B on the state ψt “ QtpA, Bqξ is just not defined. We use an obvious solution +to slice ψt as ψt,1 ‘ ¨ ¨ ¨ ‘ ψt,d, where d is the dimension of I‚, and each ψt,j can be processed by +O. Now, the complexity of B on each ψt,i is well-defined, and we can define the total complexity +as their sum. There are many different possible slicing, as they depend on the choice of an +orthonormal basis in the space of I‚. We show that the total complexity is independent of the +choice of slicing. +In Section 6, we describe our modification to the relative γ2-norm from [15], as different +settings require different versions of the bound. One thing we should account for is unidirec- +tionality. We also have to convert to multi-objective version of the bound, as we are interested +in the full complexity profile of the algorithm. The variant with multiple input oracles requires +yet another modification. We formulate the dual versions, which can be used to prove lower +bounds on worst-case complexity. +Section 7 is the main part of the paper. In it, we study Las Vegas complexity of state +conversion, and show how it can be characterised by an instance of (unidirectional) relative +γ2-bound: the adversary optimisation problem. This section is designed to be self-contained +with minimal dependency on the previous sections. We first prove a lower bound for the exact +version of the problem in Section 7.3, and then an upper bound for the approximate version in +Section 7.4. The corresponding ideas were already explained in Section 1.3. The algorithm in +7 + +Section 7.4 transforms +ξ` +x “ ξx ‘ +1 +? +T +vx +ÞÝÑ +τ ` +x “ τx ‘ +1 +? +T +vx, +where ξx ÞÑ τx is the required state conversion problem, vx is a feasible solution to the adversary +optimisation problem, and T is an arbitrary positive integer. +The Las Vegas complexity of +the algorithm on input x is }vx}2 independently from the value of T. (The total number of +queries does depend on T, though). As T increases, we can get arbitrarily close to the required +transformation, while Las Vegas complexity stays the same. In Section 7.5, we improve on this +result. We show how to perform transformation ξx ÞÑ τx exactly, while now we can get Las +Vegas complexity arbitrarily close to }vx}2. +Section 8 deals with the question of what happens if some of the input oracles Ox in the state +conversion problem are equal. If Ox “ Oy, then we get exactly the same action of the algorithm +on the inputs x, y P D. The motive of Section 8.1 is linear consistency of the feasible solution to +the adversary bound for such pairs of inputs. We show that we can assume consistency without +any loss. Moreover, this brings us to the formulation of the subspace conversion problem, and +the corresponding adversary bound in Section 8.2. In Section 8.3, we revisit the functional +composition property from Section 5 and show that we can upper bound the complexity of the +composed algorithm as the product of complexities of the constituents under the assumption +that the algorithm follows the specification of the subspace-converting subroutine. +In Section 9, we prove the relation between unidirectional and bidirectional versions of the +bound. In particular, we get back the bidirectional results from [15]. In Section 10, we prove +a separation between unidirectional and bidirectional input oracle for the unitary permutation +inversion problem. Finally, in Section 11, we make some final comments. +Ref. [60] contains an alternative exposition of some of the results in Section 7, and some +additional results on more general control problems. +2 +Preliminaries +If not said otherwise, a vector space is a finite-dimensional complex inner product space. They +are denoted by calligraphic letters. We assume that each vector space has a fixed orthonormal +basis, and we often identify an operator with the corresponding matrix. The inner product is +denoted by x¨, ¨y. A˚ stands for the adjoint linear operator, and Arri, jss for the pi, jqth entry of +the matrix A. A ˝ B stands for the Hadamard (entry-wise) product of matrices. IX stands for +the identity operator in X. All projectors are orthogonal projectors. For vectors u, v P Rn, we +write u ď v if urriss ď vrriss for all i P rns. We use A ą 0 and A ě 0 to denote positive definite +and semi-definite matrices, respectively. We use the ket-notation to emphasise that a vector is +a state of a quantum register, or to denote the elements of the computational basis. +We also need the following generalisation of the well-known parallelogram identity. We were +not able to find its statement in the existing literature. +Theorem 2.1 (Generalised Parallelogram Identity). Let v1, . . . , vd P Cn, and +U “ +¨ +˚ +˚ +˚ +˝ +α1,1 +α1,2 +. . . +α1,d +α2,1 +α2,2 +. . . +α2,d +... +... +... +... +αd,1 +αd,2 +. . . +αd,d +˛ +‹‹‹‚ +(2.1) +8 + +be a unitary matrix. Then, +}v1}2 ` }v2}2 ` ¨ ¨ ¨ ` }vd}2 “ +dÿ +j“1 +��α1,jv1 ` α2,jv2 ` ¨ ¨ ¨ ` αd,jvd +��2. +Proof. Let V be the n ˆ d matrix with vj as the columns. The above identity is equivalent to +��V +��2 +F “ +��V U +��2 +F, +where ∥¨∥F stands for the Frobenius norm. The equality follows from the fact that unitaries +preserve the Frobenius norm. +The parallelogram identity is the special case of Theorem 2.1 with U “ H, the Hadamard +matrix. +3 +Quantum-Algorithmic Definitions +In this section, we give the main definition of a quantum algorithm solving a computational +problem. Let us very briefly recall the textbook definition of a quantum query algorithm. A +standard reference is a survey by Buhrman and de Wolf [31]. (Note, however, that it only deals +with Boolean functions. See also [32].) The task is evaluation of a function f : D Ñ rℓs with +domain D Ď rqsn. The algorithm can perform arbitrary unitary transformations, as well as +access the input string x “ px1, . . . , xnq P D via the standard input oracle: +Ox : |i⟩|b⟩ ÞÑ |i⟩|b ‘ xi⟩, +where ‘ is the bit-wise XOR operation (one can also use modular addition). The algorithm is +said to compute the function f if, for all x P D, measuring the output register of the final state +of the algorithm gives fpxq with high probability. The unitary operations are free, and each +execution of Ox costs one query. The goal is to minimise the number of queries. +We separate the algorithm itself from the input and output conditions. +The algorithm +becomes a map from operators on the input register (Ox) to operators on the space of the +algorithm (the transformation performed by the algorithm). The input condition is the input +oracle Ox given on a particular input x P D, and the output condition is the set of admissible +transformations performed by the algorithm. Actually, we choose to treat input and output +conditions similarly as sets of admissible transformations, which allows algorithms to be used +as input oracles for other algorithms. We keep the set of input labels D, but it need not be +considered as the domain of a function any longer. +We describe the algorithmic part in Section 3.1, and the input/output conditions in Sec- +tion 3.2. +3.1 +Quantum Query Algorithm +The overall form of a quantum query algorithm is similar to the textbook version. When we use +the term ‘algorithm’ later in the paper, we mean a quantum query algorithm of the following +form. +Definition 3.1 (Quantum Query Algorithm). Let M and H be vectors spaces. A quantum +query algorithm in H with an oracle in M is a function which maps linear operators O: M Ñ M +into linear operators ApOq: H Ñ H, and which has the following form: +ApOq “ UT rO UT´1 rO ¨ ¨ ¨ U1 rO U0. +(3.1) +9 + +Here, each Ui : H Ñ H is a unitary that does not depend on O, and rO is some “embedding” of +O into H of the form rO “ pO b I‚q ‘ I˝, where I‚ and I˝ are identity transformations of some +size. +The operator O is called the input oracle, and each execution of rO is called a query. To +make the definition simpler, we have chosen to have one fixed embedding rO. It is often more +convenient to allow different embeddings at different queries. The two definitions are equivalent, +see Section 5.2. +The spaces M and H are called the input and the work spaces of the algorithm, respec- +tively. It is sometimes useful to consider also the output subspace K Ď H of the algorithm. +Conceptually, it contains the “interesting” part of ApOq, while its orthogonal complement in H +is the “scratch space” of the algorithm. If not specified, we may always take K “ H. +If ApOqξ “ τ for some ξ, τ P H we say that the algorithm A performs transformation ξ ÞÑ τ +on the oracle O. We call ξ the initial and τ the terminal state of the algorithm1. +Let us mention the main differences with the textbook definition. The first difference is that +we allow arbitrary input oracles O. The second difference is the ability to “skip” query: to apply +I˝ on some part of the workspace. Alternatively, in the language of circuits, we may apply a +controlled version of O, not just O. Textbook quantum query algorithms do skip queries, but it +is usually done implicitly by setting up a state that does not change by any input oracle, e.g., a +uniform superposition on the second register. Since we allow arbitrary unitaries as oracles, this +option is out of stock for us, and we have to skip queries explicitly. Interestingly, this is exactly +this feature that allows us to define quantum Las Vegas complexity. +Let us also emphasise the differences with the definition of a quantum query algorithm +in [15]. +The first one is what we call directionality. +The algorithm in [15] is bidirectional: +it allows execution of both rO as well as its inverse rO´1. The algorithm in Definition 3.1 is +unidirectional: it only allows execution of rO. For the standard input oracle, the difference is +irrelevant since each standard input oracle is its own inverse (or can be easily constructed from +it). This is not true for arbitrary unitaries. Note that unidirectionality is without any loss of +generality, as it is possible to simulate bidirectional access to a unitary O with unidirectional +access to O ‘ O˚, see Section 9. +The second difference is that, while we still require that all Ui are unitary, there is no more +need to require the input oracle O to be unitary. We allow O to be an arbitrary linear operator. +However, a more interesting choice is to consider contractions as input oracles. Note that if O +is a unitary or a contraction, then ApOq is also a unitary or a contraction, respectively. +3.2 +Input and Output Conditions +Here, we give a general take on input and output conditions imposed on a quantum algorithm, +as well as define all types of conditions we consider in this paper. In Sections 7 and 8, we +redefine the specific conditions under consideration. +The simplest way to impose requirements on an algorithm A from Definition 3.1 is to specify +its outputs ApOxq: H Ñ H on fixed inputs Ox : M Ñ M as x ranges over some set D. There +is nothing fundamentally wrong with this approach, except that it may be too specific. For +instance, the textbook definition has a very specific input oracle Ox, but a very vague output +condition. To capture this, for each x P D, we define not one, but a collection Ex of admissible +linear transformations. This gives the following very general definition. +Definition 3.2 (Computational Problem). A computational problem is given by a set of labels +D, where, for each x P D, a set of admissible inputs Ox and a set of admissible outputs Ex are +1The letters ξ and τ stand for ξεκίνημα and τέλος, respectively. +10 + +specified. A quantum algorithm A solves the problem if, for each x P D and each O P Ox, it +holds that ApOq P Ex. +We treat input and output uniformly, therefore, we use a term admissible set for both sets of +admissible inputs and outputs. We define different types of admissible sets, which are depicted +in Figure 2. We do this in terms of Ex, the output space K, and the workspace H. The definitions +for input conditions are similar with Ex replaced by Ox, and K and H by M. We say that we +have a problem of type1 with input oracles of type2, if all Ex are of type1 and all Ox are of +type2. +Figure 2 +Subspace Conversion +State Conversion +General Input Oracle +(Unitary/Contraction) +State Generation +Function Evaluation +Various types of input/output conditions. The ones at the top are more general in the +sense that the lower ones are special cases thereof as indicated by arrows. The ones to +the right are more restrictive in the sense that they impose more restrictions on the set of +admissible operators. +Most types of admissible sets considered in this paper are special cases of the following type. +Definition 3.3 (Subspace Conversion). An admissible set Ex is subspace conversion if there +exists a linear transformation Sx : Kx Ñ K defined on some linear subspace Kx Ď K such that +Ex consists of all extensions of Sx to a linear operator on H. +There are two main cases. In the isometric case, we only allow unitaries in Ex. Of course, +this makes sense only if Sx is an isometry itself. In the general non-isometric case, we assume +that Sx are contractions, and require the operators in Ex to be contractions as well. +Therefore, subspace conversion is specified by its action on the output space K, but ac- +commodates any workspace H as long as it is a superspace of K. The vectors in KzKx are +interpreted as ones where the action of the algorithm is not defined. Note that it is possible +that A P Ex maps such vectors outside of K. +There are two main special cases of subspace conversion, which are more important than +the general case itself. The first one is when Kx “ K. In this case, Sx gives a linear map from +K to itself. The algorithm can still use a larger workspace, but it is completely inaccessible +from outside, therefore, it makes sense to identify Ex with Sx. This is our default type of input +condition, which we call general input oracle. Alternatively, we call it unitary, contraction, or +linear input oracle in dependence on the type of Sx. For the output condition, we call it unitary +or contraction implementation. +The second important special case is when Kx is one-dimensional. We call it state conversion, +and denote by ξx ÞÑ τx, meaning that Aξx “ τx for all A P Ex. This is our default type of output +condition. +11 + +There are important special cases of state conversion as well. +State generation is state +conversion when all the initial states ξx are equal to some predefined state |0⟩. The most widely +used version is function evaluation, which is state generation when τx is an element of the +computational basis |fpxq⟩ for some function f : D Ñ K. +It is also possible to define approximate and non-coherent versions of above conditions. In +the ε-approximate version, we take the ε-neighbourhood of Ex. For instance, an algorithm A +solves an ε-approximate version of state conversion ξx ÞÑ τx if, for all x P D and all O P Ox, +we have }ApOqξx ´ τx} ď ε. We say that A solves the non-coherent version of the problem, if +ApOqξx “ τx b ζ for some junk state ζ that may depend on x and O. Finally, we can consider +ε-approximate non-coherent version as well, where we require that }ApOqξx ´ τx b ζ} ď ε. +Function evaluation is usually considered in the approximate non-coherent case, as it is +required that measuring the output register of the final state gives fpxq with bounded error. +However, for bidirectional oracles, coherent and non-coherent versions differ at most by a factor +of 2 in complexity. Indeed, it is possible to evaluate the function non-coherently, copy the final +output into a new register, and run the program in reverse. For unidirectional oracles, however, +this simple trick does not work, as it is impossible to run the program in reverse. It also does +not work for state generation, as it is impossible to copy general quantum state. +4 +Quantum Las Vegas Query Complexity +In this section, we define the main notion of this paper: quantum Las Vegas query complexity. +Usually query complexity of the algorithm like in Definition 3.1 is defined as T: the number +of invocations of the input oracle. We will often call it Monte Carlo query complexity in this +paper. Contrary to Monte Carlo complexity, Las Vegas complexity is input-dependent, as it +depends both on the oracle O and the initial state. +4.1 +Definition +Let A be an algorithm as in Definition 3.1, and O: M Ñ M be an input oracle. We need +the following two linear transformations on the workspace H, which can be seen as partial +executions of the algorithm. For t P rT ` 1s, let +StpA, Oq “ Ut´1 rO Ut´2 rO ¨ ¨ ¨ U1 rO U0 +(4.1) +be the transformation that maps the initial state ξ to the state just before the t-th application +of the input oracle O. In particular, S0pA, Oq “ U0 and ST`1pA, Oq “ ApOq. +Recall that the query is of the form rO “ pO b I‚q ‘ I˝. Let Π denote the projection on the +part of the space processed by O b I‚. The second transformation is +QtpA, Oq “ ΠStpA, Oq, +(4.2) +which maps ξ to the state processed by the input oracle on the t-th query. +Definition 4.1. The quantum Las Vegas query complexity of the algorithm A on the input +oracle O: M Ñ M and the initial state ξ P H is defined as +LpA, O, ξq “ +Tÿ +t“1 +��QtpA, Oqξ +��2. +(4.3) +Under usual assumptions of O being unitary and }ξ} “ 1, the term ∥QtpA, Oqξ∥2 can be +interpreted as the probability that the algorithm A actually executes the query on the t-th step, +12 + +and not skips it. Therefore, LpA, O, ξq can be seen as the expected number of queries similarly +to the definition of the randomized Las Vegas query complexity. Las Vegas complexity does not +exceed the Monte Carlo complexity T, but it can be much smaller. +The definition also encapsulates the case of algorithms with intermediate measurements as +we briefly discuss here. Assume we have a quantum algorithm B with intermediate measure- +ments. The definition is similar to Definition 3.1 with the difference that the algorithm can +perform measurements in the middle, so that the forthcoming unitaries Ui depend on the out- +come of the previous measurements. In particular, the number of queries can also depend on +the outcomes of the measurements. Let TpB, O, ξq be the expected number of queries performed +by B on oracle O and initial state ξ. Such an algorithm can be turned into a usual algorithm +A as in Definition 3.1 by deferring the measurements to the end of the algorithm [50, Section +4.4]. It is not hard to see that TpB, O, ξq ě LpA, O, ξq. Note, however, that in the absence of +measurements, the terminal state of A differs from the terminal state of B. In particular, A +computes the non-coherent version of a state conversion problem even if the original algorithm +B computes the coherent version. +4.2 +Multiple Input Oracles +Assume we have s input oracles, Op1q, Op2q, ¨ ¨ ¨ , Opsq, where Opiq acts on some space Mpiq, and +we want to provide the algorithm with access to all of them. This can be seen as a special case +of Definition 3.1, where the algorithm has access to the combined oracle +O “ Op1q ‘ Op2q ‘ ¨ ¨ ¨ ‘ Opsq +(4.4) +acting on M “ Mp1q ‘ ¨ ¨ ¨ ‘ Mpsq. Indeed, it is possible to simulate a query to Opiq using one +query to O, and it is possible to simulate a query to O using one query to each of Opiq. +Now suppose we want to measure complexity of each oracle Opiq individually. In the case of +Las Vegas complexity, this can be handled very naturally. Decompose +QtpA, Oqξ “ Qp1q +t pA, Oqξ ‘ Qp2q +t pA, Oqξ ‘ ¨ ¨ ¨ ‘ Qpsq +t pA, Oqξ, +(4.5) +where Qpiq +t pA, Oqξ is the state processed by the i-th input oracle on the t-th query. +Definition 4.2. In the above settings, the Las Vegas complexity of the i-th input oracle is +defined as +LpiqpA, O, ξq “ +Tÿ +t“1 +���Qpiq +t pA, Oqξ +��� +2 +. +The Las Vegas complexity LpA, O, ξq of the algorithm A on the composed input oracle O +from (4.4) is the vector in Rs consisting of the individual complexities LpiqpA, O, ξq. +Almost all the results in this paper can be generalised to include this variation of Las Vegas +complexity with minimal changes in the proof. To make this explicit, we introduce the following +piece of notation. Let v P M b W for some W. We have the following imposed decomposition +v “ vp1q ‘ vp2q ‘ ¨ ¨ ¨ ‘ vpsq, +with vpiq P Mpiq b W. We define +~v~2 “ +´��vp1q��2, +��vp2q��2, . . . , +��vpsq��2¯ +P Rs. +(4.6) +13 + +This notation is chosen to emphasise similarity to }v}2, and we never use ~v~ alone. This gives +us almost the same definition for Las Vegas complexity as in (4.3): +LpA, O, ξq “ +Tÿ +t“1 +‌‌‌QtpA, Oqξ +‌‌‌ +2 +. +(4.7) +The upcoming sections can be read using one of the two assumptions: +• There is a single input oracle O. In this case, definitions from Section 4.1 hold, s “ 1 +everywhere, and ~v~2 stands for }v}2. In particular, Eq. (4.3) and (4.7) are the same. +• There are multiple input oracles. In this case, we use O as in (4.4) to combine them in a +single input oracle. We use Definition 4.2, and ~v~2 is as in (4.6). +Most of the time, there is no difference between the two cases. +Let us list the properties of ~v~2 that we will need. They follow easily from the defini- +tion (4.6). +~cv~2 “ |c|2 ¨ ~v~2 +(4.8a) +~u ‘ v~2 “ ~u~2 ` ~v~2 +(4.8b) +If O is a unitary of the form in (4.4), then ~v~2 “ ~Ov~2. +(4.8c) +Finally, the generalised parallelogram identity also holds. Namely, in assumptions of Theo- +rem 2.1: +~v1~2 ` ~v2~2 ` ¨ ¨ ¨ ` ~vd~2 “ +dÿ +j“1 +‌‌α1,jvj ` α2,jvj ` ¨ ¨ ¨ ` αd,jvd +‌‌2. +(4.9) +5 +Properties of Las Vegas Complexity +Apart from functional composition, which was the main focus of previous work, algorithms can +be composed in many different ways, some of which we describe in this section. Most of them +were used before implicitly, and one of our goals was to formulate them in a more explicit way. +We also show that quantum Las Vegas complexity can handle these composition variants +naturally. Most of the results hold for linear input oracles, but we require unitary input oracles +for some. +5.1 +Basic Properties +Proposition 5.1 (Scaling). For every algorithm A, oracle O: M Ñ M, and states ξ, τ P H, +if A transforms ξ ÞÑ τ on O, then it also transforms cξ ÞÑ cτ for all c P C and +LpA, O, cξq “ |c|2LpA, O, ξq. +Proof. This follows from the definition (4.7) and (4.8a). +Note that while QtpA, Oq is linear, it distorts inner products even if O is a unitary. Hence, +there is no general way to relate LpA, O, ξ`ξ1q to LpA, O, ξq and LpA, O, ξ1q even for orthogonal +ξ and ξ1. However, we have the following result. +14 + +Proposition 5.2 (Parallelogram Identity). For every algorithm A, oracle O: M Ñ M, states +ξ1, . . . , ξd P H, and unitary U as in (2.1), we have +LpA, O, ξ1q ` LpA, O, ξ2q ` ¨ ¨ ¨ ` LpA, O, ξdq “ +dÿ +j“1 +L +` +A, O, α1,jξ1 ` α2,jξ2 ` ¨ ¨ ¨ ` αd,jξd +˘ +. +Proof. The proof is analogous to Proposition 5.1, but this time we use (4.9). +Proposition 5.3 (Inversion). For every algorithm A in H with oracles in M, there exists the +inverse algorithm A´1 in the same spaces such that for every unitary input oracle O: M Ñ M, +we have A´1pO˚q “ +` +ApOq +˘´1. Moreover, if A transforms ξ ÞÑ τ on a unitary input oracle O, +then +LpA´1, O˚, τq “ LpA, O, ξq. +Proof. The algorithm A´1 is just the inverse of (3.1): +A´1pOq “ U ˚ +0 rO U ˚ +1 rO ¨ ¨ ¨ U ˚ +T´1 rO U ˚ +T . +The relation between Las Vegas query complexities follows from the identity +QtpA´1, O˚qτ “ pO b I‚qQT`1´tpA, Oqξ +and (4.8c). +5.2 +Slicing +Let us now describe possible alternatives to the Definition 3.1 of the quantum query algorithm, +and show that they preserve Las Vegas complexity. In particular, we show that we can replace +the “embedding” rO “ pO b I‚q ‘ I˝ with a simpler construction. +Definition 5.4 (Sliced Algorithm). We call a quantum algorithm from Definition 3.1 sliced if +its query rO is of the form rO “ O ‘ I˝. +Clearly, a sliced algorithm is a special case of the general algorithm. In the other direction, +we have the following result. +Proposition 5.5 (Slicing). Every algorithm A can be transformed into a sliced algorithm A1 +such that, for every oracle O: M Ñ M and initial state ξ P H, we have ApOq “ A1pOq and +LpA, O, ξq “ LpA1, O, ξq. +Proof. Let rO “ pO b I‚q ‘ I˝ be the query of the algorithm A. We can rewrite +O b I‚ “ O ‘ O ‘ ¨ ¨ ¨ ‘ O “ pO ‘ I ‘ ¨ ¨ ¨ ‘ IqpI ‘ O ‘ ¨ ¨ ¨ ‘ Iq ¨ ¨ ¨ pI ‘ I ‘ ¨ ¨ ¨ ‘ Oq, +(5.1) +where there are d “ dim I‚ multipliers on the right-hand side.Conjugating each of them by a +unitary, we can implement rO using d queries to rO1 “ O ‘ I˝1. This does not change the action +of the algorithm. +Neither does this change its Las Vegas complexity. Indeed, let ψt “ QtpA, Oqξ be the state +processed by rO on the t-th query, and ψt,1, . . . , ψt,d be the corresponding states processed by +the oracle rO1 on the right-hand side of (5.1). Then +ψt “ ψt,1 ‘ ψt,2 ‘ ¨ ¨ ¨ ‘ ψt,d, +(5.2) +and the result follows from (4.8b). +15 + +Note that the algorithm depends on the choice of slicing in (5.1), which in turn depends +on the choice of the orthonormal basis in the space of I‚. By (5.1), this does not change the +action of the algorithm, and by (5.2), this does not change its complexity. Thus, we can further +assume, without loss of generality, that a quantum algorithm is sliced. We will use this in this +section, as it simplifies some constructions and some proofs. +Also, note that the proof of Proposition 5.5 still works if we have different embeddings of +O on each query of the algorithm in Definition 3.1. Thus, this variant of the definition is also +equivalent to Definition 3.1. +5.3 +Space Extension +The following two results formally state that we can embed an algorithm into a larger space. +The work space extension is straightforward: +Proposition 5.6 (Work Space Extension). Let A be an algorithm in H with oracles in M. +Then, for every H1, there is an algorithm A‘IH1 in H‘H1 with oracles in M such that for every +O: M Ñ M, ξ P H, and ξ1 P H1, we have pA‘IH1qpOq “ ApOq‘IH1 and LpA‘IH1, O, ξ‘ξ1q “ +LpA, O, ξq. +Proof. Let A be as in Definition 3.1. To get A ‘ IH1, replace each Ui with Ui ‘ IH1, and each +I˝ from rO with I˝ ‘ IH1. +The input space extension is also possible. For simplicity, we assume the algorithm A is +sliced. We state the extension in a rather general way.Essentially, we require that the input +oracle in the extended space agrees with the original oracle on the states actually being queried. +Let A be a sliced algorithm in H with oracle O: M Ñ M, and M ‘ M1 be a superspace +of M. We construct an algorithm A1 in H ‘ M1 with oracle O1 : M ‘ M1 Ñ M ‘ M1 in the +following way. Each unitary Ui is replaced by Ui ‘ IM1 and each query O ‘ I˝ is replaced by +O1 ‘ I˝ acting in H ‘ M1. +Proposition 5.7 (Input Space Extension). In the above assumptions, if O: M Ñ M, O1 : M‘ +M1 Ñ M ‘ M1 and ξ P H are such that +OQtpA, Oqξ “ O1QtpA, Oqξ +(5.3) +for all t, then A1pO1qξ “ ApOqξ and LpA1, O1, ξq “ LpA, O, ξq. +In particular, Eq. (5.3) holds if O1 “ O ‘ O2 for some O2 acting in M1. +Proof. Recall the operator St defined in (4.1). +By induction on t, it is easy to show that +StpA1, O1qξ “ StpA, Oqξ, from which the statement follows. +We will often identify the algorithms A and A1 above. +5.4 +Sequential Composition and Direct Sum +Proposition 5.8 (Sequential Composition). Assume there are two algorithms A and B in H +with oracles in M. Then, there exists an algorithm B ˚ A such that for all O: M Ñ M and +ξ P H we have pB ˚ AqpOq “ BpOqApOq and +LpB ˚ A, O, ξq “ L +` +B, O, ApOqξ +˘ +` LpA, O, ξq. +Proof. The algorithm B ˚ A is the algorithm B applied after A. +16 + +The condition that A and B share the same workspace seems restrictive, but it is necessary +for the formal statement of Proposition 5.8. +Usually it makes sense to assume that A and +B share the same output space K. Then, in the spirit of Definition 3.3, the initial state ξ is +assumed to be such that both ApOqξ and pB˚AqpOqξ are in K. Let W and W1 be the orthogonal +complements of K in the workspaces of A and B, respectively (the “scratch spaces”). We can +still apply Proposition 5.8 with H “ K ‘ W ‘ W1 and assuming that the algorithms A and B +are extended by the identity to H using Proposition 5.6. +Also, Proposition 5.8 assumes that A and B use the same input oracle O. This is without +loss of generality. Indeed, let A and B use different oracles O1 : M1 Ñ M1 and O2 : M2 Ñ M2. +Extend the input space of both algorithm to M “ M1 ‘ M2, and assume they both use the +input oracle O “ O1 ‘ O2. By Proposition 5.7, the action of both algorithms does not change. +The same observations also applies to Propositions 5.9 and 5.12 below. +Proposition 5.9 (Direct Sum). Let A and B be two algorithms in spaces H and H1 respectively, +and both with oracles in M. Then, there exists an algorithm A ‘ B in H ‘ H1 with oracles in +M such that for all O: M Ñ M, ξ P H, and ξ1 P H1, we have pA ‘ BqpOq “ ApOq ‘ BpOq and +LpA ‘ B, O, ξ ‘ ξ1q “ LpA, O, ξq ` LpB, O, ξ1q. +(5.4) +Proof. The algorithm A ‘ B can be implemented as pIH ‘ Bq ˚ pA ‘ IH1q. The result follows +from Propositions 5.6 and 5.8. +5.5 +Functional Composition and Tensor Product +Functional composition is a more interesting way of composing algorithms. It can be constructed +with ease assuming the outer algorithm is sliced. +Proposition 5.10 (Functional Composition). Let A be a sliced algorithm in H with oracles in +N, and B be an algorithm in N with oracles in M. Then, there exists an algorithm A ˝ B in +H with oracles in M such that for all O: M Ñ M and ξ P H, we have pA ˝ BqpOq “ ApBpOqq +and +LpA ˝ B, O, ξq “ +ÿ +t +L +` +B, O, Qt +` +A, BpOq +˘ +ξ +˘ +. +(5.5) +Proof. Denote by O1 : N Ñ N the input oracle of the outer algorithm A. Replace each query +rO1 “ O1 ‘ I˝ of A by a copy of the algorithm B ‘ I˝ obtained via Proposition 5.6. The theorem +follows from Proposition 5.8 and the observation that the copy of the algorithm B replacing the +t-th query processes the state Qt +` +A, BpOq +˘ +ξ. +This result requires a number of comments. +First, it is usually convenient to assume that N is the output space of the algorithm B, +not its workspace. This can be achieved by applying Proposition 5.7, cf. the discussion after +Proposition 5.8. +Next, Proposition 5.10 assumes the that algorithm A has a single input oracle (while the +algorithm B and, consequently, A ˝B can have multiple input oracles). Let us now consider the +case when A has multiple input oracles Opiq : N piq Ñ N piq. For each i, let Bpiq be an algorithm in +N piq with the oracle in M. Using Proposition 5.9, they can be combined into a single algorithm +B “ À +i Bpiq acting in N “ À +i N piq, which is the same space where the combined input oracle +of A acts. Thus, using (5.4), we obtain the following version of (5.5): +LpA ˝ B, O, ξq “ +ÿ +i +ÿ +t +L +´ +Bpiq, O, Qpiq +t +` +A, BpOq +˘ +ξ +¯ +. +(5.6) +17 + +We will return to the above two comments in Section 8.3. +The final comment concerns slicing. Namely, when applying Proposition 5.10 to a non-sliced +algorithm A as in Definition 3.1, it is first necessary to slice the latter using Proposition 5.5. Slic- +ing is convenient here as it allows us to use Las Vegas complexity of B on the state Qt +` +A, BpOq +˘ +ξ +directly. The downside of this approach is that the resulting algorithm depends on the way how +we slice the query O b I‚ of the algorithm A in (5.1). As discussed before, this does not change +the action of the algorithm. However, it is not clear how it affects complexity. +In order to understand this, it suffices to consider one query of the outer algorithm. That +is, we can assume the composed algorithm is of the form B b I‚. Applying Proposition 5.10 to +the sliced algorithm and using the following decomposition similar to (5.2): +ξ “ ξ1 ‘ ξ2 ‘ ¨ ¨ ¨ ‘ ξd +(5.7) +with each ξj in N, we get +LpB b I‚, O, ξq “ +dÿ +j“1 +LpB, O, ξjq. +(5.8) +Observation 5.11. The value of the right-hand side of (5.8) is independent from the choice of +a particular slicing in (5.7). +Therefore, for a non-sliced algorithm A, we can write an analogue of (5.5): +LpA ˝ B, O, ξq “ +ÿ +t +L +` +B b I‚, O, Qt +` +A, BpOq +˘ +ξ +˘ +, +(5.9) +which is well-defined due to the above observation. Similarly, in the case of multiple input +oracles, we can write the following analogue of (5.6): +LpA ˝ B, O, ξq “ +ÿ +i +ÿ +t +L +´ +Bpiq b I‚, O, Qpiq +t +` +A, BpOq +˘ +ξ +¯ +. +Proof of Observation 5.11. In (5.7), we decomposed ξ assuming some standard basis in the +space of I‚. Let u1, . . . , ud be another orthonormal basis of the same space. Thus, we have a +similar decomposition +ξ “ ξ1 +1 b u1 ` ξ1 +2 b u2 ` ¨ ¨ ¨ ` ξ1 +d b ud +(5.10) +with ξ1 +1, . . . , ξ1 +d P N, but this time based on the basis u1, . . . , ud. +Since the the basis u1, . . . , ud is orthonormal the decompositions in (5.7) and (5.10) are +connected by a unitary U in the following way, where we assume the unitary U is given by (2.1): +ξ1 +j “ α1,jξ1 ` α2,jξ2 ` ¨ ¨ ¨ ` αd,jξd. +Therefore, the complexity of the algorithm obtained when using the slicing based on u is +dÿ +j“1 +LpB, O, ξ1 +jq “ +dÿ +j“1 +LpB, O, α1,jξ1 ` α2,jξ2 ` ¨ ¨ ¨ ` αd,jξdq “ +dÿ +j“1 +LpB, O, ξjq +by Proposition 5.2. +As a by-product we get a nice expression for a tensor product of algorithms. +18 + +Proposition 5.12 (Tensor Product). Let A and B be two algorithms in spaces H and H1 +respectively, and both with oracles in M. Then, there exists an algorithm A b B in H b H1 with +oracles in M such that for all O: M Ñ M, we have pA b BqpOq “ ApOq b BpOq. Moreover, if +O is a unitary, then +LpA b B, O, ξq “ LpA b IH1, O, ξq ` LpIH b B, O, ξq, +(5.11) +where the two terms on the right-hand side are defined as in (5.8). +Proof. We can implement A b B as pIH b Bq ˚ pA b IH1q. By Proposition 5.8, we get +L +` +A b B, O, ξ +˘ +“ L +` +A b IH1, O, ξ +˘ +` L +` +IH b B, O, pApOq b IH1qξ +˘ +. +Therefore, it remains to prove that +LpIH b B, O, ξq “ L +` +IH b B, O, pApOq b IH1qξ +˘ +. +But this follows from Observation 5.11, as multiplication by a unitary ApOq b IH1 can be seen +as a change of basis in H. +If O is not unitary, we do not get such a nice expression as (5.11). +For instance, the +complexity depends on whether we implement A b B as pIH b Bq ˚ pA b IH1q or as pA b IH1q ˚ +pIH b Bq. +6 +Unidirectional Relative γ2-bound +The variants of the adversary bound in [44] and [15] are formulated in terms of generalisations +of the γ2-norm. The γ2-norm was originally developed in the context of operator factorisation +in Banach spaces [59, Section 13]. It has an independent formulation as the Schur (Hadamard) +product operator norm [26]. In the realm of theoretical computer science, it was first used in +communication complexity [47, 48, 45]. In the context of the quantum adversary, its generali- +sations appeared in [44] and [15] as filtered and relative γ2-norms, respectively. +We have to generalise the latter in several directions. First, in order to deal with unidi- +rectional access to the input oracle, we have to define the unidirectional version of the bound, +which we do in Section 6.1. +The previous (bidirectional) case can be obtained as a special +case, see Section 9. Second, in order to switch from the worst-case complexity to the complete +complexity profile, we have to introduce the multi-objective version of the bound. Finally, we +also have to modify the bound to capture the case of several input oracles. All this is done in +Section 6.2. +In Section 6.3, we prove few basic properties of the unidirectional relative γ2-bound, which +we will need later in the paper. +6.1 +Single-Objective Version +Definition 6.1 (Unidirectional relative γ2-bound). Let K and M be vector spaces, and D be a +set of labels. Let E “ tExyu and ∆ “ t∆xyu, where x, y P D, be two families of linear operators: +Axy : K Ñ K and ∆xy : M Ñ M that satisfy Exy “ E˚ +yx and ∆xy “ ∆˚ +yx for all x, y P D. +The unidirectional relative γ2-bound +Ð +γ2pE|∆q “ +Ð +γ2pExy | ∆xyqx,yPD, +19 + +is defined as the optimal value of the following optimisation problem, where Vx are linear +operators: +minimise +maxxPD∥Vx∥2 +(6.1a) +subject to +Exy “ V ˚ +x p∆xy b IWqVy +for all x, y P D; +(6.1b) +W is a vector space, +Vx : K Ñ M b W. +(6.1c) +Depending on the context, we will denote by +Ð +γ2pE|∆q both the optimal value and the +optimization problem itself. +We will be mostly using the following one-dimensional version, where each Ex,y “ ex,y is a +scalar. Then, the bound reads as follows: +minimise +maxxPD∥vx∥2 +(6.2a) +subject to +exy “ +@ +vx, p∆xy b IWqvy +D +for all x, y P D; +(6.2b) +W is a vector space, +vx P M b W. +(6.2c) +The version (6.2) is the one mentioned in Figure 1 in the introduction. Its feasible solutions +correspond to the algorithms solving the problem. In order to prove lower bounds, we need +another closely related notion. +Let us define the following generalisation of the Hadamard +product. Assume X and Y be some sets of labels, and ∆ “ p∆x,yq, where x P X and y P Y , be +a set of matrices of the same dimensions. For Γ, an X ˆ Y matrix, we define Γ ˝∆ as an X ˆ Y +block matrix, where the block corresponding to x P X and y P Y is given by Γrrx, yss∆x,y. +Definition 6.2 (Unidirectional subrelative γ2-bound). In assumptions of Definition 6.1, the +unidirectional subrelative γ2-bound +ó +γ2pE|∆q “ +ó +γ2pExy | ∆xyqx,yPD, +is defined as the optimal value of the following optimisation problem: +maximise +λmaxpΓ ˝ Eq +(6.3a) +subject to +λmaxpΓ ˝ ∆q ď 1, +(6.3b) +where Γ ranges over D ˆ D Hermitian matrices. Here λmax stands for the largest eigenvalue of +a Hermitian matrix. +This version is similar to the dual of the relative γ2-norm from [15], except that it has +λmax instead of the spectral norm. The latter, in its turn, is similar to the negative-weighted +adversary from [37]. It is easy to show that (6.3) lower bounds (6.1). +Theorem 6.3 (Weak Duality). For E and ∆ as in Definitions 6.1 and 6.2, we have +Ð +γ2pE|∆q ě +ó +γ2pE|∆q. +Proof. Assume we have a feasible solution Vx to +Ð +γ2pE|∆q. From (6.1b), we get that for every +D ˆ D-matrix Γ: +Γ ˝ E “ V ˚“ +pΓ ˝ ∆q b IW +‰ +V, +where V “ À +xPD Vx is the block-diagonal matrix with the blocks Vx on the diagonal. Hence, +λmaxpΓ ˝ Eq “ max +v:}v}“1 v˚pΓ ˝ Eqv “ max +v:}v}“1pV vq˚“ +pΓ ˝ ∆q b IW +‰ +V v +ď }V }2 ¨ λmax +` +pΓ ˝ ∆q b IW +˘ +“ max +xPD }Vx}2 ¨ λmaxpΓ ˝ ∆q ď +Ð +γ2pA|∆qλmaxpΓ ˝ ∆q. +20 + +Let us note, although we will not need it in this paper, that in the one-dimensional case it +is possible to strengthen the previous theorem. +Theorem 6.4. If all Ex,y “ ex,y are one-dimensional, then +Ð +γ2pE|∆q “ +ó +γ2pE|∆q. +Therefore, the lower bound (6.3) is tight in this case. The proof follows from strong duality +and is a variant of the proof in [15]. It can be found in Appendix A. Note that Theorem 6.3 is +not true in general, when Ex,y are not one-dimensional. +6.2 +Multi-Objective Version +Since we consider Las Vegas complexity of each individual input, considering a single number as +an output of an optimisation problem like (6.1) and (6.2) is too restrictive. Here we define the +multi-objective version of the same optimisation problem. Additionally, we consider the version +of the bound with multiple ∆, which corresponds to the multiple-oracle case of Section 4.2. +For the latter, assume that M from Definition 6.1 is decomposed as M “ Mp1q ‘ Mp2q ‘ +¨ ¨ ¨ ‘ Mpsq, and each ∆xy has a similar decomposition: +∆xy “ ∆p1q +xy ‘ ∆p2q +xy ‘ ¨ ¨ ¨ ‘ ∆psq +xy +(6.4) +with ∆piq +xy : Mpiq Ñ Mpiq. Additionally, we write Vx : K Ñ M b W from (6.1) as a vertical stack +of matrices +Vx “ +¨ +˚ +˚ +˚ +˚ +˝ +V p1q +x +V p2q +x +... +V psq +x +˛ +‹‹‹‹‚ +with V piq +x +: K Ñ Mpiq b W. We also generalise (4.6) to such matrices: +~Vx~2 “ +´ +}V p1q +x +}2, }V p2q +x +}2, . . . , }V psq +x +}2¯ +. +Definition 6.5 (Multi-objective unidirectional relative γ2 optimisation problem). In notation +of Definition 6.1 and the above assumptions on M and ∆, the multi-objective unidirectional +relative γ2 optimisation problem +Ð +γ2pE|∆q “ +Ð +γ2pExy | ∆xyqx,yPD, +is defined as follows: +minimise +p~Vx~2qxPD +(6.5a) +subject to +Exy “ V ˚ +x p∆xy b IWqVy +for all x, y P D; +(6.5b) +W is a vector space, +Vx : K Ñ M b W. +(6.5c) +The bound (6.5) is equivalent to (6.1) with the only difference in the objective, which justifies +the use of the same notation +Ð +γ2pE|∆q. Later we will almost exclusively use the multi-objective +version. +In the multiple-oracle case, we assume that the decomposition in (6.4) is implicit. Note that +it only changes the objective, and does not change the constraints. Also, the constraint (6.5b) +in this case is equivalent to +Exy “ +sÿ +i“1 +` +V piq +x +˘˚` +∆piq +xy b IW +˘ +V piq +y . +21 + +Definition 6.6. For a feasible solution Vx of (6.5), we call p~Vx~2qxPD the objective profile of +the feasible solution. In the single-oracle case, it is a vector in RD. In the multiple-oracle case, +it is a vector in RD b Rs. The feasible objective space of the optimization problem (6.5) is the +set of all objective profiles over all feasible solutions Vx of (6.5). +Claim 6.7. If all Ex,y “ ex,y are one-dimensional, the feasible objective space of (6.5) is a +topologically closed subset of RD b Rs. +This is also true in general, but we only need the one-dimensional case, which we prove in +Appendix A. +Finally, for the multi-oracle case, we have the following variant of Theorem 6.3, which binds +the matrix Γ to the individual }V piq +x }2. It can be used to prove trade-offs between input oracles. +Theorem 6.8. For every feasible solution Vx to (6.5) and every D ˆ D Hermitian matrix Γ, +we have +λmaxpΓ ˝ Eq ď +sÿ +i“1 +λmaxpΓ ˝ ∆piqq max +xPD +��V piq +x +��2. +Proof. Again, let V “ À +xPD Vx and V piq “ À +xPD V piq +x . The proof follows the proof of Theo- +rem 6.3 with the following change at the last step: +λmaxpΓ ˝ Eq “ max +v:}v}“1 v˚pΓ ˝ Eqv “ max +v:}v}“1pV vq˚“ +pΓ ˝ ∆q b IW +‰ +V v +“ max +v:}v}“1 +sÿ +i“1 +pV piqvq˚“ +pΓ ˝ ∆piqq b IW +‰ +V piqv +ď +sÿ +i“1 +λmaxpΓ ˝ ∆piqq max +xPD +��V piq +x +��2. +6.3 +Properties +Let us list some properties of the unidirectional relative γ2-bound. We are mostly interested in +the case when the right-hand side ∆x,y is fixed, and the left-hand side ex,y is variable. +Proposition 6.9. Assume that w and w1 are in the feasible objective spaces of optimization +problems +Ð +γ2 +` +ex,y|∆x,y +˘ +x,yPD and +Ð +γ2 +` +e1 +x,y|∆x,y +˘ +x,yPD, respectively. Then, for all real c, c1 ě 0, the +vector cw ` c1w1 is in the feasible objective space of +Ð +γ2 +` +c1ep1q +x,y ` c2ep2q +x,y | ∆x,y +˘ +x,yPD. +Proof. Assume that +` +vx +˘ +xPD is a feasible solution to +Ð +γ2 +` +ex,y | ∆x,y +˘ +x,yPD with objective profile +w, and v1 +x is defined similarly for w1. Then, +`?cvx‘ +? +c1v1 +x +˘ +xPD is a feasible solution to +Ð +γ2 +` +cex,y` +c1e1 +x,y | ∆x,y +˘ +x,yPD with objective profile cw ` c1w1 by (4.8a) and (4.8b). +We will often have that ∆xx “ 0 for all x P D. In this case, it is easy to specify all families +of ex,y that have a feasible solution. +Proposition 6.10. Assume that ∆x,y in addition to ∆x,y “ ∆˚ +y,x satisfy ∆x,x “ 0 for all x. +Let pex,yqx,yPD be any collection of complex numbers such that ex,y “ e˚ +y,x for all x, y P D, and +ex,y “ 0 whenever ∆x,y “ 0. Then the optimisation problem +Ð +γ2 +` +ex,y | ∆x,y +˘ +x,yPD has a feasible +solution. +22 + +Proof. Due to Proposition 6.9, it suffices to consider the case when there exist distinct x0, y0 P D +such that ex0,y0 “ e˚ +y0,x0 are the only non-zero ex,y. By the assumption, ∆x0,y0 ‰ 0. Hence, there +exist vectors u, v such that u˚∆x0,y0v “ 1. Define the feasible solution as vx0 “ u, vy0 “ ex0,y0v, +and vx “ 0 otherwise. +Describing the set of ex,y that have feasible solution in the general case (when ∆x,x ‰ 0) is +more complicated, and we do not do it here. +Proposition 6.11. If ∆x,x “ 0 for all x, then the feasible objective space of +Ð +γ2pE|∆q is upwards +closed, i.e., if w P RD b Rs is in the feasible objective space, and w1 ě w (component-wise), +then w1 is also in the feasible objective space. +Proof. Let Vx be a feasible solution such that ~Vx~ “ wx for all x. +Let Mx be pairwise +orthogonal copies of M that are also orthogonal to MbW. There exist V 1 +x : K Ñ pMbWq‘Mx +such that their projection to M b W agree to Vx and ~V 1 +x~ “ w1 +x. +They also satisfy the constraints (6.5b) with the properly enlarged W. Indeed, for x ‰ y this +follows from the orthogonality of Mx and My, and for x “ y this follows from ∆x,x “ 0. +7 +Adversary Bound for State Conversion +This is the central section of the paper, in which we define the adversary bound for state conver- +sion with general input oracles, and prove that it equals Las Vegas complexity. In Section 7.1, +we restate the state conversion problem, define its Las Vegas complexity, and formulate the +corresponding adversary optimisation problem. In Section 7.2, we explain the intuition behind +the latter definition. Sections 7.3 and 7.4 are devoted to the two main technical results: a lower +bound for exact, and an upper bound for approximate state conversion. They are the corner- +stones of what comes next. In Section 7.5, we prove an upper bound for exact state conversion, +thus showing that the adversary bound is precisely equal to Las Vegas complexity. We finish the +section with two examples. In Section 7.6, we consider a simple example of a state conversion +problem with |D| “ 2, and in Section 7.7 we obtain the adversary bound of Boolean function +evaluation. +The main results in Sections 7.3 and 7.4 hold even for general linear input oracles. +In +Section 7.5, we have to assume that the input oracles are unitary. +7.1 +Definitions +Our choice of problem for this section is state conversion with general input oracles. +The +motivation for this initial choice is as follows. First, we want more control on the input oracle: +we require that, for each x P D, we have only one input oracle. Second, we would like to have +larger flexibility on the side of the algorithm, that is why we choose the state conversion problem, +where we have to map one state into another. In the beginning, we even do it approximately. +Going to more specific tasks, like state generation, does not give us anything. We will extend +the output and the input conditions to subspace conversion in the next section. +Let us give an explicit definition of state conversion, which follows from the general consid- +eration of Section 3.2. +Definition 7.1 (State Conversion with General Input Oracles). Let D be a set of labels, and +M and K vector spaces. For each x P D, let Ox : M Ñ M be a linear transformation. A +state conversion problem is given by a collection of tuples ξx ÞÑ τx where x ranges over D +and ξx, τx P K. Assume that K is embedded in the space H of a quantum algorithm A. We +23 + +say that the algorithm A solves the state conversion problem ξx ÞÑ τx on input oracles Ox, if +ApOxqξx “ τx for all x P D. +Some of the results in this section hold even if we only assume that Ox are linear trans- +formations. However, we will usually assume that Ox are contractions or unitaries. The lower +bound result hold even for infinite D, but for the upper bounds it is crucial that D is finite. +This definition also includes the case of multiple input oracles as described in Section 4.2. +Then, as in (4.4), each Ox “ Op1q +x +‘ Op2q +x +‘ ¨ ¨ ¨ Opsq +x +with Opiq +x +acting in Mpiq. +We derive Las Vegas complexity of this problem from the general definition of Section 4. +We study not only the worst-case complexity, but consider each input x P D individually. +Definition 7.2 (Las Vegas complexity of State Conversion). Assume we have a state conversion +problem is as in Definition 7.1, and an algorithm A that solves it. The Las Vegas complexity +of the algorithm A on input x P D, is defined as LxpAq “ LpA, Ox, ξxq. +The worst-case +Las Vegas complexity is defined as maxxPD LxpAq, in which case, we assume we have a single +input oracle. +The complexity profile of the algorithm A is the vector in RD b Rs given by +LDpAq “ pLxpAqqxPD. The feasible complexity space of the problem is a subset of RD b Rs +which is the set of all complexity profiles of the algorithms solving the problem. +Let us now define the adversary optimisation problem corresponding to the state conversion +problem. It is a generalisation of the version of the adversary bound from [15] to the case of +unidirectional input oracles. +Definition 7.3 (Adversary Optimisation Problem). Assume ξx ÞÑ τx is a state conversion +problem with unidirectional input oracles Ox : M Ñ M, as x P D. Its adversary optimisation +problem is the following unidirectional γ2 optimisation problem: +Ð +γ2 +´ +xξx, ξyy ´ xτx, τyy | IM ´ O˚ +xOy +¯ +x,yPD. +(7.1) +Here IM stands for the identity on M, but we often omit this subscript. It is easy to see +that the constraints of Definition 6.1 are satisfied, and this is a legitimate unidirectional γ2- +optimisation problem. Let us write it down explicitly as we will be using it quite extensively. +We consider it as a multi-objective optimisation problem. +minimise +` +~vx~2˘ +xPD +(7.2a) +subject to +xξx, ξyy ´ xτx, τyy “ +@ +vx, ppI ´ O˚ +xOyq b IWqvy +D +for all x, y P D; +(7.2b) +W is a vector space, +vx P M b W. +(7.2c) +7.2 +Intuition +Let us describe the intuition behind the bound (7.1). For a collection of vectors pξxqxPD, let +Gξ denote the corresponding Gram matrix: Gξrrx, yss “ xξx, ξyy. Two collections of vectors can +be transformed one into another by a unitary transformation if and only if they have the same +Gram matrix. Since unitary transformations are free in quantum query algorithms, we may +replace collections of vectors by the corresponding Gram matrices. For instance, rather than +saying that an algorithm solves state conversion ξx ÞÑ τx, we can say that it transforms Gξ into +Gτ, or write Gτ ÞÑ Gξ. +Then, the left-hand side of (7.2b) gives the difference of the corresponding Gram matrices +Gξ ´ Gτ. The right-hand side +@ +vx, ppI ´ O˚ +xOyq b IWqvy +D +“ +@ +vx, vy +D +´ +@ +pOx b IWqvx, pOy b IWqvy +D +(7.3) +24 + +gives the change in the Gram matrix when the state vx is processed by the oracle Ox. The +objective value ~vx~2 can be interpreted as the corresponding Las Vegas complexity. Therefore, +the optimisation problem seeks for the best possible states vx to be processed by the oracle to +get the required change in the Gram matrix. +The issue of how to get the states vx to the +input oracle is ignored here. Therefore, one can see the adversary optimisation problem as a +semi-definite relaxation of a quantum query algorithm. +To get the lower bound in Section 7.3, we accumulate changes in the Gram matrix like +in (7.3) over all the queries made by the algorithm. +The proof closely follows the proof of +Theorem 10 from [15]. To get the algorithm in Section 7.4, we repeatedly apply a scaled down +version of the query in (7.3). The result is that the Gram matrix slowly slides close to the line +connecting Gξ to Gτ in the cone of D ˆ D semi-definite matrices. +7.3 +Lower Bound (For Exact Version) +Theorem 7.4. Assume A is an algorithm that performs state conversion ξx ÞÑ τx with unidi- +rectional access to general linear oracles Ox as x P D. Then, its complexity profile LDpAq is in +the feasible objective space of the adversary optimization problem (7.1). +Proof. Denote for brevity +ψt,x “ StpA, Oxqξx “ Ut´1 rOx Ut´2 rOx ¨ ¨ ¨ U1 rOxU0ξx, +and let ψ1 +t,x “ QtpA, Oxqξx be the state processed on step t by the input oracle. The operators +St and Qt are defined in (4.1) and (4.2), respectively. +We have xψ1,x, ψ1,yy “ xξx, ξyy, and ψT`1,x “ τx. This gives +xξx, ξyy ´ xτx, τyy “ +Tÿ +t“1 +´ +xψt,x, ψt,yy ´ xψt`1,x, ψt`1,yy +¯ +. +Next, for the effect of one query rOx “ pOx b I‚q ‘ I˝: +xψt,x, ψt,yy ´ xψt`1,x, ψt`1,yy “ xψt,x, ψt,yy ´ +A +rOxψt,x, rOyψt,y +E +“ +@ +ψ1 +t,x, ψ1 +t,y +D +´ +@ +pOx b I‚qψ1 +t,x, pOy b I‚qψ1 +t,y +D +“ +@ +ψ1 +t,x, ψ1 +t,y +D +´ +@ +ψ1 +t,x, pO˚ +xOy b I‚qψ1 +t,y +D +“ +@ +ψ1 +t,x, ppIM ´ O˚ +xOyq b I‚qψ1 +t,y +D +. +(7.4) +This means that we can take +vx “ +T +à +t“1 +ψ1 +t,x +(7.5) +as a feasible solution to (7.1). By (4.7) and (4.8b), ~vx~2 is equal to the Las Vegas complexity +Lx, hence, this feasible solution has LDpAq as its objective profile. +The theorem is proven for exact and coherent state conversion. However, it can be used for +approximate or non-coherent state conversion ξx ÞÑ τx as well. Indeed, the latter is equivalent +to exact coherent state conversion ξx ÞÑ τ 1 +x for some τ 1 +x satisfying the corresponding closeness +requirements to τx as explained in Section 3.2. See Section 10 for an example. +The map ξx ÞÑ vx is important enough, so that we introduce a special notation for it: +VpA, Oq: ξ ÞÑ +T +à +t“1 +QtpA, Oqξ, +(7.6) +which is a linear transformation. +25 + +7.4 +Upper Bound (For Approximate Version) +Theorem 7.5. Let ξx ÞÑ τx be a state conversion problem and Ox a general linear oracle, where +x ranges over a finite set D. Assume pvxqxPD is a feasible solution to the adversary optimization +problem (7.1) with L “ maxxPD }vx}2. Then, for every ε ą 0, there exists an algorithm A with +the following properties: +• it solves state conversion ξ` +x ÞÑ τ ` +x with unidirectional access to Ox, where ξ` +x and τ ` +x are +some states (not necessarily in K) satisfying }ξ` +x ´ ξx}, }τ ` +x ´ τx} ď ε for all x P D; +• its Monte Carlo query complexity is T “ +P +L{ε2T +; +• for each x, its Las Vegas query complexity Lx is ~vx~2. +Specifically, the algorithm transforms +ξ` +x “ ξx ‘ +1 +? +T +vx +ÞÝÑ +τ ` +x “ τx ‘ +1 +? +T +vx +(7.7) +in T queries, and the state processed by the input oracle on each query is vx{ +? +T. +Proof. Denote v1 +x “ pOx b IWqvx. As in (7.3), we obtain +xξx, ξyy ´ xτx, τyy “ +@ +vx, ppI ´ O˚ +xOyq b IWqvy +D +“ xvx, vyy ´ +@ +v1 +x, v1 +y +D +for all x, y P D. This is equivalent to +@ +v1 +x, v1 +y +D +` xξx, ξyy “ xvx, vyy ` xτx, τyy. +This means that there exists a unitary transformation U satisfying +Upv1 +x ‘ ξxq “ vx ‘ τx +for all x P D. +Let us now describe the algorithm. It depends on an integer parameter T, which is also its +query complexity. Its space is of the form V ‘ K b J , where V is isomorphic to M b W and J +is an T-qudit. +Up to unitaries, the transformation in (7.7) is equivalent to +1 +? +T +|vx⟩V ` |ξx⟩K b +ˆ 1 +? +T +Tÿ +j“1 +|j⟩J +˙ +ÞÝÑ +1 +? +T +|vx⟩V ` |τx⟩K b +ˆ 1 +? +T +Tÿ +j“1 +|j⟩J +˙ +. +(7.8) +The algorithm performs this transformation by going through the states +ψt,x “ +1 +? +T |vx⟩V ` |τx⟩K b +ˆ 1 +? +T +t´1 +ÿ +j“1 +|j⟩J +˙ +` |ξx⟩K b +ˆ 1 +? +T +Tÿ +j“t +|j⟩J +˙ +(7.9) +just before the t-th query. Note that ψ1,x and ψT`1,x are the states on the left- and the right- +hand sides of (7.8), respectively. On the t-th query, apply the input oracle Ox b IW to the +register V in ψt,x. This results in the state +1 +? +T +ˇˇv1 +x +� +V ` |τx⟩K b +ˆ 1 +? +T +t´1 +ÿ +j“1 +|j⟩J +˙ +` |ξx⟩K b +ˆ 1 +? +T +Tÿ +j“t +|j⟩J +˙ +. +26 + +Next, apply U to the space V ‘ K b |t⟩J , which gives ψt`1,x. After T iterations, we get the +required transformation. +The differences ξ` +x ´ ξx and τ ` +x ´ τx are both vx{ +? +T. The norm of this vector is less than +ε as long as T ě L{ε2, as required. Finally, the Las Vegas complexity on input x is exactly +T ¨ +‌‌‌‌ +vx +? +T +‌‌‌‌ +2 +“ ~vx~2. +Figure 3 +Gξ` +Gξ +Gτ ` +Gτ +Gτ 1 +Visualisation of the algorithm A used in Theorem 7.5 and Corollary 7.6. The algorithm +follows the straight line from Gξ` to Gτ `, as the Gram matrices of the states in (7.9) for +various t are uniformly placed on this line. The wiggly line indicates the application of A +to Gξ in Corollary 7.6. The algorithm follows closely to the line connecting Gξ` and Gτ ` +and terminates in a point Gτ 1 close to Gτ `, but not, generally, Gτ. +An immediate corollary is that for contraction oracles we can replace ξ` +x with the original +ξx and get essentially the same bound on Monte Carlo complexity. +Corollary 7.6. Assume the premises of Theorem 7.5, where the input oracles Ox are con- +tractions. Then, for every ε ą 0, there exists a quantum algorithm with Monte Carlo query +complexity +P +4L{ε2T +that ε-approximately and coherently solves state conversion ξx ÞÑ τx with +unidirectional access to Ox. +This is a unidirectional version of the main technical result of [15]. This version has slightly +better dependence on ε, compared to [15], which had O +` +ε´2 log 1 +ε +˘ +. By the example due to +Kothari [41], see [15], the dependence on ε is tight up to constant factors. +Proof of Corollary 7.6. Consider the same algorithm A as in Theorem 7.5. Denote by τ 1 +x its final +state when executed on the initial state ξx and the oracle Ox. See Figure 3 for an illustration. +Since Ox is a contraction, the whole algorithm ApOxq is a contraction as well. Hence, +}τ 1 +x ´ τ ` +x } “ }ApOxqξx ´ ApOxqξ` +x } ď }ξx ´ ξ` +x } ď ε. +Since }τ ` +x ´ τx} ď ε, the triangle inequality gives us }τx ´ τ 1 +x} ď 2ε. Dividing ε by 2, we get the +required algorithm. +We also get a relation between Las Vegas and Monte Carlo complexities, which is a direct +consequence of Theorem 7.4 and Corollary 7.6. +Corollary 7.7. Assume there is an algorithm that solves state conversion ξx ÞÑ τx with con- +traction input oracles Ox exactly and has worst-case Las Vegas complexity L. Then, there exists +an algorithm that ε-approximately and coherently solves the same problem and has Monte Carlo +complexity OpL{ε2q. +27 + +Since the distance }τx ´ τ 1 +x} ď ε converts to error ε2 after measurement, it is reasonable to +say that the complexity of the algorithm is inversely linear in the error, which is similar to the +randomised case. This is the result mentioned in the introduction. +7.5 +Upper Bound For Exact Version +The results of the previous two subsections are good enough for most purposes. In particular, +in Theorem 7.5, we can take ξ` +x and τ ` +x as close to ξx and τx as we want and the Las Vegas +complexity stays ~vx~2. +However, in this section we improve this result and show how to +perform exact state conversion ξx ÞÑ τx essentially in the same budget. Together with the lower +bound, Theorem 7.4, this shows that Las Vegas complexity is exactly equal to the adversary +bound. +This is not only mathematically more satisfying and follows the convention of Las Vegas +complexity to describe exact computation. One of the motivations behind this result is that +a priori there is no good way to bind Las Vegas complexity on close initial states. If ξx and +ξ` +x are at a distance ε, then, from general principles, it can be deduced that their Las Vegas +complexities differ by at most εT. But this is useless because T generally is not bounded. This +poses problems if, for example, we want to compose Las Vegas programs using Propositions 5.8 +or 5.10 and we only have approximate versions of the subroutines. +In this section, we extensively use the language of Gram matrices introduced in Section 7.2. +Our first observation is that if both Gram matrices Gξ and Gτ are full rank, then state conversion +can be performed exactly. +Lemma 7.8. Assume the premises of Theorem 7.5. If we additionally have that Gξ and Gτ are +positive definite, then state conversion ξx ÞÑ τx can be solved exactly with Las Vegas complexity +~vx~2. Moreover, the state processed by the oracle on each query is vx{ +? +T, where T is the +number of queries. +Proof. Since Gξ, Gτ ą 0, there exists an integer T such that Gξ, Gτ ě 1 +T Gv. Choose ξ´ +x and +τ ´ +x so that their Gram matrices are Gξ´ “ Gξ ´ 1 +T Gv and Gτ ´ “ Gτ ´ 1 +T Gv. Note that, for all +x, y P D: +@ +ξ´ +x , ξ´ +y +D +´ +@ +τ ´ +x , τ ´ +y +D +“ xξx, ξyy ´ xτx, τyy. +Hence, vx is a feasible solution to the adversary bound (7.1) for state conversion ξ´ +x ÞÑ τ ´ +x as +well. Applying Theorem 7.5 to the latter and using (7.7), we get an algorithm performing exact +state conversion +ξ1 +x “ ξ´ +x ‘ +1 +? +T +vx +ÞÝÑ +τ 1 +x “ τ ´ +x ‘ +1 +? +T +vx, +where on each step the state vx{ +? +T is processed by the oracle. These collections of vectors have +Gram matrices Gξ, and Gτ, respectively. Hence, they can be turned into ξx and τx, respectively, +by unitaries. +In the illustration in Figure 3, the ξx and τx are equivalent to ξ` +x and τ ` +x , and the algorithm +follows the straight line connecting Gξ` and Gτ `. +The assumptions Gξ ą 0 and Gτ ą 0 are very strong, and almost never hold. For instance, +the state generation problem has Gξ of rank 1. For (exact and coherent) Boolean function +evaluation, the rank of Gτ is 2. However, we will be able to apply this lemma by first “pushing” +both Gξ and Gτ into the space of positive-definite matrices. +But for that we will need some additional assumptions. First, we assume that all Ox are +unitaries. Second, we get the point p}vx}2qxPD only as a limit of points in the feasible complexity +space. This is because pushing Gξ and Gτ takes complexity, which can be made arbitrary small, +28 + +but cannot be made zero. In Section 7.6, we will show that the above two assumptions are +necessary. Finally, for simplicity we assume that all Ox are pairwise distinct. We will lift this +restriction in Section 8.1. +Let us start with the question when a state conversion ξx ÞÑ τx is possible at all. If there +is an algorithm performing the transformation, we say that Gτ is achievable from Gξ. Denote +by Rξ the real affine space of D ˆ D Hermitian matrices A satisfying Arrx, xss “ }ξx}2 for all +x P D. +Lemma 7.9. Assuming that the input oracles Ox are unitary are pairwise distinct, we have: +(a) If the state conversion ξx ÞÑ τx is possible, then Gτ P Rξ and Rτ “ Rξ; +(b) For any two M, M1 P Rξ, the optimisation problem +Ð +γ2 +` +Mrrx, yss ´ M1rrx, yss | I ´ O˚ +xOy +˘ +x,yPD +has a feasible solution. +Proof. The point (a) merely says that unitaries ApOxq do not change the norm of a vector. +The point (b) follows from Proposition 6.10. Indeed, in notation of the this proposition, +∆x,y “ I ´O˚ +xOy “ 0 if and only if x “ y, and ex,x “ Mrrx, xss´M1rrx, xss “ 0 for all x P D. +We can now describe our algorithm for exact state conversion. +Theorem 7.10. Let ξx ÞÑ τx be a state conversion problem with pairwise distinct unitary input +oracles Ox, where x ranges over a finite set D. Assume pvxqxPD is a feasible solution to the +adversary optimization problem (7.1). Then, for every δ ą 0, there exists a quantum algorithm +A with the following properties: +• A solves state conversion ξx ÞÑ τx exactly with unidirectional access to Ox; +• for each x P D, we have +��LxpAq ´ ~vx~2�� ď δ. +Together with Theorem 7.4 and Claim 6.7, this gives the following main result of the paper: +Theorem 7.11. In the assumptions of Theorem 7.10, the following two sets are equal: +• the topological closure of the feasible complexity space of the state conversion problem; and +• the feasible objective space of the corresponding adversary optimisation problem (7.1). +Proof of Theorem 7.10. The idea is as follows. First, we show that we can transform Gξ and +Gτ into some M1, M2 ą 0. As soon as we get into the space of full-rank Gram matrices, we can +use Lemma 7.8. Namely, we use it to perform the two middle steps in the following chain of +transformations: +Gξ ÞÑ p1 ´ εqGξ ` εM1 ÞÑ p1 ´ εqGξ ` εM2 ÞÑ p1 ´ εqGτ ` εM2 ÞÑ Gτ, +(7.10) +where ε ą 0 is some small number. See Figure 4(b). The main work happens in the third step. +We use Propositions 5.6 and 5.1 to show that the complexity of the other steps vanishes with +ε Ñ 0. The final idea is that we perform the last transformation in reverse using Proposition 5.3. +Let us proceed with the proof. We may assume that Gτ P Rξ. +Otherwise, neither the +state conversion is possible, nor the optimisation problem has a feasible solution. We may also +assume that neither of ξx is zero, since then Gτ P Rξ implies τx “ 0 and any algorithm always +transforms 0 ÞÑ 0, so we may drop this input. Consider the matrix M defined by +Mrrx, yss “ +# +Gξrrx, xss “ Gτrrx, xss, +if x “ y; +0, +otherwise. +(7.11) +29 + +Clearly, M P Rξ and M ą 0. By Lemma 7.9(b), the adversary bound (7.1) corresponding to the +transformation Gξ ÞÑ M has a feasible solution. Using Corollary 7.6 and continuity of the inner +product, we can get Gram matrices achievable from Gξ that are arbitrarily close to M. Since +M is positive definite, there exists a positive definite M1 among them. Let B be the algorithm +that performs the transformation Gξ ÞÑ M1. +Using the same argument but with ξx replaced by τx and Ox replaced by O˚ +x, we get M2 ą 0 +and an algorithm E that transforms Gτ ÞÑ M2 using the input oracles O˚ +x. +By Lemma 7.9(a), both M1 and M2 are in Rξ. They are both positive definite. By point +(b) of the same lemma and Lemma 7.8, there exists a quantum algorithm C that transforms +M1 ÞÑ M2 exactly. These algorithms are depicted in Figure 4(a). +Figure 4 +Gξ +Gτ +(a) +M +M1 +M2 +B +C +E +Gξ +Gτ +(b) +M +M1 +M2 +Bε +Cε +Dε +E´1 +ε +The algorithms used in the proof of Theorem 7.10. The Gram matrices Gξ and Gτ are +not of full rank here, therefore are depicted on the edge of the cone of positive semidefinite +matrices. The matrix M is positive definite, and so are all the matrices in a sufficiently +small circle around M. In (b), the algorithms Bε and Cε are the scaled-down versions of B +and C. The algorithm E´1 is additionally reversed. As ε Ñ 0, the algorithm Dε approaches +the line connecting Gξ and Gτ. +For small enough ε ą 0, the following table lists the algorithms performing the transforma- +tions in (7.10) with their Las Vegas complexities. A graphical representation of the algorithms +30 + +is given in Figure 4(b). +Algorithm +Transformation +Complexity +Bε +Gξ +ÞÑ +p1 ´ εqGξ ` εM1 +LxpBεq “ εLxpBq +Cε +p1 ´ εqGξ ` εM1 +ÞÑ +p1 ´ εqGξ ` εM2 +LxpCεq “ εLxpCq +Dε +p1 ´ εqGξ ` εM2 +ÞÑ +p1 ´ εqGτ ` εM2 +LxpDεq “ p1 ´ εq~vx~2 +E´1 +ε +p1 ´ εqGτ ` εM2 +ÞÑ +Gτ +LxpEεq “ εLxpEq +The algorithm Bε is I ‘ B of Proposition 5.6, where B transforms εGξ ÞÑ εM1. +The +complexity follows from Proposition 5.1. +The algorithm Cε is analogous with C transform- +ing εM1 ÞÑ εM2. The algorithm E´1 +ε +is I ‘ E´1, where E´1 transforms εM2 ÞÑ εGτ due to +Propositions 5.3 and 5.1. +To get Dε, note that both matrices are positive definite, and their difference is p1 ´ εqpGξ ´ +Gτq, hence, we can use Lemma 7.8 with ?1 ´ ε vx as a feasible solution to the corresponding +adversary bound (7.1). +The algorithm A is the sequential composition of these subroutines, hence, by Proposi- +tion 5.8, we have +LxpAq “ εLxpBq ` εLxpCq ` p1 ´ εq~vx~2 ` εLxpEq Ñ ~vx~2 +as ε Ñ 0. +7.6 +Example with Two Labels +Here we consider an example when D “ t0, 1u and the input oracles O0 ‰ O1 are unitary. For +normalized states, Gram matrices can be parametrized by a single off-diagonal parameter a. +We write +Ga “ +ˆ 1 +a +a˚ +1 +˙ +. +(7.12) +Consider a transformation Ga ÞÑ Gb. In other words, we have that xξ0, ξ1y “ a and xτ0, τ1y “ b. +Claim 7.12. The feasible objective space of the corresponding adversary optimisation prob- +lem (7.1) is the epigraph of a hyperbola: +" +pw0, w1q +ˇˇˇ w0, w1 ě 0 and ?w0w1 ě +|a ´ b| +}O0 ´ O1} +* +. +(7.13) +Proof. First, since O0 and O1 are unitaries, we get that +Garrx, xss ´ Gbrrx, xss “ 0 “ xvx, ppI ´ O˚ +i Oiq b IWqvxy +for all vx and x “ 0, 1. So we only have to analyse the off-diagonal term +Garr0, 1ss ´ Gbrr0, 1ss “ a ´ b. +Denote d “ }I ´ O˚ +0O1} “ }O0 ´ O1}. +If v0, v1 is a feasible solution to the adversary +optimisation problem, then from (7.2b), we get +|a ´ b| “ +��xv0, ppI ´ O˚ +0O1q b IWqv1y +�� ď d }v0} ¨ }v1}, +implying the lower bound in (7.13). +31 + +In the opposite direction, assume ?w0w1 “ |a ´ b|{d. Let u and v be the normalised left +and right singular vectors of I ´ O˚ +0O1 with the singular value d. Then, we have +a ´ b “ p?w0uq˚pI ´ O˚ +0O1qpa ´ bq?w1v +|a ´ b| +, +implying that pw0, w1q is in the feasible objective space. The claim follows from Proposition 6.11. +By Theorem 7.11, the topological closure of the feasible complexity space of the correspond- +ing state conversion problem equals (7.13). We show that, in general, not all points in this set +are attained as complexity profiles of the algorithms performing the transformation. +Claim 7.13. Let O0 “ 1 and O1 “ ´1 be 1-dimensional unitaries. Consider the transformation +G1 ÞÑ Gi (where i is the imaginary unit) on these input oracles. The point p1{ +? +2, 1{ +? +2q is in +the feasible objective space of the corresponding adversary optimisation problem, but not in the +feasible complexity space of the problem. +Proof. The first statement follows from Claim 7.12. It remains to prove there is no algorithm +solving the problem with this complexity profile. +Consider an algorithm A that performs this transformation in T queries, and assume that +it goes through the following Gram matrices during its execution: +G1 ÞÑ Gc1 ÞÑ Gc2 ÞÑ ¨ ¨ ¨ ÞÑ GcT ´1 ÞÑ Gi. +(7.14) +First, we claim that only Gb with b P R are achievable from G1 in one query. Indeed, G1 +corresponds to a state collection ξ0, ξ1 with ξ0 “ ξ1. Therefore, the state processed by the input +oracle is the same for x “ 0 and x “ 1. Denote it ψ1. For the states τ0 and τ1 after the query, +we have +xτ0, τ1y “ 1 ´ 2}ψ1}2 P R. +Therefore, among c1, . . . , cT´1 there exists cj P Rzt1u. Write the algorithm as a sequential +composition A “ C ˚ B, where B performs the transformation G1 ÞÑ Gcj in (7.14), and C the +transformation Gcj ÞÑ Gi. +Using Claim 7.12, we get that for any point pw0, w1q in the feasible objective space for +transformation Ga ÞÑ Gb with our choice of input oracles +w0 ` w1 ě 2?w0w1 ě |a ´ b|. +Hence, by Theorem 7.4: +L0pBq ` L1pBq ě |1 ´ cj| +and +L0pCq ` L1pCq ě |cj ´ i|. +Combining this with Proposition 5.8 and the triangle inequality in C, we get +L0pAq ` L1pAq “ L0pBq ` L0pCq ` L1pBq ` L1pCq ě |1 ´ cj| ` |cj ´ i| ą |1 ´ i| “ +? +2. +Thus, p1{ +? +2, 1{ +? +2q is not in the feasible complexity space. +Let us now move to the case when O0 and O1 are contractions. Our goal is to show that +Theorems 7.10 and 7.11 are false in this case. For that, consider the transformation G1 ÞÑ G0 +with the input oracles in C2 given by +O0 “ +ˆ +1 +0 +0 +0 +˙ +and +O1 “ +ˆ +0 +´1 +0 +0 +˙ +. +32 + +Claim 7.14. For the above problem, the adversary optimisation problem has a feasible solution, +but there is no algorithm performing the required transformation exactly. +Proof. The feasible solution is v0 “ |0⟩ and v1 “ |1⟩. Let us prove there is no algorithm solving +the problem. +The initial Gram matrix G1 means that we have equal initial states ξ0 “ ξ1 of unit norm. +As G0 ‰ G1, the algorithm has to make at least one query. Consider the first query. Since +ξ0 “ ξ1, the state given to the oracle is the same for both inputs, denote it ψ1. We may assume +ψ1 ‰ 0, as otherwise this query can be ignored. Let ψx be the state of the algorithm after the +query on input x. +Without loss of generality, the algorithm is sliced, hence, ψ1 “ α|0⟩ `β|1⟩ for some α, β P C. +As ψ1 ‰ 0, either α ‰ 0, or β ‰ 0. If α ‰ 0, then ∥ψ1∥ ă 1. If β ‰ 0, then ∥ψ0∥ ă 1. In either +case, it is impossible to get both terminal states τ0 and τ1 to have unit norm. Therefore, there +is no algorithm solving the problem. +7.7 +Boolean Function Evaluation +Here, we derive the adversary bound for Boolean function evaluation as a simple special case +of Theorem 7.11. +Let us specify exactly what we mean by Boolean function evaluation in this context. Let +f : D Ñ t0, 1u with D Ď t0, 1un be a (partial) Boolean function. We assume the input oracle +Ox encodes x P D in the phase, and we consider the multi-oracle settings. That is, there are n +unitary input oracles Opiq +x : C Ñ C defined by Opiq +x “ p´1qxi. Since Ox is Hermitian, there is no +difference between unidirectional and bidirectional access to this oracle. We also assume that +the function is evaluated in the phase, exactly and coherently. That is, the output space K “ C +and the goal is to map |0⟩ ÞÑ p´1qfpxq|0⟩. Since the space is one-dimensional, this can also be +seen as an instance of unitary implementation. +We get that xξx, ξyy ´ xτx, τyy “ 2 ¨ 1fpxq‰fpyq, where 1P is the indicator variable. Similarly, +I ´ O˚ +xOy “ 2 Àn +j“1 1xj‰yj, where À is a direct sum of n matrices, each of size 1 ˆ 1, resulting +in an n ˆ n diagonal matrix. Dividing by 2, we get the adversary optimisation problem +Ð +γ2 +´ +1fpxq‰fpyq +ˇˇ +n +à +j“1 +1xj‰yj +¯ +. +(7.15) +It is similar to the corresponding expression in [15], except that it is unidirectional. In Section 9, +we will show that since Ox “ O˚ +x, the unidirectional version is equal to the usual bidirectional +relative γ2-bound. +The multi-objective optimisation problem (7.15) exactly characterises the Las Vegas com- +plexity of each of the n individual input symbols on every input x P D. +It is also possible to substitute the input oracle with the one that encodes xj in the phase. +Namely, with the oracle rOpiq +x : C2 Ñ C2 given by |b⟩ ÞÑ |b ‘ xj⟩, where ‘ is XOR here. Indeed, +in the Fourier basis, rOpiq +x +“ I1 ‘ Opiq +x , hence, the algorithm can just ignore the I1 part. The +same holds for the output, if we require that the algorithm has to perform the transformation +|b⟩ ÞÑ |b ‘ fpxq⟩ in the output space K “ C2. +8 +Subspace Conversion +In Section 8.1, we continue the settings of Theorem 7.10 but without the assumption that input +oracles are pairwise distinct. This leads to our investigation of the linear consistency of feasible +33 + +solutions. +The corresponding problem can be formulated as subspace conversion, which we +analyse in Section 8.2, define the corresponding notion of complexity and extend the connection +to the adversary bound. Finally, in Section 8.3, we revisit the functional composition property +of Section 5.5. The notion of complexity we introduced for the subspace conversion problem +will allow us to formulate and prove a simpler estimate on the complexity of the composed +algorithm. +8.1 +Linear Consistency +Throughout the section, we assume we have a state conversion problem ξx ÞÑ τx in K with input +oracles Ox, where x ranges over D. We will be particularly interested in pairs of inputs x, y +with Ox “ Oy. +For O P tOx | x P Du, let DO “ tx P D | Ox “ Ou and KO “ spantξx | x P DOu. The +important point is that the algorithm performs the same linear transformation ApOq on all +x P DO. Therefore, the pairs ξx ÞÑ τx for x P DO should be linearly consistent. The adversary +optimisation problem is in accord with this requirement as shown in the next result. +Proposition 8.1. Assume that the adversary optimisation problem (7.1) for a state conversion +problem ξx ÞÑ τx with contraction oracles Ox has a feasible solution. Then, for each O, there +exists a linear transformation TO : KO Ñ K such that τx “ TOξx for all x P DO. Moreover, if +O is unitary, TO is unitary. +Proof. Fix O, and restrict the optimisation problem to DO. Since O is a contraction, I ´ O˚O +exists semi-definite, and S “ ppI ´ O˚Oq b IWq1{2 is defined. By (7.2b), we have +xξx, ξyy ´ xτx, τyy “ +@ +vx, ppI ´ O˚Oq b IWqvy +D +“ +@ +Svx, Svy +D +for all x, y P DO. Thus, there exists a unitary that maps ξx ÞÑ τx ‘ Svx for all x. Hence, the +mapping TO : ξx ÞÑ τx is linear. If O is unitary, then S “ 0, and TO is unitary. +Note that the latter result is false for general linear transformations. For instance, if O “ 2I +it is easy to construct a feasible solution for a non-linear state conversion 0 ÞÑ |0⟩. This does +not contradict Theorem 7.5 though, because there the initial state is perturbed. Effects like +this is the main reason why we focus on contraction oracles. +Let us also consider linear consistency of feasible solutions. +Definition 8.2 (Linear consistency of feasible solutions). We say that a feasible solution vx +to the adversary optimisation problem (7.1) is linearly consistent if, for each O, there exists a +linear transformation VO : KO Ñ M b W such that vx “ VOξx for all x P Dx. +One way to ensure this condition is to impose the following. +Definition 8.3 (Linear independence assumption). We say that a state conversion problem +satisfies the linear independence assumption if, for each O, the vectors in tξx | x P DOu are +linearly independent. +Under this assumption, we are losing nothing in relation to the transformation performed. +For each O, we can uniquely extend this state conversion to all ξ P KO by linearity. We will +call the latter the linearly extended state conversion problem. +Proposition 8.4. We have +(a) Any feasible solution obtained via Theorem 7.4 is linearly consistent. +34 + +(b) Linear independence assumption implies linear consistency for all feasible solutions. +(c) Moreover, under linear independence assumption, any feasible solution can be uniquely +extended to a linearly consistent feasible solution to the linearly extended state conversion +problem. +Proof. For Point (a), use VO “ VpA, Oq from (7.6). Point (b) is obvious. Point (c) follows by +linearity. Let xi range over DO and yj over DO1. If +@ +ξxi, ξyj +D +´ +@ +τxi, τyj +D +“ +@ +vxi, ppI ´ O˚O1q b IWqvyj +D +for all xi and yj, then +Aÿ +i +aiξxi, +ÿ +j +bjξyj +E +´ +Cÿ +i +aiτxi, +ÿ +j +bjτyj +G +“ +Aÿ +i +aivxi, ppI ´ O˚O1q b IWq +ÿ +j +bjvyj +E +for all complex ai and bj. +Contrary to Propositions 8.1 and 8.4(a), feasible solutions to the adversary optimisation +problem need not satisfy linear consistency. For example, let D “ t0, 1, `, ´u, K be a qubit, +ξ0 “ |0⟩, ξ1 “ |1⟩, ξ` “ p|0⟩ ` |1⟩q{ +? +2, and ξ´ “ p|0⟩ ´ |1⟩q{ +? +2. We let Ox “ I and τx “ ξx for +all x P D. This problem in trivially solvable in 0 queries. But the following is a feasible solution +to (7.1), which is not linearly consistent: v0 “ v1 “ 0, and v` “ v´ “ |0⟩. +This poses a problem for strengthening Theorem 7.11. The feasible solution above gives +us a point p0, 0, 1, 1q in the feasible objective space. On the other hand, by the parallelogram +identity, Proposition 5.2, we have that for every algorithm A: +L`pAq ` L´pAq “ L0pAq ` L1pAq, +implying that the point p0, 0, 1, 1q is not in the topological closure of the feasible complexity +space. +One can say that this example is artificial. There is no need to deteriorate the solution +v0 “ v1 “ v` “ v´ “ 0 by increasing v` and v´. The following result states that this is a +general observation. Recall that a solution to a multi-objective optimisation problem is called +Pareto optimal if it is not strictly dominated by any other solution. In our case that means that +there is no other feasible solution v1 +x to the same optimisation problem such that ~v1 +x~2 ď ~vx~2 +for all x and ~v1 +x~2 ă ~vx~2 for some x. +Proposition 8.5. Any Pareto optimal solution to the adversary optimisation problem with +contraction input oracles is linearly consistent. +Proof. Let vx be a feasible solution. Take any O. We will show that either vx is linear consistent +on DO, or there is a feasible solution that strictly dominates vx. +By an argument like in Proposition 8.4(b) and (c), there exists a feasible solution v1 +x that +is linearly consistent on DO and equal to vx outside of DO. That is, there exists a linear map +V 1 : KO Ñ M b W such that v1 +x “ V 1ξx for all x P DO. Recall the conditions (7.2b): +xξx, ξyy ´ xτx, τyy “ +@ +vx, ppI ´ O˚ +xOyq b IWqvy +D +. +First, let us consider these constraints for x P DO and y R DO. (The constraints with x R DO +and y P DO are equivalent to these ones due to the symmetry imposed on a unidirectional +relative γ2-optimisation problem, Definition 6.1). Let Π1 denote the projector onto the span of +35 + +ppI ´ OOyq b IWqvy as y ranges over DzDO. The constraints (7.2b) are linear in vx and define +Π1vx uniquely. Therefore, Π1vx “ Π1v1 +x, and the mapping ξx ÞÑ Π1vx is linear for x P DO. +Next, consider the constraints (7.2b) for x, y P DO. Similarly to the proof of Proposition 8.1, +the operator pI ´ O˚Oq b IW is positive semi-definite. Let S “ ppI ´ O˚Oq b IW q1{2 and Π2 +denote the projector onto its range. We claim that the mapping ξx ÞÑ Π2vx is linear on DO as +well. Indeed, for x, y P DO, we have: +@ +Sv1 +x, Sv1 +y +D +“ xξx, ξyy ´ xτx, τyy “ xSΠ2vx, SΠ2vyy +Hence, there is a unitary U that maps Sv1 +x ÞÑ SΠ2vx. Then, Π2vx “ S`USV 1ξx, where S` is +the Moore-Penrose pseudoinverse. +Let Π denote the projector onto the span of Π1 and Π2. Since both ξx ÞÑ Π1vx and ξx ÞÑ Π2vx +are linear on DO, the same holds for ξx ÞÑ Πvx. If we replace vx by Πvx for each x P DO, we +get a feasible solution that is linearly consistent on DO and dominates vx. +Thus, by imposing the linear consistency condition of Definition 8.2 we are only losing +solutions where some of the objectives ~vx~2 are artificially inflated. +In the following, we +will only consider linearly consistent solutions. +By Proposition 8.4(c), we may assume the +problem satisfies the linear independence condition. We have the following generalisation of +Theorem 7.10. +Theorem 8.6. Let ξx ÞÑ τx be a state conversion problem with unitary oracles Ox, where +x ranges over a finite set D, and which satisfies the linear independence condition. Assume +pvxqxPD is a feasible solution to the adversary optimization problem (7.1), and let TO and VO +be like in Proposition 8.1 and Definition 8.2. Then, for every δ ą 0, there exists a quantum +algorithm A with the following properties: +• for every O, and ξ P KO, A transforms ξ ÞÑ TOξ on input oracle O; +• moreover, +��LpA, O, ξq ´ ~VOξ~2�� ď δ. +Proof. The proof is a modification of the proof of Theorem 7.10. Redefine Rξ as the real affine +space of DˆD Hermitian matrices A satisfying Arrx, yss “ xξx, ξyy for all x, y P D with Ox “ Oy. +The two points of Lemma 7.9 still hold. The matrix M is defined by +Mrrx, yss “ +# +Gξrrx, yss “ Gτrrx, yss, +if Ox “ Oy; +0, +otherwise. +(8.1) +Again, M P Rξ, as well as M ą 0, since it is a block-diagonal matrix whose blocks are Gram +matrices of linearly independent collections of vectors. Other than that, the algorithm is exactly +the same as in Theorem 7.10. +The algorithm transforms ξx ÞÑ τx on the input oracle O for all x P DO. By linearity, it +transforms ξ ÞÑ TOξ for all ξ P KO. The same linearity property holds for all queries made by +the algorithm. Therefore, the Las Vegas query complexity of the Dε subroutine of the algorithm +when the A is executed on the initial state ξ P KO is p1 ´ εq~VOξ~2. The complexities of other +subroutines tend to zero as ε Ñ 0, which gives the required result. +8.2 +Subspace Conversion Problem +As one can see, what Theorem 8.6 actually solves is the subspace conversion problem from +Definition 3.3. Let us restate the problem assuming general input oracles. +36 + +Definition 8.7 (Subspace Conversion, Restated). Let D be a set of labels, and M and K +be vector spaces. For each x P D, let Ox : M Ñ M be a linear transformation. A subspace +conversion problem is given by a collection of linear maps Tx : Kx Ñ K with Kx Ď K, where x +ranges over D. Assume that K is embedded in the space H of a quantum algorithm A. We +say that the algorithm A solves the subspace conversion problem Tx on input oracles Ox, if, for +every x P D, the map ApOxq agrees with Tx on Kx. We define the complexity LxpAq on the +input x as the supremum of LpA, O, ξq as ξ ranges over the unit vectors in Kx. The remaining +complexity-related definitions are as in Definition 7.2. +In the case of a single input oracle, the supremum in the definition of LxpAq is just the +maximum. In the case of multiple input oracles of Section 4.2, the supremum is understood +with respect to the dominance relation u ď v. In other words, it is the entry-wise maximum. +Note that it is not true, in general, that there exists a unit ξ P Kx such that LxpAq “ LpA, Ox, ξq, +since different input oracles can attain their maxima at different ξ. +The corresponding adversary optimisation problem is as follows: +Definition 8.8 (Adversary for Subspace Conversion). Consider a subspace conversion problem +as defined above with input oracles Ox, and let Kx be the projector onto Kx. The corresponding +adversary optimisation problem is given by +Ð +γ2 +` +K˚ +xKy ´ T ˚ +x Ty | I ´ O˚ +xOy +˘ +x,yPD. +(8.2) +Note that while (6.5) states that Vx : K Ñ MbW, in this optimisation problem we actually +have Vx : Kx Ñ M b W, as Tx is defined on Kx, and the coimage of Kx is Kx as well. In +particular, for the state conversion problem, where each Kx is one-dimensional, we get back the +definition from (7.1). On the other extreme, for the unitary implementation problem, where +Kx “ K for all x, Eq. (8.2) reads as +Ð +γ2 +` +I ´ T ˚ +x Ty | I ´ O˚ +xOy +˘ +x,yPD. +Theorem 8.6 can be reformulated as follows. +Theorem 8.9. Let Tx : Kx Ñ K be an isometric subspace conversion problem with unitary input +oracles Ox and finite set of labels D. Then, the topological closure of the feasible complexity +space of this problem coincides with the feasible objective space of the corresponding adversary +optimisation problem (8.2). +Proof. If Vx is a feasible solution, then by (6.5b): +K˚ +xKy ´ T ˚ +x Ty “ V ˚ +x ppI ´ O˚ +xOyq b IWqVy. +Multiplying by ξ˚ on the left and ξ1 on the right gives +@ +Kxξ, Kyξ1D +´ +@ +Txξ, Txξ1D +“ +@ +Vxξ, ppI ´ O˚ +xOyq b IWqVyξ1D +. +(8.3) +Hence, Vxξ is a feasible solution to the adversary optimisation problem of state conversion ξ ÞÑ +Txξ with oracles Ox as x ranges over D and ξ over Kx. The theorem follows from Theorem 8.6. +In other words, Eq. (8.2) is just a way to write down a linearly consistent feasible solution +to an adversary optimisation problem, where Vx acts like VO in Definition 8.2. The objective +~Vx~2 is then the largest (entry-wise) complexity on the oracle Ox as ξ ranges over unit vectors +in Kx. +37 + +8.3 +Composition, Revisited +Here we revisit the composition properties from Section 5.5 using the notions from Section 8.2. +Let Tx : Nx Ñ N be a subspace conversion problem as x ranges over D, and B be an +algorithm solving this problem on input oracles Ox : M Ñ M. For each x P D, let O1 +x : N Ñ N +agree with Tx on Nx. Let ξx ÞÑ τx be a state conversion problem with the input oracles O1 +x. +Assume that a sliced algorithm A solves this problem and has the following property: +@x P D @t: QtpA, O1 +xqξx P Nx. +(8.4) +We can estimate the complexity of the composed algorithm as the product of complexities +of its constituents. +Proposition 8.10. Under the above assumption, the composed algorithm A ˝ B defined in +Proposition 5.10 solves the state conversion problem ξx ÞÑ τx with input oracles Ox. Moreover, +LxpA ˝ Bq ď LxpAqLxpBq. +(8.5) +Proof. By (8.4) and the fact that B solves the subspace conversion problem, we have that O1 +x +and BpOxq satisfy the condition (5.3) of Proposition 5.7. Therefore, we can replace the input +oracle O1 +x of A by BpOxq. The first statement then follows from Proposition 5.10. +Concerning complexity, we have: +LpA ˝ B, Ox, ξxq “ +ÿ +t +L +´ +B, Ox, Qt +` +A, BpOxq +˘ +ξx +¯ +“ +ÿ +t +L +´ +B, Ox, Qt +` +A, O1 +x +˘ +ξx +¯ +ď LxpBq +ÿ +t +��Qt +` +A, O1 +x +˘��2 “ LxpBqLxpAq. +Here, we used (5.5) on the first step, Proposition 5.7 on the second step, and the definition of +LxpBq and Proposition 5.1 on the third step. +In the above proposition it is assumed that the algorithm A has a single input oracle (while +B can have multiple input oracles). Similarly, it is possible to get an analogue of (5.6). Again, +assume A has multiple input oracles Opiq : N piq Ñ N piq. For each i, let T piq +x : N piq +x +Ñ N piq be a +subspace conversion problem with x ranging over D, and Bpiq be an algorithm that solves the +above problem with input oracle Ox : M Ñ M. By Proposition 5.9, the algorithm B “ À +i Bpiq +solves state conversion Tx “ À +i T piq +x : Nx Ñ N, where Nx “ À +i N piq +x +and N “ À +i N piq. Let, +for each x P D and i, O1 +x +piq be a linear map on N piq that agrees with T piq +x +on N piq +x , and denote +O1 +x “ À +i O1 +x +piq. Then, using a similar estimate as in the proof of Proposition 8.10, but with (5.6) +instead of (5.5), we get: +LxpA ˝ Bq ď +ÿ +i +LxpAqrriss ¨ LxpBpiqq. +Above we assumed for simplicity that all the subspace conversion problems T piq +x +have the same +set of labels D. This is without loss of generality. If the i-th problem has the set of labels Dpiq, +it is possible to take D as the Cartesian product D “ ś +i Dpiq or some subset thereof. +38 + +9 +Bidirectionality +In this section, we consider aspects specific to bidirectional access to the input oracle. +In +particular, we show how one can obtain the main results from [15]. +As mentioned in the +introduction, bidirectional case is just a special case of the unidirectional case. +Proposition 9.1. For each state conversion problem with unitary input oracle pOxqxPD, the +feasible complexity space assuming bidirectional access to the oracle Ox coincides to the fea- +sible complexity space assuming unidirectional access to the oracle Ox ‘ O˚ +x. +Moreover, the +corresponding Monte Carlo complexities differ at most by a factor of 2. +Proof. Each algorithm A with bidirectional access to Ox can be simulated with unidirectional +access to Ox ‘ O˚ +x by using the parts Ox and O˚ +x of the oracle to process direct and reverse +queries of A. Both Monte Carlo and Las Vegas complexities do not change. +On the other hand, if A has unidirectional access to Ox ‘ O˚ +x, it can be simulated with +bidirectional access to Ox by first processing the Ox-part with the direct query, and then the +O˚ +x-part with the reverse query. Las Vegas complexity does not change, and the Monte Carlo +complexity grows by a factor of 2. +Let us define the (bidirectional) relative γ2-norm. We start with the single-objective version, +which is the version used in [15]. +Definition 9.2 (Bidirectional relative γ2-bound). Let K, and M be vector spaces, and D be a +set of labels. Let E “ tExyu and ∆ “ t∆xyu, where x, y P D be two families of linear operators: +Axy : K Ñ K and ∆xy : M Ñ M. +The relative γ2-norm +Ø +γ2pE|∆q “ +Ø +γ2pExy | ∆xyqx,yPD, +is defined as the optimal value of the following optimisation problem, where Ux and Vx are +linear operators, +minimise +maxxPD maxt∥Ux∥2, ∥Vx∥2u +(9.1a) +subject to +Exy “ U ˚ +x p∆xy b IWqVy +for all x, y P D; +(9.1b) +W is a vector space, +Ux, Vx : K Ñ M b W. +(9.1c) +The one-dimensional version is : +minimise +maxxPD maxt∥ux∥2, ∥vx∥2u +(9.2a) +subject to +exy “ +@ +ux, p∆xy b IWqvy +D +for all x, y P D; +(9.2b) +W is a vector space, +ux, vx P M b W. +(9.2c) +The relative γ2-norm can be also defined in terms of the unidirectional γ2-bound as +Ø +γ2pExy | ∆xyqx,yPD “ +Ð +γ2p rExy | r∆xyqx,yPDYD1. +(9.3) +Here D1 “ tx1 | x P Du is a disjoint copy of D, and rE and r∆ are defined as +rEx,y1 “ rE˚ +x1,y “ Ex,y, +rEx,y “ rEx1,y1 “ 0, +r∆x,y1 “ r∆˚ +x1,y “ ∆x,y, +r∆x,y “ r∆x1,y1 “ 0 +for all x, y P D. This instantly gives a dual for the the one-dimensional version of the bound, +which was already proven in [15]: +39 + +Theorem 9.3. The optimal value of (9.2) is equal to the optimal value of the following opti- +mization problem: +maximise +}Γ ˝ E} +(9.4a) +subject to +}Γ ˝ ∆} ď 1, +(9.4b) +where Γ ranges over D ˆ D matrices. +Proof. Use the above representation, Theorem 6.4, and the fact that +}A} “ λmax +ˆ +0 +A +A˚ +0 +˙ +. +Now let us move to the connection between unidirectional and bidirectional oracles. By +Proposition 9.1, unidirectional access to Ox is equivalent to bidirectional access to Ox ‘ O˚ +x. +The following proposition shows that we can substitute unidirectional γ2 bound with oracle +Ox ‘ O˚ +x with bidirectional γ2-norm with oracle Ox. +Proposition 9.4. Let Ox “ Op1q +x ‘¨ ¨ ¨‘Opsq +x +be unitary oracles as x ranges over D, and assume +that ey,x “ e˚ +x,y and ex,x “ 0 are complex numbers for all x, y P D. Consider the following two +optimization problems +Ø +γ2pex,y | I ´ O˚ +xOyqx,yPD +and +Ð +γ2 +` +ex,y | I ´ pO˚ +xOy ‘ OxO˚ +yq +˘ +x,yPD. +Then, for every collection pLxqxPD, with Lx P Rs, the following statements are equivalent: +(a) there exists a feasible solution ux, vx to the first optimization problem with Lx “ p~ux~2 ` +~vx~2q{2; +(b) there exists a feasible solution ux, vx to the first optimization problem with Lx “ ~ux~2 “ +~vx~2; +(c) there exists a feasible solution ˜vx to the second optimization problem with Lx “ ~˜vx~2. +Proof. First, let us prove paq ñ pcq. Assume we have a feasible solution ux, vx to (9.2) with +∆x,y “ I ´ O˚ +xOy. Let us denote rOx “ Ox b IW. In particular, we have +ex,y “ +A +ux, pI ´ rO˚ +x rOyqvy +E +ùñ ex,y “ xux, vyy ´ +A +rOxux, rOyvy +E +, +and +ey,x “ +A +uy, pI ´ rO˚ +y rOxqvx +E +ùñ ex,y “ xvx, uyy ´ +A +rOxvx, rOyuy +E +. +Consider the following two equalities +� +ux ` vx, +` +I ´ rO˚ +x rOy +˘ +puy ` vyq +� +“ +xux, uyy +` +xux, vyy +` +xvx, uyy +` +xvx, vyy +´ +A +rOxux, rOyuy +E +´ +A +rOxux, rOyvy +E +´ +A +rOxvx, rOyuy +E +´ +A +rOxvx, rOyvy +E +, +and +� +rOxux ´ rOxvx, +` +I ´ rOx rO˚ +y +˘ +p rOyuy ´ rOyvyq +� +“ +A +rOxux, rOyuy +E +´ +A +rOxux, rOyvy +E +´ +A +rOxvx, rOyuy +E +` +A +rOxvx, rOyvy +E +´ +xux, uyy +` +xux, vyy +` +xvx, uyy +´ +xvx, vyy. +40 + +Hence, +˜vx “ pux ` vxq ‘ p rOxux ´ rOxvxq +2 +is a feasible solution to (6.2) with ∆x,y “ I ´ pO˚ +xOy ‘ OxO˚ +yq. We have +~˜vx~2 “ +‌‌ux ` vx +‌‌2 ` +‌‌ rOxux ´ rOxvx +‌‌2 +4 +“ +‌‌ux ` vx +‌‌2 ` +‌‌ux ´ vx +‌‌2 +4 +“ +‌‌ux +‌‌2 ` +‌‌vx +‌‌2 +2 +, +where we used (4.8a), (4.8b), (4.8c) and (4.9), respectively. This proves that paq ñ pcq. +Now let us prove pcq ñ pbq. Assume ˜vx is a feasible solution to the second optimization +problem. Let ˜v1 +x and ˜v2 +x be the parts of ˜vx processed by I ´ O˚ +xOy and I ´ OxO˚ +y, respectively. +Set +ux “ ˜v1 +x ‘ rO˚ +x˜v2 +x +and +vx “ ˜v1 +x ‘ r´ rO˚ +x˜v2 +xs. +Then, +xux, ppI ´ O˚ +xOyq b pIW ‘ IWqqvyy “ +A +˜v1 +x, pI ´ rO˚ +x rOyq˜v1 +y +E +` +A +rO˚ +x˜v2 +x, p rO˚ +x rOy ´ Iq rO˚ +y ˜v2 +y +E +“ +A +˜v1 +x, pI ´ rO˚ +x rOyq˜v1 +y +E +` +A +˜v2 +x, pI ´ rOx rO˚ +yq˜v2 +y +E +“ ex,y. +Again, using (4.8b) and (4.8c), ~ux~ “ ~vx~ “ ~˜vx~. This proves pcq ñ pbq. The remaining +implication pbq ñ paq is obvious. +Therefore, we can make the following definition. +Definition 9.5. The multi-objective bidirectional relative γ2-optimisation problem +Ø +γ2 +` +ex,y | ∆x,y +˘ +x,yPD +is defined as +minimise +´~ux~2 ` ~vx~2 +2 +¯ +xPD +(9.5a) +subject to +exy “ +@ +ux, p∆xy b IWqvy +D +for all x, y P D; +(9.5b) +W is a vector space, +ux, vx P M b W. +(9.5c) +Alternatively, one may substitute (9.5a) with +minimise +´ +max +␣ +~ux~2, ~vx~2(¯ +xPD. +Definition 9.6 (Bidirectional Adversary Optimisation Problem). Assume ξx ÞÑ τx is a state +conversion problem with bidirectional input oracles Ox : M Ñ M, as x P D. Its adversary +optimisation problem is +Ø +γ2 +´ +xξx, ξyy ´ xτx, τyy | IM ´ O˚ +xOy +¯ +x,yPD. +(9.6) +An important corollary is as follows. +Corollary 9.7. Assuming bidirectional access, the adversary bound (7.1) can be replaced with +the corresponding bidirectional version (9.6), and the results of the corresponding Theorems 7.4, +7.5, 7.10, 7.11, and Corollary 7.6 still hold. +41 + +For instance, Corollary 7.6 after this transformation is the main technical result from [15] +with slightly better dependence on ε. And the adversary bound for Boolean function evalua- +tion (7.15) equals +Ø +γ2 +´ +1fpxq‰fpyq +ˇˇ +n +à +j“1 +1xj‰yj +¯ +, +(9.7) +which is equivalent to the known bound from [52]. The corresponding dual (9.4) is the lower +bound from [37]. +10 +Unitary Permutation Inversion +The goal of this section is to prove a separation between unidirectional and bidirectional access +to an oracle on a natural problem. We will achieve this using the following problem. +Definition 10.1 (Unitary Permutation Inversion). The set of labels is the set of permutations +on n elements D “ Sn. For each π P Sn, let Oπ : Cn Ñ Cn be the input oracle defined by +Oπ|i⟩ “ |πpiq⟩ for all i P rns The task is to find π´1p1q. +First note that this problem is different from the usual permutation inversion problem. In +the latter, the permutation π is encoded using the standard input oracle |i⟩|b⟩ ÞÑ |i⟩|b ‘ πpiq⟩. +The latter is a well-known problem, first defined in [24]. It is similar to Grover’s search, but +different enough to complicate direct reductions from the lower bound for unstructured search. +Ambainis [1] gave a tight lower bound of Ωp?nq. +Nayak [49] gave a direct reduction from +unstructured search. See also a recent paper by Rosmanis [56]. +Since the unidirectional and bidirectional access are equivalent for standard oracle, we resort +to the unitary oracle. The reason of requiring π to be a permutation is solely to ensure that Oπ +is a unitary. +The problem can be trivially solved in one query with bidirectional access: apply O˚ +π to +|1⟩ and read out the result. Since unitary inversion using the standard oracle requires Ωp?nq +queries, this means that the unitary permutation oracle |i⟩ ÞÑ |πpiq⟩ cannot be simulated by +the standard oracle. +Intuitively, it seems the problem should be hard for unidirectional input oracles. We show +that this is indeed the case. +Theorem 10.2. Any quantum query algorithm solving the unitary inversion problem (with +bounded error and non-coherently) with unidirectional access to the input oracles has to make +Ωp?nq queries. +Note, however, that there is no matching upper bound. Grover’s search cannot be directly +applied here because of the unidirectional access. The remaining part of this section is devoted +to the proof of this theorem. The proof relies on Theorem 6.3, and we have to find the adversary +matrix Γ from (6.3). +Interestingly, the analysis is a variant of the usual positive-weighted adversary, but it is +different from the one used by Ambainis in the lower bound proof of the usual permutation +inversion problem [1]. We need the following technical result, which was used [58] to reduce +the combinatorial formulation of the positive-weighted adversary like in [1] to the spectral +formulation as in [9, 37]. We give a slightly modified version. +Lemma 10.3. Let A be a matrix with entries 0, ˘1. Then, +}A} ď +max +i,j : Arri,jss‰0 +a +RiCj, +where Ri and Cj is the number of non-zero elements in the i-th row and j-column, respectively. +42 + +Proof. Taking the absolute value of each entry can only increase the norm, hence, we can assume +the matrix A only has 0,1 entries. Then, this is a special case of Lemma 4.2 of [58]. +Assume |0⟩ ÞÑ |τπ⟩ is a state-generating problem such that measuring τπ gives π´1p1q with +probability at least 2{3. We use this property to ensure that +Rexτπ, τσy ď 2 +? +2 +3 +for π, σ P Sn such that π´1p1q ‰ σ´1p1q. +(10.1) +Define the corresponding output object, which is an Sn ˆ Sn-matrix E with +Errπ, σss “ 1 ´ xτπ, τσy. +Let us define the adversary matrix Γ. Denote by Cn the subset of Sn formed by permutations +having a single cycle of length n. We will only consider permutations in Cn. +We say that π, σ P Cn are in relation, denoted π ú σ, if π and σ have cyclic structures of +the following form: +π: 1 ÞÑ ¨ ¨ ¨ ÞÑ pk ÞÑ pk`1 ÞÑ ¨ ¨ ¨ pℓ ÞÑ pℓ`1 ÞÑ ¨ ¨ ¨ pn ÞÑ 1, +σ: 1 ÞÑ ¨ ¨ ¨ ÞÑ pk ÞÑ pℓ`1 ÞÑ ¨ ¨ ¨ pn ÞÑ pk`1 ÞÑ ¨ ¨ ¨ pℓ ÞÑ 1. +(10.2) +for some 1 ď k ă ℓ ă n. In other words, the interval pk`1 ÞÑ ¨ ¨ ¨ ÞÑ pℓ is taken out and put at +the end of the cycle. Alternatively, one can say that the suffix pk`1 ÞÑ ¨ ¨ ¨ ÞÑ pn is cyclically +shifted. This is a symmetric relation, but neither reflexive, nor transitive. +As usual for the positive-weighted adversary, define an CnˆCn matrix Γ by Γrrπ, σss “ 1πúσ. +Lemma 10.4. We have the following properties of the matrix Γ: +• Γ is a Hermitian matrix; +• Γrrπ, σss “ 0 if π´1p1q “ σ´1p1q; +• λmaxpΓq “ Ωpn2q with the principal eigenvector given by the all-1 vector; +• λmaxp´Γq ď n ´ 2. +Proof. The first two properties follow from the definition of the relation: If π ú σ, then +σ ú π. Also, in this case, π´1p1q ‰ σ´1p1q. The third property follows from the fact that +each row has exactly pn ´ 1qpn ´ 2q{2 ones. +Now, let us prove the fourth property. It is equivalent to pn ´ 2qI ` Γ ě 0. Let us prove +the latter. Fix 1 ď k ď n ´ 2. Say that π „k σ if π “ σ or π and σ are in relation like +in (10.2) with this fixed value of k. Note that „k is an equivalence relation. Define the matrix +Γk by Γkrrπ, σss “ 1π„kσ. It is a block-diagonal matrix with all-1 blocks on the diagonal. Hence, +Γk ě 0, which gives +n´2 +ÿ +k“1 +Γk “ pn ´ 2qI ` Γ ě 0. +Let u be the normalised all-1 vector. Then, +λmaxpΓ ˝ Eq ě u˚pΓ ˝ Equ ě +ˆ +1 ´ 2 +? +2 +3 +˙ +u˚Γu “ Ωpn2q, +(10.3) +where we used the second point of Lemma 10.4 and (10.1) on the second step, and the third +point of Lemma 10.4 on the third. +43 + +It remains to estimate Γ ˝ ∆, where +∆π,σ “ I ´ O˚ +πOσ “ I ´ Oπ´1σ. +By the definition of Γ, we can restrict our attention to the pairs π, σ, which are in relation (10.2). +In notation of (10.2), we have that π´1σ is a single cycle of length 3 +π´1σ: pk ÞÑ pℓ ÞÑ pn ÞÑ pk +and identity elsewhere. Hence, +∆π,σ “ +¨ +˝ +1 +0 +´1 +´1 +1 +0 +0 +´1 +1 +˛ +‚ +(10.4) +where the rows and columns are labelled by pk, pℓ, pn in this order, and the matrix has zeroes +everywhere else. +The matrix Γ ˝ ∆ is labelled by the elements pπ, iq P Cn ˆ rns. +The block (10.4) when +embedded in the latter has +rows pπ, pkq, pπ, pℓq, pπ, pnq +and +columns pσ, pkq, pσ, pℓq, pσ, pnq. +We would like to apply Lemma 10.3 to Γ ˝ ∆. For instance, we see that there are at most n +choices of ρ P Cn such that ∆π,ρ has non-zero elements in row pπ, pkq, since pk has to be one of +the two elements used to define the relation π ú ρ for this to happen. Similarly, in notation +of Lemma 10.3, we get the following estimates: +Rπ,pk, Rπ,pℓ, Cσ,pk, Cσ,pn ď 2n. +(10.5a) +For one row and one column we get a worse estimate, where we count the total number of ρ in +relation with π (or σ, respectively): +Rπ,pn, Cσ,pℓ ď 2n2. +(10.5b) +Therefore, we should treat the element on the intersection of the latter row and the latter +column separately. Rewrite (10.4): +∆π,σ “ +¨ +˝ +1 +0 +´1 +´1 +1 +0 +0 +´1 +1 +˛ +‚“ +¨ +˝ +1 +0 +´1 +´1 +1 +0 +0 +0 +1 +˛ +‚` +¨ +˝ +0 +0 +0 +0 +0 +0 +0 +´1 +0 +˛ +‚, +Let us denote the first and the second matrices in the last sum by ∆1 +π,σ and ∆2 +π,σ, respectively, +and the corresponding families by ∆1 and ∆2. +Claim 10.5. }Γ ˝ ∆1} “ Opn3{2q. +Proof. This follows from Lemma 10.3 using the estimates in (10.5). +Claim 10.6. λmaxpΓ ˝ ∆2q “ Opnq. +Proof. The matrix Γ ˝∆2 is just the matrix ´Γ where the row and the column label π becomes +` +π, π´1p1q +˘ +and the matrix is extended by zeroes elsewhere. Hence, the claim follows from the +fourth point of Lemma 10.4. +Combining Claims 10.5 and 10.6, we get that +λmaxpΓ ˝ ∆q “ Opn3{2q. +Together with (10.3), this gives the required lower bound. +44 + +11 +Discussion and Future Work +In this paper, we defined a natural notion of Las Vegas complexity, and demonstrated its +versatility for various composition results. We proved that a Las Vegas algorithm can be turned +into an approximate Monte Carlo algorithm with a slight increase in complexity. +We have +shown that Las Vegas complexity is equal to the adversary bound for exact state and subspace +conversion. +The latter is exciting as the same object is shown to have two different facets. For some +problems, intuition gathered from quantum algorithms might be helpful in coming up with +good Las Vegas algorithms. For other problems, it might be easier to forget about limitations of +quantum algorithms and work directly with optimisation problems. Our algorithm of Section 7.4 +can be seen as a way to guess arbitrarily large states to process by the input oracle. +It is +interesting to understand consequences of such a subroutine. +Due to its exactness, Las Vegas complexity results in “cleaner” algorithms without necessity +to worry about error reduction. Similar results have been already obtained for function evalu- +ation using compositional properties of the adversary bound. But having “clean” subroutines +for various state-generating and state-converting problems might be helpful as well, especially, +given that they not always have built-in tools for error reduction. +This paper should be seen as a prequel to Ref. [15] since it gives a more general and simple +exposition of the first half of Ref. [15], but mostly ignores the second half, which deals with +applications to function and relation evaluation. Complete reconciliation of the results from [15] +with the current paper is left as important future work. Let us just mention two results that +can be easily obtained in this way. +• Purifiers of [15] imply that, assuming bidirectional access, a Monte Carlo algorithm for +approximate and non-coherent function evaluation can be turned into an exact coherent +Las Vegas algorithm for the same function with constant increase in complexity. +As +mentioned in the introduction, this is in contrast to randomised Las Vegas complexity. +• The bound (9.7) is equal to Las Vegas complexity of bidirectional function evaluation +also for non-Boolean functions. However, we only get this result up to a constant factor. +Understanding the exact relation between the two is still an open problem. +We list just a few other open problems. What is Las Vegas complexity of various important +subroutines, for instance, amplitude amplification? Can the adversary bound for contraction +oracles be applied for some problems like faulty oracles? Is there an nice formulation of the +adversary bound for (approximate and non-coherent) function evaluation with unidirectional +input oracles? In particular, what is the true complexity of the unitary permutation inversion +problem? The purifiers mentioned above seem to crucially depend on bidirectionality. +Finally, an interesting research direction is to obtain analogues of some of the results in this +paper for time complexity. +Acknowledgements +We thank anonymous reviewers for their comments on an earlier draft of the paper. +A.B. is supported by the ERDF project number 1.1.1.5/18/A/020 “Quantum algorithms: +from complexity theory to experiment”. +References +[1] A. Ambainis. Quantum lower bounds by quantum arguments. Journal of Computer and System +Sciences, 64(4):750–767, 2002. Earlier: STOC’00, arXiv:quant-ph/0002066. +45 + +[2] A. Ambainis. A new quantum lower bound method, with an application to a strong direct product +theorem for quantum search. Theory of Computing, 6(1):1–25, 2010. arXiv:quant-ph/0508200. +[3] A. Ambainis. Quantum search with variable times. Theory of Computing Systems, 47(3):786–807, +2010. Earlier: STACS’08, arXiv:quant-ph/0609188. +[4] A. Ambainis. +Variable time amplitude amplification and quantum algorithms for linear alge- +bra problems. +In Proc. of 29th STACS, volume 14 of LIPIcs, pages 636–647. Dagstuhl, 2012. +arXiv:1010.4458. +[5] A. Ambainis, K. Balodis, A. Belovs, T. Lee, M. Santha, and J. Smotrovs. Separations in query +complexity based on pointer functions. Journal of the ACM, 64(5):32, 2017. Earlier: STOC’16, +arXiv:1506.04719. +[6] A. Ambainis, A. M. Childs, B. W. Reichardt, R. ˇSpalek, and S. Zhang. Any AND-OR formula of +size N can be evaluated in time N 1{2`op1q on a quantum computer. SIAM Journal on Computing, +39(6):2513–2530, 2010. +[7] A. Ambainis, L. Magnin, M. R¨otteler, and J. Roland. Symmetry-assisted adversaries for quantum +state generation. In Proc. of 26th IEEE CCC, pages 167–177, 2011. arXiv:1012.2112. +[8] H. Barnum. Semidefinite programming characterization and spectral adversary method for quantum +complexity with noncommuting unitary queries. In Algorithms and Complexity for Continuous Prob- +lems, volume 6391 of Dagstuhl Seminar Proceedings, pages 1–25, 2007. arXiv:quant-ph/0703141. +[9] H. Barnum, M. Saks, and M. Szegedy. Quantum decision trees and semi-definite programming. In +Proc. of 18th IEEE CCC, pages 179–193, 2003. +[10] R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. +Journal of the ACM, 48(4):778–797, 2001. Earlier: FOCS’98, arXiv:quant-ph/9802049. +[11] A. Belovs. Learning-graph-based quantum algorithm for k-distinctness. In Proc. of 53rd IEEE +FOCS, pages 207–216, 2012. arXiv:1205.1534. +[12] A. Belovs. Span programs for functions with constant-sized 1-certificates. In Proc. of 44th ACM +STOC, pages 77–84, 2012. arXiv:1105.4024. +[13] A. Belovs. Quantum walks and electric networks. arXiv:1302.3143, 2013. +[14] A. Belovs. Quantum algorithms for learning symmetric juntas via the adversary bound. Computa- +tional Complexity, 24(2):255–293, 2015. Earlier: CCC’14, arXiv:1311.6777. +[15] A. Belovs. Variations on quantum adversary. arXiv:1504.06943, 2015. +[16] A. Belovs and E. Blais. Quantum algorithm for monotonicity testing on the hypercube. Theory of +Computing, 11(16):403–412, 2015. arXiv:1503.02868. +[17] A. Belovs, A. M. Childs, S. Jeffery, R. Kothari, and F. Magniez. Time-efficient quantum walks for +3-distinctness. In Proc. of 40th ICALP, Part I, volume 7965 of LNCS, pages 105–122. Springer, +2013. +[18] A. Belovs and B. W. Reichardt. Span programs and quantum algorithms for st-connectivity and +claw detection. +In Proc. of 20th ESA, volume 7501 of LNCS, pages 193–204. Springer, 2012. +arXiv:1203.2603. +[19] A. Belovs and A. Rosmanis. On the power of non-adaptive learning graphs. Computational Com- +plexity, 23(2):323–354, 2014. Earlier: CCC’13, arXiv:1210.3279. +[20] A. Belovs and A. Rosmanis. Adversary lower bounds for the collision and the set equality problems. +Quantum Information & Computation, 18(3-4):200–224, 2018. arXiv:1310.5185. +[21] A. Belovs and A. Rosmanis. Tight quantum lower bound for approximate counting with quantum +states. arXiv:2002.06879, 2020. +[22] A. Belovs and R. ˇSpalek. Adversary lower bound for the k-sum problem. In Proc. of 4th ACM +ITCS, pages 323–328, 2013. arXiv:1206.6528. +46 + +[23] S. Ben-David and R. Kothari. +Quantum distinguishing complexity, zero-error algorithms, and +statistical zero knowledge. +In Proc. of 14th TQC, volume 135 of LIPIcs, pages 2:1–2:23, 2019. +arXiv:1902.03660. +[24] C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum +computing. SIAM Journal on Computing, 26(5):1510–1523, 1997. arXiv:quant-ph/9701001. +[25] A. Berzina, A. Dubrovsky, R. Freivalds, L. Lace, and O. Scegulnaja. Quantum query complexity for +some graph problems. In Proc. of 30th SOFSEM, volume 2932 of LNCS, pages 140–150. Springer, +2004. +[26] R. Bhatia. Positive definite matrices. Princeton University Press, 2009. +[27] M. Brandeho and J. Roland. A Universal Adiabatic Quantum Query Algorithm. In Proc. of 10th +TQC, volume 44 of LIPIcs, pages 163–179, 2015. arXiv:1409.3558. +[28] H. Buhrman, R. Cleve, R. de Wolf, and C. Zalka. Bounds for small-error and zero-error quantum +algorithms. In Proc. of 40th IEEE FOCS, pages 358–368, 1999. arXiv:cs/9904019. +[29] H. Buhrman and R. De Wolf. Quantum zero-error algorithms cannot be composed. Information +processing letters, 87(2):79–84, 2003. arXiv:quant-ph/0211029. +[30] H. Buhrman and R. ˇSpalek. Quantum verification of matrix products. In Proc. of 17th ACM-SIAM +SODA, pages 880–889, 2006. arXiv:quant-ph/0409035. +[31] H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey. Theo- +retical Computer Science, 288:21–43, 2002. +[32] R. Cleve. An introduction to quantum complexity theory. arXiv:quant-ph/9906111, 1999. +[33] S. D¨orn and T. Thierauf. The quantum query complexity of algebraic properties. In Proc. of 16th +FCT, volume 4639 of LNCS, pages 250–260. Springer, 2007. arXiv:0705.1446. +[34] C. D¨urr, M. Heiligman, P. Høyer, and M. Mhalla. +Quantum query complexity of some graph +problems. +In Proc. of 31st ICALP, volume 3142 of LNCS, pages 481–493. Springer, 2004. +arXiv:quant-ph/0401091. +[35] A. C. Elitzur and L. Vaidman. Quantum mechanical interaction-free measurements. Foundations +of Physics, 23(7):987–997, 1993. arXiv:hep-th/9305002. +[36] E. Farhi, J. Goldstone, and S. Gutmann. A quantum algorithm for the Hamiltonian NAND tree. +Theory of Computing, 4:169–190, 2008. +[37] P. Høyer, T. Lee, and R. ˇSpalek. Negative weights make adversaries stronger. In Proc. of 39th ACM +STOC, pages 526–535, 2007. arXiv:quant-ph/0611054. +[38] S. Jeffery. Quantum subroutine composition. arXiv:2209.14146, 2022. +[39] S. Kimmel, C. Y.-Y. Lin, and H.-H. Lin. Oracles with Costs. In Proc. of 10th TQC, volume 44 of +LIPIcs, pages 1–26, 2015. arXiv:1502.02174. +[40] A. Kondacs and J. Watrous. On the power of quantum finite state automata. In Proc. of 38th IEEE +FOCS, pages 66–75, 1997. arXiv:cs/9904019. +[41] R. Kothari. Semidefinite programming characterization of bounded-error quantum query complexity. +Manuscript. +[42] F. Le Gall, H. Nishimura, and S. Tani. +Quantum algorithms for finding constant-sized sub- +hypergraphs. In Proc. of 20th COCOON, volume 8591 of LNCS, pages 429–440. Springer, 2014. +arXiv:1310.4127. +[43] T. Lee, F. Magniez, and M. Santha. Improved quantum query algorithms for triangle finding and +associativity testing. In Proc. of 24th ACM-SIAM SODA, pages 1486–1502, 2013. arXiv:1210.1014. +[44] T. Lee, R. Mittal, B. W. Reichardt, R. ˇSpalek, and M. Szegedy. Quantum query complexity of state +conversion. In Proc. of 52nd IEEE FOCS, pages 344–353, 2011. arXiv:1011.3020. +47 + +[45] T. Lee, A. Shraibman, and R. ˇSpalek. A direct product theorem for discrepancy. In Proc. of 23rd +IEEE CCC, pages 71–80, 2008. +[46] C. Y.-Y. Lin and H.-H. Lin. +Upper Bounds on Quantum Query Complexity Inspired by the +Elitzur-Vaidman Bomb Tester. In Proc. of 30th CCC, volume 33 of LIPIcs, pages 537–566, 2015. +arXiv:1410.0932. +[47] N. Linial, S. Mendelson, G. Schechtman, and A. Shraibman. Complexity measures of sign matrices. +Combinatorica, 27(4):439–463, 2007. +[48] N. Linial and A. Shraibman. Lower bounds in communication complexity based on factorization +norms. Random Structures & Algorithms, 34:368–394, 2009. Earlier: STOC’07. +[49] A. Nayak. Inverting a permutation is as hard as unordered search. Theory of Computing, 7(1):19–25, +2011. arXiv:arXiv:1007.2899. +[50] M. A. Nielsen and I. L. Chuang. Quantum computation and quantum information: 10th anniversary +edition. Cambridge University Press, 2010. +[51] O. Regev and L. Schiff. Impossibility of a quantum speed-up with a faulty oracle. In Proc. of 35th +ICALP, volume 5125 of LNCS, pages 773–781, 2008. arXiv:1202.1027. +[52] B. W. Reichardt. Span programs and quantum query complexity: The general adversary bound +is nearly tight for every Boolean function. In Proc. of 50th IEEE FOCS, pages 544–551, 2009. +arXiv:0904.2759. +[53] B. W. Reichardt. Reflections for quantum query algorithms. In Proc. of 22nd ACM-SIAM SODA, +pages 560–569, 2011. arXiv:1005.1601. +[54] B. W. Reichardt. Span-program-based quantum algorithm for evaluating unbalanced formulas. In +Proc. of 6th TQC, volume 6745 of LNCS, pages 73–103. Springer, 2014. arXiv:0907.1622. +[55] B. W. Reichardt and R. ˇSpalek. Span-program-based quantum algorithm for evaluating formulas. +Theory of Computing, 8:291–319, 2012. Earlier: STOC’08, arXiv:0710.2630. +[56] A. Rosmanis. +Tight bounds for inverting permutations via compressed oracle arguments. +arXiv:2103.08975, 2021. +[57] R. ˇSpalek. The multiplicative quantum adversary. In Proc. of 23rd IEEE CCC, pages 237–248, +2008. arXiv:quant-ph/0703237. +[58] R. ˇSpalek and M. Szegedy. All quantum adversary methods are equivalent. Theory of Computing, +2:1–18, 2006. Earlier: ICALP’05, arXiv:quant-ph/0409116. +[59] N. Tomczak-Jaegermann. Banach-Mazur distances and finite-dimensional operator ideals, volume 38 +of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Tech- +nical, 1989. +[60] D. Yolcu. +The adversary bound revisited: From optimal query algorithms to optimal control. +arXiv:2211.16293, 2022. +[61] B. Zhan, S. Kimmel, and A. Hassidim. Super-polynomial quantum speed-ups for Boolean evaluation +trees with hidden structure. In Proc. of 3rd ACM ITCS, pages 249–265, 2012. arXiv:1101.0796. +A +Duality +We use semi-definite duality. The dual is constructed by explicitly writing down the Lagrangian +and transforming it. Thus, weak duality (the maximisation problem bounds the minimisation +problem from below) is apparent. To prove strong duality (their optimal values are equal), we +rely on Slater’s condition. The latter says that strong duality holds if one of the optimisation +problems is convex and strictly feasible, i.e. +there exists a feasible solution making all the +inequalities in the problem strict. +48 + +It turns out that the calculations are concise using multidimensional tensors with con- +tractions given by the inner product formula between Hermitian matrices: xA, By “ tr A˚B. +However, given that matrices are tensors themselves, this notation might be confusing, so we +opted to use the following one, that we find more intuitive. +We assume the matrices are square and are labelled by elements of direct products of some +sets. If A is a matrix labelled by X ˆ Y , and B is a matrix labelled by X ˆ Z, then A ˝ B is a +matrix labelled by X ˆ Y ˆ Z given by +A ˝ Brrpx, y, zq, px1, y1, z1qss “ Arrpx, yq, px1, y1qss Brrpx, zq, px1, z1qss. +This includes the usual Hadamard product (when |Y | “ |Z| “ 1), the tensor product (when +|X| “ 1) and the version of the Hadamard product used in (6.3b) (when |Y | “ 1). +For the matrix A as above, let ř +Y A be the X ˆ X matrix given by +`ř +Y A +˘ +rrx, x1ss “ +ÿ +y,y1PY Arrpx, yq, px1, y1qss. +ř without the subindex stands for the total sum of all entries. In particular, we have xA, A1y “ +řpA ˝ A1q and the partial trace is trY pAq “ ř +Y pA ˝ IY q, where A is complex conjugate and IY +is the Y ˆ Y identity matrix. +Proof of Theorem 6.4. We have three sets of labels: D, and the bases of M and W, for which +we use letters M and W. By (6.2b): +exy “ tr +“ +v˚ +xp∆xybIW qvy +‰ +“ tr +“ +vyv˚ +xp∆xy bIWq +‰ +“ tr +“ +pvxv˚ +yq˚p∆xy bIWq +‰ +“ ř` +vxv˚y ˝∆xy ˝IW +˘ +. +Let us merge all these conditions into one. Let E be the D ˆD matrix given by pexyq, and ∆ be +the pD ˆ Mq ˆ pD ˆ Mq matrix with the blocks ∆x,y. Both these matrices are Hermitian. Let +also v be the vector in CDˆMˆW obtained by joining all vx. Then, all the constraints in (6.2b) +can be concisely written as +E “ ř +M,Wpvv˚ ˝ ∆ ˝ IWq “ ř +M +`ř +Wpvv˚ ˝ IW q ˝ ∆ +˘ +“ ř +M +` +X ˝ ∆ +˘ +, +where X is a positive semi-definite pD ˆ Mq ˆ pD ˆ Mq-matrix given by X “ trW pvv˚q. +Conversely, any positive semi-definite matrix can be written in this way for a large enough W. +Also, the matrix ř +MpX ˝ ID,Mq is the diagonal matrix with }vx}2 on the diagonal. Therefore, +we get the following equivalent formulation of the optimisation problem (6.2): +minimise +t +(A.1a) +subject to +tID ě ř +MpX ˝ ID,Mq +(A.1b) +E “ ř +MpX ˝ ∆q +(A.1c) +X ě 0, +t P R. +(A.1d) +We introduce two Lagrangian multipliers Y ě 0 and Λ which are DˆD Hermitian matrices, +resulting in the following Lagrangian: +t ` ř +D +” +Y ˝ +`ř +MpX ˝ ID,Mq ´ tID +˘ı +` ř +D +” +Λ ˝ +` +E ´ ř +MpX ˝ ∆q +˘ı +After rearrangement: +ř +DpΛ ˝ Eq ` t +“ +1 ´ tr Y +‰ +` ř +D,M +“ +X ˝ pY ˝ ID,M ´ Λ ˝ ∆q +‰ +49 + +This gives the following dual: +maximise +ř +DpΛ ˝ Eq +(A.2a) +subject to +tr Y “ 1 +(A.2b) +Λ ˝ ∆ ď Y ˝ ID,M +(A.2c) +Y ě 0, +Λ Hermitian. +(A.2d) +Note that this optimisation problem is strictly feasible as it suffices to take Λ “ 0 and Y a +multiple of the identity matrix satisfying tr Y “ 1. Therefore, by Slater’s condition, the optimal +values of (A.1) and (A.2) are equal. +By studying (A.2c), we see that we can assume that Y is rank-1 (by extending the diagonal +matrix Y ˝ID,M), and we can write Λ as Γ˝Y for some Hermitian DˆD-matrix Γ. Then (A.2c) +becomes +Y ˝ Γ ˝ ∆ ď Y ˝ ID,M, +(A.3) +and the objective (A.2a) becomes +ř +DpY ˝ Γ ˝ Eq. +(A.4) +This is clearly continuous in Y for fixed Γ and E, thus, we can additionally assume that Y +has non-zero diagonal. Then, the Hadamard inverse of Y is defined and positive semi-definite, +hence, Eq. (A.3) is equivalent to +Γ ˝ ∆ ď ID,M. +(A.5) +Altogether, the objective (A.4) with conditions (A.5), (A.2b), and Y is positive semi-definite +rank-1 gives us the dual +maximise +λmaxpΓ ˝ Eq +subject to +λmaxpΓ ˝ ∆q ď 1 +as required. +Proof of Claim 6.7. The set of feasible solutions of the optimisation problems (6.1) and (6.5) is +the same so we can use the same characterisation (A.1) as in the proof of Theorem 6.4. +Let W denote the feasible objective space of the optimisation problem, and BR denote +the set of vectors in RD b Rs with the sum of entries bounded by R. The objective profile +w “ p~vx~qxPD can be obtained by summing the diagonal entries of the corresponding matrix +X. Thus, W X BR is the image under a continuous map of the set of feasible solutions X +to (A.1) with tr X ď R. The latter set is easily seen to be compact, hence, W X BR is closed. +As R is arbitrary, W is closed as well. +50 + diff --git a/QdA0T4oBgHgl3EQfDf80/content/tmp_files/load_file.txt b/QdA0T4oBgHgl3EQfDf80/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd5469bd457f3a2f51bcab17f70fb8b87fbd5d18 --- /dev/null +++ b/QdA0T4oBgHgl3EQfDf80/content/tmp_files/load_file.txt @@ -0,0 +1,2437 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf,len=2436 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='02003v1 [quant-ph] 5 Jan 2023 One-Way Ticket to Las Vegas and the Quantum Adversary Aleksandrs Belovs∗ Duyal Yolcu† Abstract We propose a new definition of quantum Las Vegas query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We show that it is exactly equal to the quantum adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is achieved by a new and very simple way of transforming a feasible solution to the adversary optimisation problem into a quantum query algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This allows us to generalise the bound to include unidirectional access, multiple input oracles, and input oracles that are not unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As an application, we demonstrate a separation between unidirectional and bidirectional access to an input oracle for a rather natural unitary permutation inversion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 1 Introduction This paper combines two topics: Las Vegas query complexity and the quantum adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 Las Vegas Complexity There are two main types of randomised query algorithms, with different complexity measures: A Monte Carlo query algorithm, also known as bounded-error, is allowed to output an incorrect answer with some small probability ε, usually 1{3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm is allowed to make a certain number of queries, which cannot be exceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This number is the query complexity of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A Las Vegas query algorithm, also known as zero-error, always has to give the correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the other hand, it has no strict limit on the number of queries it can make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Sometimes it can make few queries, sometimes a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Its query complexity is defined as the expected number of queries it makes on a certain input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Las Vegas algorithms have a number of nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, complexity is independent of the choice of the error parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, one can talk about the exact value of complexity for a particular problem on a particular input, which can even not be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Second, Las Vegas algorithms can be nicely composed as there is no need for error reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For Monte Carlo algorithms, one usually gets extra logarithmic factors due to the necessity to reduce the error of the inner subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' One can terminate a Las Vegas algorithm after a certain number of queries, turning it into a Monte Carlo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Markov’s inequality, a Las Vegas algorithm with complexity L can be turned into a Monte Carlo algorithm with error parameter ε and complexity OpL{εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the other hand, there exist functions whose Las Vegas complexity is much larger than their Monte Carlo complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [5] features a quadratic separation for a total Boolean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For partial functions, the separation can be even larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ∗Faculty of Computing, University of Latvia †https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='com/qudent 1 Let us now turn to quantum query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the overwhelming majority of cases, the complexity under consideration is Monte Carlo: the number of queries is fixed, and the algorithm can output an incorrect output with small probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Zero-error quantum algorithms have also been defined and studied [10, 28, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A zero- error quantum algorithm is not allowed to give an incorrect output, but it can output ’?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' with probability at most 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The answer ’?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' means that the algorithm has not figured out what the answer is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This model is indeed a quantum counterpart of one way of defining randomised Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, it lacks the nice features of the randomised Las Vegas complexity outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The definition depends on the value with which ’?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' can be outputted, and it also does not compose nicely [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Another related notion is variable-time model introduced by Ambainis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In this case, a quantum subroutine can run for an unpredicted number of steps, and the average running time is the corresponding quadratic mean ař t ptt2, where pt is the probability the subroutine runs for t steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis showed how to perform search [3] and amplitude amplification [4] on such subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A very recent paper by Jeffery [38] also considers quantum walks with such subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Up to our knowledge, this notion has not been studied as a complexity measure per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Also, these results are mostly concerned with time complexity, while we study query complexity in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 Adversary Bound The quantum adversary bound was first developed as a powerful tool for proving quantum query lower bounds, but it has been later extended to include upper bounds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The adversary bound originates from the hybrid method by Bennett, Bernstein, Brassard, and Vazirani [24], which was further refined by Ambainis in the first version of the adver- sary bound [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Due to its attractive combinatorial formulation, it fostered a large number of applications [34, 25, 30, 33] to name just a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The bound was strengthened by Høyer, Lee, and ˇSpalek [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Using the semidefinite for- mulation of the adversary bound by Barnum, Saks, and Szegedy [9], they showed that the same expression still yields a lower bound if one replaces non-negative entries by arbitrary real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This negative-weighted version of the bound is strictly more powerful than the positive-weighted one, but it is also harder to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In a series of papers [55, 52, 53], Reichardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' surprisingly proved that the negative-weighted version of the bound is tight: The dual formulation of the bound (which is equal to the primal formulation due to strong duality) can be transformed into a quantum query algorithm with the same complexity up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The negative-weighted adversary bound has been used to prove lower bounds [22, 19, 20], but more frequently to prove upper bounds, in particular using the learning graph approach [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, the adversary bound (sometimes in the equivalent form of span programs) was used to construct quantum algorithms for formula evaluation [55, 54, 61], finding subgraphs [43, 18, 42], k-distinctness problem [11], and in learning and property testing [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The next steps came when the adversary bound was extended to state generation by Ambai- nis, Magnin, R¨otteler, and Roland [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' and state conversion by Lee, Mittal, Reichardt, ˇSpalek, and Szegedy [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs [15] extended the bound for various types of input oracles, including the case when the input oracle can be an arbitrary unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' These generalisations came with a twist, as the bound became semi-tight: a lower bound for the exact version of the problem and an upper bound for the approximate version of the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us briefly touch on techniques used in the above papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis [1] and Høyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [37] only proved lower bounds, which they did considering a so-called progress function of 2 the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The upper bounds by Reichardt [55, 52, 53] used a rather complicated quantum walk, which was inspired by previous work on evaluating NAND-trees [36, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The (discrete) quantum walk comprises two reflections, one simple and input-dependent, and the other one complicated and input-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The analysis of the algorithm required technically involved spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The paper by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [44] featured many important technical innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, the problem was generalised to state conversion, where the task of the algorithm is to transform one vector ξx into another τx on every input x in the domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This turned out to be a very fruitful approach, as the algorithm can be broken into smaller steps, which can be then analysed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Second, a very simple proof of the lower bound was presented, which worked by a direct conversion of the algorithm into the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This essentially established the adversary bound as a semi-definite relaxation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Third, the bound was formu- lated as an instance of filtered γ2-norm, which is a generalisation of γ2-norm used previously in other context, see Section 6 for more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, the proof of the upper bound was signif- icantly simplified by introducing easy and powerful Effective Spectral Gap Lemma to analyse the resulting quantum walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The lemma can be also used independently [13, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [44] assumed the standard input oracle that encodes a string x P rqsn for some alphabet rqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The problem of choice by Belovs [15] was still state conversion ξx ÞÑ τx but this time with general input oracles Ox, which is just an arbitrary unitary transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This removed the oracle-specific details from the proof, thus making it more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (Barnum [8] already considered the problem of function evaluation with unitary input oracles, but that paper went unnoticed at the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=') The bound was formulated as an instance of relative γ2-norm, which further generalises filtered γ2-norm, and which, in our opinion, is more natural than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs used the adversary bound for this problem to construct various adversaries for function and relation evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, let us note that all the versions considered above are that of the so-called additive adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We leave out of consideration the multiplicative version by ˇSpalek [57] based on the earlier work of Ambainis [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 Our Results and Techniques Las Vegas Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We propose a different definition of quantum Las Vegas query complexity, which is very natural and more quantum in spirit than the previous notion of zero- error quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We define it as the total sum of the squared norms of all the states processed by the input oracle during the execution of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since the square of the norm means probability in the quantum world, this quantity can be interpreted as the expected number of queries performed by the algorithm on input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Our variant of quantum Las Vegas complexity possesses all the nice properties mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It does not feature any artificial constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It composes nicely as we will show in Sections 5 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, as we will show in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4, every quantum Las Vegas algorithm with complexity L can be turned into a Monte Carlo algorithm with error ε and complexity OpL{εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since the term ‘zero-error’ is standard for the previous definition, we call our version ‘Las Vegas’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Contrary to Monte Carlo complexity, Las Vegas complexity is input-dependent: different inputs can have different complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, we can study not only the worst-case complexity, but also track complexity on each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We capture this by introducing complexity profile, which is the vector recording the complexity of the algorithm on all inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Very recently, and independently of our paper, Jeffery [38] came up with essentially the same notion, which was combined with variable-time quantum algorithm to get a composition result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 3 The results of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 can be seen as a query analogue of the latter composition result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' New Simplified Upper Bound Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, our paper continues the line of work relating quantum query algorithms and the adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Fol- lowing Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [44], the relation between the two can be depicted as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Figure 1 Computational problem ÐÑ Adversary optimisation problem Input of the problem ÐÑ Variable-vector in the problem Query algorithm solving the problem ÐÑ Feasible solution to the problem Complexity of the algorithm ÐÑ Objective value of the feasible solution Correspondence between quantum query algorithms and the adversary bound After formulating the adversary optimisation problem in Row 1 of Figure 1, the main issue is to prove the corresponding lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A lower bound is a transformation of an algorithm into a feasible solution in Row 3 of Figure 1, which respects the fourth row of the same diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' An upper bound is a transformation in the opposite direction, which turns a feasible solution into an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For the lower bound, we follow the same approach that was developed in [44] and used in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The variable-vector is the direct sum of all the queries made to the oracle on the corresponding input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, its squared norm lower bounds query complexity, as the state processed on one query has norm at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The crucial novel ingredient in our paper is a new construction of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The idea behind it is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' What we would like to do is to reverse the above process and to give the variable-vector from the feasible solution as a query to the input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' At the first sight, it is not clear how to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, the algorithm does not know the input, hence, does not know which vector to give.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' And even if it knew, the latter vector generally has norm much larger than 1, making it impossible to use it as a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have found a very simple way around these complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume we add a small “catalyst” to the state of algorithm, which is just a scaled down variable-vector from the feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We process the catalyst by the input oracle, as we wanted, and use the result to change a tiny part of the state in the required direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The constraints of the adversary optimisation problem ensure that the latter step can be implemented by an input-independent unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' What is remarkable, however, is that we get the catalyst back after this unitary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' So we can use it again, and again, until we perform the required transformation on all of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, it suffices for the algorithm to “guess” the catalyst just once to perform the transformation described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The “guessing” ability is folklore in quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' What is meant here is that if the catalysis is small, the distance between the original state and the state with the catalyst is also small, and it gets preserved during the execution of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, we end up close to the target state even if we started without the catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The smaller the catalyst, the larger the number of queries needed, but the smaller the error induced by guessing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us compare our algorithm with two previous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' They are the aforementioned quantum-walk-based algorithm by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [44] and an adiabatic algorithm by Brandeho and Roland [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Both of them use the guessing ability to extend the initial state with a small state incorporating the feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' After that the algorithm uses a quantum walk or an adiabatic transformation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' What we demonstrated is that the same effect can be obtained by a very simple unitary 4 transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This substantially simplifies the analysis and makes the algorithm more trans- parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, we see what queries are being made by the algorithm: it repeatedly calls the input oracle on the scaled down variable-vector from the feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This allows us to make various improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, we see that the Las Vegas complexity of the algorithm is exactly the objective value of the optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Second, we can easily incorporate multiple input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Third, the upper bound works assuming unidirectional access to the input oracle, while the previous algorithm in [15] required bidirectional access (the algorithm can query both the input oracle Ox and its inverse O˚ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Fourth, we do not even need the input oracle to be unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, we get a slightly better dependence of the number of queries (in the traditional sense) on the error parameter ε, which is now tight up to constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us further discuss these improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Relation to Las Vegas Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' With the new upper bound, it becomes easy to cal- culate the Las Vegas complexity of the algorithm, which leads to the main result of this paper: The quantum adversary bound is equal to the Las Vegas complexity of state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We note that this is a threefold tighter connection between the adversary bound and the usual (Monte Carlo) query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, the bound is tight, while connection to Monte Carlo complexity is semi-tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Second, the bound is exactly equal to Las Vegas complexity, and not just up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Third, the bound holds for all inputs simultaneously, and not just worst-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This result automatically carries over to all special cases of state conversion, including state generation and function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Multiple Input Oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Considering multiple input oracles is often useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, even the standard input oracle Ox : |i⟩|b⟩ ÞÑ |i⟩|b ‘ xi⟩ is a direct sum of multiple input oracles Oxi : |b⟩ ÞÑ |b ‘ xi⟩, which encode individual symbols of the input string x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' These settings were investigated previously, most notably in the context of compositional re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt and ˇSpalek [55] considered span programs with costs, where costs were assigned to individual symbols, and which were meant to capture the complexity of the corresponding subproblems, and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [52] similarly consider the adversary bound with costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Multiple oracles are also necessary in the study of trade-offs between different input re- sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Kimmel, Lin, and Lin [39] used an adversary-based approach to show a trade-off between two input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Again, the adversary featured costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs and Rosmanis [21] used a similar approach with general input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Actually, the whole notion of Las Vegas complexity, including the multiple oracle case, is greatly inspired by the latter paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the case of Las Vegas complexity, dealing with several input oracles is easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We can use the same definition (the sum of the squared norms of the states processed by the oracle) for each oracle independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The complexity profile becomes a matrix, where, for each input, the complexity of each of the input oracles is listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the adversary bound, the variable-vector is similarly broken down into parts corresponding to different input oracles, and all the results carry over with minimal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Having a vector of complexities of all the input oracles, it is easy to get compositional results as well as trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Unidirectionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The upper bound in [15] used quantum walk, which imposed bidirectional access to the input oracle to implement the required reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the new upper bound, there are no reflections, hence, there is no need for this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Allowing bidirectional access has a lot of rationale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, oracles are usually thought as quantum subroutines, and each quantum subroutine can be easily inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Also, many basic 5 quantum algorithms like Grover’s search and amplitude amplification often require bidirectional access to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the other hand, unidirectional access also naturally comes up in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, if we send the state to some other party to apply the input oracle, we may trust the recipient to perform the required query, but we might not be able to ask them to perform it in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, the assumption that we have unidirectional access to the input oracle simplifies the upper and the lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The bidirectional case can be obtained as a special case, see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' General Input Oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As there is no bidirectionality assumption, we can replace the unitary oracle assumed in [15] by an arbitrary linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It turns out that many of results hold still hold even under such assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It seems, however, that contraction oracles, which are linear transformations of norm not exceeding 1, might be a good choice to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Contraction oracles seem to contradict the unitarity condition usually imposed on quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Nonetheless, such oracles naturally come up in practise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For example, the input oracle can perform some measurement and continue only if the outcome is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Similar settings appear in interaction-free measurements by Elitzur and Vaidman [35] and subsequent bomb query algorithm by Lin and Lin [46], measure-many quantum finite automata [40], and faulty oracles [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Subspace Conversion Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, we define and study the subspace conversion prob- lem, which is in between state conversion ξx ÞÑ τx, which we assume for the action of the algorithm, and unitary (or contraction) Ox, which we use for the input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In this problem, the task is to implement a linear transformation Tx : Kx Ñ K defined on a linear subspace Kx of the workspace K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If Kx is one-dimensional, this is state conversion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' if Kx “ K, this is unitary (or contraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We introduce a complexity notion for this problem, which is the largest Las Vegas complexity of the algorithm achieved when executed on a unit vector in Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The definition turns out to be natural for composition, and it is still exactly characterised by the corresponding version of the adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Unitary Permutation Inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, we use this occasion to demonstrate a separation between unidirectional and bidirectional access to the input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We consider the unitary permutation inversion problem, where the oracle is a unitary that implements a permutation |i⟩ ÞÑ |πpiq⟩, and the task is to find π´1p1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We prove an Ωp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='nq lower bound, where n is the size of the domain of π, whereas the problem is trivially solvable with 1 query to the inverse oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Up to our knowledge, these types of questions have not been addressed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 Overview of the Paper In this subsection, we give a very brief overview of the paper, highlighting the most important points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The main part of the paper starts with Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, we define a quantum query algorithm as a sequence of linear transformations UT rO UT´1 rO ¨ ¨ ¨ U1 rO U0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) where Ui are some unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The operator rO “ pO b I‚q ‘ I˝ is a query, where O is the input oracle, and I‚ and I˝ are some identity transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, the algorithm implements a transformation O ÞÑ ApOq: from the input oracle to the linear operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, 6 we describe problems solved by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We first give a general definition, capable of describing a wide range of problems, but for the purposes of this paper, the most important problem is state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Given a family of input oracles Ox and pairs ξx ÞÑ τx, where x ranges overs some set D, the task is to develop an algorithm A such that ApOxq maps ξx into τx for all x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The remaining part of the paper follows a similar division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Sections 4 and 5 are devoted to the study of Las Vegas complexity of algorithms without connection to any particular problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7, we consider the state conversion problem, and in Section 8, subspace conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, the adversary bound makes its first appearance in Section 7 as it is tied to a particular problem being solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Other problems can be studied as well, for instance, in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7, we consider the problem of Boolean function evaluation, and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [15] considers a wide range of other problems, which we leave outside the confines of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Section 6 is an intermission, and Sections 9 and 10 contain complementary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us describe these sections in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 4, we define Las Vegas complexity a quantum query algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let QtpA, Oqξ be the state processed by the input oracle (the O bI‚ part of the query operator rO) on the t-th query when executed on the input oracle O and the initial state ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We define the Las Vegas complexity LpA, O, ξq as the sum of }QtpA, Oqξ}2 over all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that it depends both on the input oracle O and the initial state ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, we define the same notion for multiple input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In essence, the complexity LpA, O, ξq becomes a tuple which accounts for the total squared norm of the states processed by each of the input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The difference between the single-oracle and the multiple-oracles variants is mostly cosmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The reader may choose to assume the single-oracle variant throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 5, we study various properties of Las Vegas complexity without relation to any particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We consider various ways algorithms can be composed: inversion, direct sum, tensor product, sequential and functional composition, and show that our definition of Las Vegas complexity encompasses these composition variants naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The results are pretty straightforward, but there is one subtlety involving functional composition, where one algorithm B is used as an input oracle for another algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The thing is that the algorithm A executes the input oracle as O b I‚, while we assume the algorithm B implements O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This means that the complexity of B on the state ψt “ QtpA, Bqξ is just not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We use an obvious solution to slice ψt as ψt,1 ‘ ¨ ¨ ¨ ‘ ψt,d, where d is the dimension of I‚, and each ψt,j can be processed by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Now, the complexity of B on each ψt,i is well-defined, and we can define the total complexity as their sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' There are many different possible slicing, as they depend on the choice of an orthonormal basis in the space of I‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We show that the total complexity is independent of the choice of slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 6, we describe our modification to the relative γ2-norm from [15], as different settings require different versions of the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' One thing we should account for is unidirec- tionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We also have to convert to multi-objective version of the bound, as we are interested in the full complexity profile of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The variant with multiple input oracles requires yet another modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We formulate the dual versions, which can be used to prove lower bounds on worst-case complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Section 7 is the main part of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In it, we study Las Vegas complexity of state conversion, and show how it can be characterised by an instance of (unidirectional) relative γ2-bound: the adversary optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This section is designed to be self-contained with minimal dependency on the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We first prove a lower bound for the exact version of the problem in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, and then an upper bound for the approximate version in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The corresponding ideas were already explained in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm in 7 Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 transforms ξ` x “ ξx ‘ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T vx ÞÝÑ τ ` x “ τx ‘ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T vx, where ξx ÞÑ τx is the required state conversion problem, vx is a feasible solution to the adversary optimisation problem, and T is an arbitrary positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The Las Vegas complexity of the algorithm on input x is }vx}2 independently from the value of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (The total number of queries does depend on T, though).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As T increases, we can get arbitrarily close to the required transformation, while Las Vegas complexity stays the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5, we improve on this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We show how to perform transformation ξx ÞÑ τx exactly, while now we can get Las Vegas complexity arbitrarily close to }vx}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Section 8 deals with the question of what happens if some of the input oracles Ox in the state conversion problem are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If Ox “ Oy, then we get exactly the same action of the algorithm on the inputs x, y P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The motive of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 is linear consistency of the feasible solution to the adversary bound for such pairs of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We show that we can assume consistency without any loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Moreover, this brings us to the formulation of the subspace conversion problem, and the corresponding adversary bound in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, we revisit the functional composition property from Section 5 and show that we can upper bound the complexity of the composed algorithm as the product of complexities of the constituents under the assumption that the algorithm follows the specification of the subspace-converting subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 9, we prove the relation between unidirectional and bidirectional versions of the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, we get back the bidirectional results from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 10, we prove a separation between unidirectional and bidirectional input oracle for the unitary permutation inversion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, in Section 11, we make some final comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [60] contains an alternative exposition of some of the results in Section 7, and some additional results on more general control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2 Preliminaries If not said otherwise, a vector space is a finite-dimensional complex inner product space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' They are denoted by calligraphic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We assume that each vector space has a fixed orthonormal basis, and we often identify an operator with the corresponding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The inner product is denoted by x¨, ¨y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A˚ stands for the adjoint linear operator, and Arri, jss for the pi, jqth entry of the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A ˝ B stands for the Hadamard (entry-wise) product of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' IX stands for the identity operator in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' All projectors are orthogonal projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For vectors u, v P Rn, we write u ď v if urriss ď vrriss for all i P rns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We use A ą 0 and A ě 0 to denote positive definite and semi-definite matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We use the ket-notation to emphasise that a vector is a state of a quantum register, or to denote the elements of the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We also need the following generalisation of the well-known parallelogram identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We were not able to find its statement in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 (Generalised Parallelogram Identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , vd P Cn, and U “ ¨ ˚ ˚ ˚ ˝ α1,1 α1,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' α1,d α2,1 α2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' α2,d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' αd,1 αd,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' αd,d ˛ ‹‹‹‚ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) 8 be a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, }v1}2 ` }v2}2 ` ¨ ¨ ¨ ` }vd}2 “ dÿ j“1 ��α1,jv1 ` α2,jv2 ` ¨ ¨ ¨ ` αd,jvd ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let V be the n ˆ d matrix with vj as the columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The above identity is equivalent to ��V ��2 F “ ��V U ��2 F, where ∥¨∥F stands for the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The equality follows from the fact that unitaries preserve the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The parallelogram identity is the special case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 with U “ H, the Hadamard matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 3 Quantum-Algorithmic Definitions In this section, we give the main definition of a quantum algorithm solving a computational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us very briefly recall the textbook definition of a quantum query algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A standard reference is a survey by Buhrman and de Wolf [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (Note, however, that it only deals with Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' See also [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=') The task is evaluation of a function f : D Ñ rℓs with domain D Ď rqsn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm can perform arbitrary unitary transformations, as well as access the input string x “ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , xnq P D via the standard input oracle: Ox : |i⟩|b⟩ ÞÑ |i⟩|b ‘ xi⟩, where ‘ is the bit-wise XOR operation (one can also use modular addition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm is said to compute the function f if, for all x P D, measuring the output register of the final state of the algorithm gives fpxq with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The unitary operations are free, and each execution of Ox costs one query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The goal is to minimise the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We separate the algorithm itself from the input and output conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm becomes a map from operators on the input register (Ox) to operators on the space of the algorithm (the transformation performed by the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The input condition is the input oracle Ox given on a particular input x P D, and the output condition is the set of admissible transformations performed by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Actually, we choose to treat input and output conditions similarly as sets of admissible transformations, which allows algorithms to be used as input oracles for other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We keep the set of input labels D, but it need not be considered as the domain of a function any longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We describe the algorithmic part in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, and the input/output conditions in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 Quantum Query Algorithm The overall form of a quantum query algorithm is similar to the textbook version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' When we use the term ‘algorithm’ later in the paper, we mean a quantum query algorithm of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 (Quantum Query Algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let M and H be vectors spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A quantum query algorithm in H with an oracle in M is a function which maps linear operators O: M Ñ M into linear operators ApOq: H Ñ H, and which has the following form: ApOq “ UT rO UT´1 rO ¨ ¨ ¨ U1 rO U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) 9 Here, each Ui : H Ñ H is a unitary that does not depend on O, and rO is some “embedding” of O into H of the form rO “ pO b I‚q ‘ I˝, where I‚ and I˝ are identity transformations of some size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The operator O is called the input oracle, and each execution of rO is called a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To make the definition simpler, we have chosen to have one fixed embedding rO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is often more convenient to allow different embeddings at different queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The two definitions are equivalent, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The spaces M and H are called the input and the work spaces of the algorithm, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is sometimes useful to consider also the output subspace K Ď H of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Conceptually, it contains the “interesting” part of ApOq, while its orthogonal complement in H is the “scratch space” of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If not specified, we may always take K “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If ApOqξ “ τ for some ξ, τ P H we say that the algorithm A performs transformation ξ ÞÑ τ on the oracle O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We call ξ the initial and τ the terminal state of the algorithm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us mention the main differences with the textbook definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The first difference is that we allow arbitrary input oracles O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The second difference is the ability to “skip” query: to apply I˝ on some part of the workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Alternatively, in the language of circuits, we may apply a controlled version of O, not just O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Textbook quantum query algorithms do skip queries, but it is usually done implicitly by setting up a state that does not change by any input oracle, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=', a uniform superposition on the second register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since we allow arbitrary unitaries as oracles, this option is out of stock for us, and we have to skip queries explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Interestingly, this is exactly this feature that allows us to define quantum Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us also emphasise the differences with the definition of a quantum query algorithm in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The first one is what we call directionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm in [15] is bidirectional: it allows execution of both rO as well as its inverse rO´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 is unidirectional: it only allows execution of rO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For the standard input oracle, the difference is irrelevant since each standard input oracle is its own inverse (or can be easily constructed from it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is not true for arbitrary unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that unidirectionality is without any loss of generality, as it is possible to simulate bidirectional access to a unitary O with unidirectional access to O ‘ O˚, see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The second difference is that, while we still require that all Ui are unitary, there is no more need to require the input oracle O to be unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We allow O to be an arbitrary linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, a more interesting choice is to consider contractions as input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that if O is a unitary or a contraction, then ApOq is also a unitary or a contraction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 Input and Output Conditions Here, we give a general take on input and output conditions imposed on a quantum algorithm, as well as define all types of conditions we consider in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Sections 7 and 8, we redefine the specific conditions under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The simplest way to impose requirements on an algorithm A from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 is to specify its outputs ApOxq: H Ñ H on fixed inputs Ox : M Ñ M as x ranges over some set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' There is nothing fundamentally wrong with this approach, except that it may be too specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, the textbook definition has a very specific input oracle Ox, but a very vague output condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To capture this, for each x P D, we define not one, but a collection Ex of admissible linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This gives the following very general definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 (Computational Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A computational problem is given by a set of labels D, where, for each x P D, a set of admissible inputs Ox and a set of admissible outputs Ex are 1The letters ξ and τ stand for ξεκίνημα and τέλος, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 10 specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A quantum algorithm A solves the problem if, for each x P D and each O P Ox, it holds that ApOq P Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We treat input and output uniformly, therefore, we use a term admissible set for both sets of admissible inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We define different types of admissible sets, which are depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We do this in terms of Ex, the output space K, and the workspace H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The definitions for input conditions are similar with Ex replaced by Ox, and K and H by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We say that we have a problem of type1 with input oracles of type2, if all Ex are of type1 and all Ox are of type2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Figure 2 Subspace Conversion State Conversion General Input Oracle (Unitary/Contraction) State Generation Function Evaluation Various types of input/output conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The ones at the top are more general in the sense that the lower ones are special cases thereof as indicated by arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The ones to the right are more restrictive in the sense that they impose more restrictions on the set of admissible operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Most types of admissible sets considered in this paper are special cases of the following type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 (Subspace Conversion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' An admissible set Ex is subspace conversion if there exists a linear transformation Sx : Kx Ñ K defined on some linear subspace Kx Ď K such that Ex consists of all extensions of Sx to a linear operator on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' There are two main cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the isometric case, we only allow unitaries in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Of course, this makes sense only if Sx is an isometry itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the general non-isometric case, we assume that Sx are contractions, and require the operators in Ex to be contractions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, subspace conversion is specified by its action on the output space K, but ac- commodates any workspace H as long as it is a superspace of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The vectors in KzKx are interpreted as ones where the action of the algorithm is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that it is possible that A P Ex maps such vectors outside of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' There are two main special cases of subspace conversion, which are more important than the general case itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The first one is when Kx “ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In this case, Sx gives a linear map from K to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm can still use a larger workspace, but it is completely inaccessible from outside, therefore, it makes sense to identify Ex with Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is our default type of input condition, which we call general input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Alternatively, we call it unitary, contraction, or linear input oracle in dependence on the type of Sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For the output condition, we call it unitary or contraction implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The second important special case is when Kx is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We call it state conversion, and denote by ξx ÞÑ τx, meaning that Aξx “ τx for all A P Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is our default type of output condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 11 There are important special cases of state conversion as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' State generation is state conversion when all the initial states ξx are equal to some predefined state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The most widely used version is function evaluation, which is state generation when τx is an element of the computational basis |fpxq⟩ for some function f : D Ñ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is also possible to define approximate and non-coherent versions of above conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the ε-approximate version, we take the ε-neighbourhood of Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, an algorithm A solves an ε-approximate version of state conversion ξx ÞÑ τx if, for all x P D and all O P Ox, we have }ApOqξx ´ τx} ď ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We say that A solves the non-coherent version of the problem, if ApOqξx “ τx b ζ for some junk state ζ that may depend on x and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, we can consider ε-approximate non-coherent version as well, where we require that }ApOqξx ´ τx b ζ} ď ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Function evaluation is usually considered in the approximate non-coherent case, as it is required that measuring the output register of the final state gives fpxq with bounded error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, for bidirectional oracles, coherent and non-coherent versions differ at most by a factor of 2 in complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, it is possible to evaluate the function non-coherently, copy the final output into a new register, and run the program in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For unidirectional oracles, however, this simple trick does not work, as it is impossible to run the program in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It also does not work for state generation, as it is impossible to copy general quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 4 Quantum Las Vegas Query Complexity In this section, we define the main notion of this paper: quantum Las Vegas query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Usually query complexity of the algorithm like in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 is defined as T: the number of invocations of the input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will often call it Monte Carlo query complexity in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Contrary to Monte Carlo complexity, Las Vegas complexity is input-dependent, as it depends both on the oracle O and the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 Definition Let A be an algorithm as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, and O: M Ñ M be an input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We need the following two linear transformations on the workspace H, which can be seen as partial executions of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For t P rT ` 1s, let StpA, Oq “ Ut´1 rO Ut´2 rO ¨ ¨ ¨ U1 rO U0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) be the transformation that maps the initial state ξ to the state just before the t-th application of the input oracle O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, S0pA, Oq “ U0 and ST`1pA, Oq “ ApOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Recall that the query is of the form rO “ pO b I‚q ‘ I˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Π denote the projection on the part of the space processed by O b I‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The second transformation is QtpA, Oq “ ΠStpA, Oq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) which maps ξ to the state processed by the input oracle on the t-th query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The quantum Las Vegas query complexity of the algorithm A on the input oracle O: M Ñ M and the initial state ξ P H is defined as LpA, O, ξq “ Tÿ t“1 ��QtpA, Oqξ ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) Under usual assumptions of O being unitary and }ξ} “ 1, the term ∥QtpA, Oqξ∥2 can be interpreted as the probability that the algorithm A actually executes the query on the t-th step, 12 and not skips it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, LpA, O, ξq can be seen as the expected number of queries similarly to the definition of the randomized Las Vegas query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Las Vegas complexity does not exceed the Monte Carlo complexity T, but it can be much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The definition also encapsulates the case of algorithms with intermediate measurements as we briefly discuss here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume we have a quantum algorithm B with intermediate measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The definition is similar to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 with the difference that the algorithm can perform measurements in the middle, so that the forthcoming unitaries Ui depend on the out- come of the previous measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, the number of queries can also depend on the outcomes of the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let TpB, O, ξq be the expected number of queries performed by B on oracle O and initial state ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Such an algorithm can be turned into a usual algorithm A as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 by deferring the measurements to the end of the algorithm [50, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is not hard to see that TpB, O, ξq ě LpA, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note, however, that in the absence of measurements, the terminal state of A differs from the terminal state of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, A computes the non-coherent version of a state conversion problem even if the original algorithm B computes the coherent version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 Multiple Input Oracles Assume we have s input oracles, Op1q, Op2q, ¨ ¨ ¨ , Opsq, where Opiq acts on some space Mpiq, and we want to provide the algorithm with access to all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This can be seen as a special case of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, where the algorithm has access to the combined oracle O “ Op1q ‘ Op2q ‘ ¨ ¨ ¨ ‘ Opsq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) acting on M “ Mp1q ‘ ¨ ¨ ¨ ‘ Mpsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, it is possible to simulate a query to Opiq using one query to O, and it is possible to simulate a query to O using one query to each of Opiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Now suppose we want to measure complexity of each oracle Opiq individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the case of Las Vegas complexity, this can be handled very naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Decompose QtpA, Oqξ “ Qp1q t pA, Oqξ ‘ Qp2q t pA, Oqξ ‘ ¨ ¨ ¨ ‘ Qpsq t pA, Oqξ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) where Qpiq t pA, Oqξ is the state processed by the i-th input oracle on the t-th query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the above settings, the Las Vegas complexity of the i-th input oracle is defined as LpiqpA, O, ξq “ Tÿ t“1 ���Qpiq t pA, Oqξ ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The Las Vegas complexity LpA, O, ξq of the algorithm A on the composed input oracle O from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) is the vector in Rs consisting of the individual complexities LpiqpA, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Almost all the results in this paper can be generalised to include this variation of Las Vegas complexity with minimal changes in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To make this explicit, we introduce the following piece of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let v P M b W for some W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have the following imposed decomposition v “ vp1q ‘ vp2q ‘ ¨ ¨ ¨ ‘ vpsq, with vpiq P Mpiq b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We define ~v~2 “ ´��vp1q��2, ��vp2q��2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , ��vpsq��2¯ P Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6) 13 This notation is chosen to emphasise similarity to }v}2, and we never use ~v~ alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This gives us almost the same definition for Las Vegas complexity as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3): LpA, O, ξq “ Tÿ t“1 \u200c\u200c\u200cQtpA, Oqξ \u200c\u200c\u200c 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) The upcoming sections can be read using one of the two assumptions: There is a single input oracle O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In this case, definitions from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 hold, s “ 1 everywhere, and ~v~2 stands for }v}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' There are multiple input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In this case, we use O as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) to combine them in a single input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We use Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, and ~v~2 is as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Most of the time, there is no difference between the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us list the properties of ~v~2 that we will need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' They follow easily from the defini- tion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ~cv~2 “ |c|2 ¨ ~v~2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8a) ~u ‘ v~2 “ ~u~2 ` ~v~2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8b) If O is a unitary of the form in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4), then ~v~2 “ ~Ov~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8c) Finally, the generalised parallelogram identity also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Namely, in assumptions of Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1: ~v1~2 ` ~v2~2 ` ¨ ¨ ¨ ` ~vd~2 “ dÿ j“1 \u200c\u200cα1,jvj ` α2,jvj ` ¨ ¨ ¨ ` αd,jvd \u200c\u200c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9) 5 Properties of Las Vegas Complexity Apart from functional composition, which was the main focus of previous work, algorithms can be composed in many different ways, some of which we describe in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Most of them were used before implicitly, and one of our goals was to formulate them in a more explicit way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We also show that quantum Las Vegas complexity can handle these composition variants naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Most of the results hold for linear input oracles, but we require unitary input oracles for some.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 Basic Properties Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 (Scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For every algorithm A, oracle O: M Ñ M, and states ξ, τ P H, if A transforms ξ ÞÑ τ on O, then it also transforms cξ ÞÑ cτ for all c P C and LpA, O, cξq “ |c|2LpA, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This follows from the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that while QtpA, Oq is linear, it distorts inner products even if O is a unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, there is no general way to relate LpA, O, ξ`ξ1q to LpA, O, ξq and LpA, O, ξ1q even for orthogonal ξ and ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 14 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 (Parallelogram Identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For every algorithm A, oracle O: M Ñ M, states ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , ξd P H, and unitary U as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1), we have LpA, O, ξ1q ` LpA, O, ξ2q ` ¨ ¨ ¨ ` LpA, O, ξdq “ dÿ j“1 L ` A, O, α1,jξ1 ` α2,jξ2 ` ¨ ¨ ¨ ` αd,jξd ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The proof is analogous to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, but this time we use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 (Inversion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For every algorithm A in H with oracles in M, there exists the inverse algorithm A´1 in the same spaces such that for every unitary input oracle O: M Ñ M, we have A´1pO˚q “ ` ApOq ˘´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Moreover, if A transforms ξ ÞÑ τ on a unitary input oracle O, then LpA´1, O˚, τq “ LpA, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm A´1 is just the inverse of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1): A´1pOq “ U ˚ 0 rO U ˚ 1 rO ¨ ¨ ¨ U ˚ T´1 rO U ˚ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The relation between Las Vegas query complexities follows from the identity QtpA´1, O˚qτ “ pO b I‚qQT`1´tpA, Oqξ and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 Slicing Let us now describe possible alternatives to the Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 of the quantum query algorithm, and show that they preserve Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, we show that we can replace the “embedding” rO “ pO b I‚q ‘ I˝ with a simpler construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 (Sliced Algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We call a quantum algorithm from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 sliced if its query rO is of the form rO “ O ‘ I˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Clearly, a sliced algorithm is a special case of the general algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the other direction, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 (Slicing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Every algorithm A can be transformed into a sliced algorithm A1 such that, for every oracle O: M Ñ M and initial state ξ P H, we have ApOq “ A1pOq and LpA, O, ξq “ LpA1, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let rO “ pO b I‚q ‘ I˝ be the query of the algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We can rewrite O b I‚ “ O ‘ O ‘ ¨ ¨ ¨ ‘ O “ pO ‘ I ‘ ¨ ¨ ¨ ‘ IqpI ‘ O ‘ ¨ ¨ ¨ ‘ Iq ¨ ¨ ¨ pI ‘ I ‘ ¨ ¨ ¨ ‘ Oq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) where there are d “ dim I‚ multipliers on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='Conjugating each of them by a unitary, we can implement rO using d queries to rO1 “ O ‘ I˝1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This does not change the action of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Neither does this change its Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, let ψt “ QtpA, Oqξ be the state processed by rO on the t-th query, and ψt,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , ψt,d be the corresponding states processed by the oracle rO1 on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then ψt “ ψt,1 ‘ ψt,2 ‘ ¨ ¨ ¨ ‘ ψt,d, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) and the result follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 15 Note that the algorithm depends on the choice of slicing in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1), which in turn depends on the choice of the orthonormal basis in the space of I‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1), this does not change the action of the algorithm, and by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2), this does not change its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, we can further assume, without loss of generality, that a quantum algorithm is sliced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will use this in this section, as it simplifies some constructions and some proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Also, note that the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 still works if we have different embeddings of O on each query of the algorithm in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, this variant of the definition is also equivalent to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 Space Extension The following two results formally state that we can embed an algorithm into a larger space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The work space extension is straightforward: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 (Work Space Extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let A be an algorithm in H with oracles in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for every H1, there is an algorithm A‘IH1 in H‘H1 with oracles in M such that for every O: M Ñ M, ξ P H, and ξ1 P H1, we have pA‘IH1qpOq “ ApOq‘IH1 and LpA‘IH1, O, ξ‘ξ1q “ LpA, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let A be as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To get A ‘ IH1, replace each Ui with Ui ‘ IH1, and each I˝ from rO with I˝ ‘ IH1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The input space extension is also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For simplicity, we assume the algorithm A is sliced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We state the extension in a rather general way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='Essentially, we require that the input oracle in the extended space agrees with the original oracle on the states actually being queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let A be a sliced algorithm in H with oracle O: M Ñ M, and M ‘ M1 be a superspace of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We construct an algorithm A1 in H ‘ M1 with oracle O1 : M ‘ M1 Ñ M ‘ M1 in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Each unitary Ui is replaced by Ui ‘ IM1 and each query O ‘ I˝ is replaced by O1 ‘ I˝ acting in H ‘ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7 (Input Space Extension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the above assumptions, if O: M Ñ M, O1 : M‘ M1 Ñ M ‘ M1 and ξ P H are such that OQtpA, Oqξ “ O1QtpA, Oqξ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) for all t, then A1pO1qξ “ ApOqξ and LpA1, O1, ξq “ LpA, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) holds if O1 “ O ‘ O2 for some O2 acting in M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Recall the operator St defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By induction on t, it is easy to show that StpA1, O1qξ “ StpA, Oqξ, from which the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will often identify the algorithms A and A1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 Sequential Composition and Direct Sum Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 (Sequential Composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume there are two algorithms A and B in H with oracles in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, there exists an algorithm B ˚ A such that for all O: M Ñ M and ξ P H we have pB ˚ AqpOq “ BpOqApOq and LpB ˚ A, O, ξq “ L ` B, O, ApOqξ ˘ ` LpA, O, ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm B ˚ A is the algorithm B applied after A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 16 The condition that A and B share the same workspace seems restrictive, but it is necessary for the formal statement of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Usually it makes sense to assume that A and B share the same output space K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, in the spirit of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, the initial state ξ is assumed to be such that both ApOqξ and pB˚AqpOqξ are in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let W and W1 be the orthogonal complements of K in the workspaces of A and B, respectively (the “scratch spaces”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We can still apply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 with H “ K ‘ W ‘ W1 and assuming that the algorithms A and B are extended by the identity to H using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Also, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 assumes that A and B use the same input oracle O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, let A and B use different oracles O1 : M1 Ñ M1 and O2 : M2 Ñ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Extend the input space of both algorithm to M “ M1 ‘ M2, and assume they both use the input oracle O “ O1 ‘ O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7, the action of both algorithms does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The same observations also applies to Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9 (Direct Sum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let A and B be two algorithms in spaces H and H1 respectively, and both with oracles in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, there exists an algorithm A ‘ B in H ‘ H1 with oracles in M such that for all O: M Ñ M, ξ P H, and ξ1 P H1, we have pA ‘ BqpOq “ ApOq ‘ BpOq and LpA ‘ B, O, ξ ‘ ξ1q “ LpA, O, ξq ` LpB, O, ξ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm A ‘ B can be implemented as pIH ‘ Bq ˚ pA ‘ IH1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The result follows from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 Functional Composition and Tensor Product Functional composition is a more interesting way of composing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It can be constructed with ease assuming the outer algorithm is sliced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 (Functional Composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let A be a sliced algorithm in H with oracles in N, and B be an algorithm in N with oracles in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, there exists an algorithm A ˝ B in H with oracles in M such that for all O: M Ñ M and ξ P H, we have pA ˝ BqpOq “ ApBpOqq and LpA ˝ B, O, ξq “ ÿ t L ` B, O, Qt ` A, BpOq ˘ ξ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote by O1 : N Ñ N the input oracle of the outer algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Replace each query rO1 “ O1 ‘ I˝ of A by a copy of the algorithm B ‘ I˝ obtained via Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The theorem follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 and the observation that the copy of the algorithm B replacing the t-th query processes the state Qt ` A, BpOq ˘ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This result requires a number of comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, it is usually convenient to assume that N is the output space of the algorithm B, not its workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This can be achieved by applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' the discussion after Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Next, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 assumes the that algorithm A has a single input oracle (while the algorithm B and, consequently, A ˝B can have multiple input oracles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us now consider the case when A has multiple input oracles Opiq : N piq Ñ N piq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each i, let Bpiq be an algorithm in N piq with the oracle in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9, they can be combined into a single algorithm B “ À i Bpiq acting in N “ À i N piq, which is the same space where the combined input oracle of A acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4), we obtain the following version of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5): LpA ˝ B, O, ξq “ ÿ i ÿ t L ´ Bpiq, O, Qpiq t ` A, BpOq ˘ ξ ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6) 17 We will return to the above two comments in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The final comment concerns slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Namely, when applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 to a non-sliced algorithm A as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, it is first necessary to slice the latter using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Slic- ing is convenient here as it allows us to use Las Vegas complexity of B on the state Qt ` A, BpOq ˘ ξ directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The downside of this approach is that the resulting algorithm depends on the way how we slice the query O b I‚ of the algorithm A in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As discussed before, this does not change the action of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, it is not clear how it affects complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In order to understand this, it suffices to consider one query of the outer algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' That is, we can assume the composed algorithm is of the form B b I‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 to the sliced algorithm and using the following decomposition similar to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2): ξ “ ξ1 ‘ ξ2 ‘ ¨ ¨ ¨ ‘ ξd (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) with each ξj in N, we get LpB b I‚, O, ξq “ dÿ j“1 LpB, O, ξjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8) Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The value of the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8) is independent from the choice of a particular slicing in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, for a non-sliced algorithm A, we can write an analogue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5): LpA ˝ B, O, ξq “ ÿ t L ` B b I‚, O, Qt ` A, BpOq ˘ ξ ˘ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9) which is well-defined due to the above observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Similarly, in the case of multiple input oracles, we can write the following analogue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6): LpA ˝ B, O, ξq “ ÿ i ÿ t L ´ Bpiq b I‚, O, Qpiq t ` A, BpOq ˘ ξ ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof of Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7), we decomposed ξ assuming some standard basis in the space of I‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , ud be another orthonormal basis of the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, we have a similar decomposition ξ “ ξ1 1 b u1 ` ξ1 2 b u2 ` ¨ ¨ ¨ ` ξ1 d b ud (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10) with ξ1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , ξ1 d P N, but this time based on the basis u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , ud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since the the basis u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , ud is orthonormal the decompositions in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10) are connected by a unitary U in the following way, where we assume the unitary U is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1): ξ1 j “ α1,jξ1 ` α2,jξ2 ` ¨ ¨ ¨ ` αd,jξd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, the complexity of the algorithm obtained when using the slicing based on u is dÿ j“1 LpB, O, ξ1 jq “ dÿ j“1 LpB, O, α1,jξ1 ` α2,jξ2 ` ¨ ¨ ¨ ` αd,jξdq “ dÿ j“1 LpB, O, ξjq by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As a by-product we get a nice expression for a tensor product of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 18 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='12 (Tensor Product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let A and B be two algorithms in spaces H and H1 respectively, and both with oracles in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, there exists an algorithm A b B in H b H1 with oracles in M such that for all O: M Ñ M, we have pA b BqpOq “ ApOq b BpOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Moreover, if O is a unitary, then LpA b B, O, ξq “ LpA b IH1, O, ξq ` LpIH b B, O, ξq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11) where the two terms on the right-hand side are defined as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We can implement A b B as pIH b Bq ˚ pA b IH1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8, we get L ` A b B, O, ξ ˘ “ L ` A b IH1, O, ξ ˘ ` L ` IH b B, O, pApOq b IH1qξ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, it remains to prove that LpIH b B, O, ξq “ L ` IH b B, O, pApOq b IH1qξ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' But this follows from Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11, as multiplication by a unitary ApOq b IH1 can be seen as a change of basis in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If O is not unitary, we do not get such a nice expression as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, the complexity depends on whether we implement A b B as pIH b Bq ˚ pA b IH1q or as pA b IH1q ˚ pIH b Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 6 Unidirectional Relative γ2-bound The variants of the adversary bound in [44] and [15] are formulated in terms of generalisations of the γ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The γ2-norm was originally developed in the context of operator factorisation in Banach spaces [59, Section 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It has an independent formulation as the Schur (Hadamard) product operator norm [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the realm of theoretical computer science, it was first used in communication complexity [47, 48, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the context of the quantum adversary, its generali- sations appeared in [44] and [15] as filtered and relative γ2-norms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have to generalise the latter in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, in order to deal with unidi- rectional access to the input oracle, we have to define the unidirectional version of the bound, which we do in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The previous (bidirectional) case can be obtained as a special case, see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Second, in order to switch from the worst-case complexity to the complete complexity profile, we have to introduce the multi-objective version of the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, we also have to modify the bound to capture the case of several input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' All this is done in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, we prove few basic properties of the unidirectional relative γ2-bound, which we will need later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 Single-Objective Version Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 (Unidirectional relative γ2-bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let K and M be vector spaces, and D be a set of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let E “ tExyu and ∆ “ t∆xyu, where x, y P D, be two families of linear operators: Axy : K Ñ K and ∆xy : M Ñ M that satisfy Exy “ E˚ yx and ∆xy “ ∆˚ yx for all x, y P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The unidirectional relative γ2-bound Ð γ2pE|∆q “ Ð γ2pExy | ∆xyqx,yPD, 19 is defined as the optimal value of the following optimisation problem, where Vx are linear operators: minimise maxxPD∥Vx∥2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1a) subject to Exy “ V ˚ x p∆xy b IWqVy for all x, y P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1b) W is a vector space, Vx : K Ñ M b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1c) Depending on the context, we will denote by Ð γ2pE|∆q both the optimal value and the optimization problem itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will be mostly using the following one-dimensional version, where each Ex,y “ ex,y is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, the bound reads as follows: minimise maxxPD∥vx∥2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2a) subject to exy “ @ vx, p∆xy b IWqvy D for all x, y P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) W is a vector space, vx P M b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2c) The version (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) is the one mentioned in Figure 1 in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Its feasible solutions correspond to the algorithms solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In order to prove lower bounds, we need another closely related notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us define the following generalisation of the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume X and Y be some sets of labels, and ∆ “ p∆x,yq, where x P X and y P Y , be a set of matrices of the same dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For Γ, an X ˆ Y matrix, we define Γ ˝∆ as an X ˆ Y block matrix, where the block corresponding to x P X and y P Y is given by Γrrx, yss∆x,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 (Unidirectional subrelative γ2-bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In assumptions of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, the unidirectional subrelative γ2-bound ó γ2pE|∆q “ ó γ2pExy | ∆xyqx,yPD, is defined as the optimal value of the following optimisation problem: maximise λmaxpΓ ˝ Eq (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3a) subject to λmaxpΓ ˝ ∆q ď 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3b) where Γ ranges over D ˆ D Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Here λmax stands for the largest eigenvalue of a Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This version is similar to the dual of the relative γ2-norm from [15], except that it has λmax instead of the spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The latter, in its turn, is similar to the negative-weighted adversary from [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is easy to show that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) lower bounds (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 (Weak Duality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For E and ∆ as in Definitions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, we have Ð γ2pE|∆q ě ó γ2pE|∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume we have a feasible solution Vx to Ð γ2pE|∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1b), we get that for every D ˆ D-matrix Γ: Γ ˝ E “ V ˚“ pΓ ˝ ∆q b IW ‰ V, where V “ À xPD Vx is the block-diagonal matrix with the blocks Vx on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, λmaxpΓ ˝ Eq “ max v:}v}“1 v˚pΓ ˝ Eqv “ max v:}v}“1pV vq˚“ pΓ ˝ ∆q b IW ‰ V v ď }V }2 ¨ λmax ` pΓ ˝ ∆q b IW ˘ “ max xPD }Vx}2 ¨ λmaxpΓ ˝ ∆q ď Ð γ2pA|∆qλmaxpΓ ˝ ∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 20 Let us note, although we will not need it in this paper, that in the one-dimensional case it is possible to strengthen the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If all Ex,y “ ex,y are one-dimensional, then Ð γ2pE|∆q “ ó γ2pE|∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, the lower bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) is tight in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The proof follows from strong duality and is a variant of the proof in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 is not true in general, when Ex,y are not one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 Multi-Objective Version Since we consider Las Vegas complexity of each individual input, considering a single number as an output of an optimisation problem like (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) is too restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Here we define the multi-objective version of the same optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Additionally, we consider the version of the bound with multiple ∆, which corresponds to the multiple-oracle case of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For the latter, assume that M from Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 is decomposed as M “ Mp1q ‘ Mp2q ‘ ¨ ¨ ¨ ‘ Mpsq, and each ∆xy has a similar decomposition: ∆xy “ ∆p1q xy ‘ ∆p2q xy ‘ ¨ ¨ ¨ ‘ ∆psq xy (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) with ∆piq xy : Mpiq Ñ Mpiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Additionally, we write Vx : K Ñ M b W from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) as a vertical stack of matrices Vx “ ¨ ˚ ˚ ˚ ˚ ˝ V p1q x V p2q x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' V psq x ˛ ‹‹‹‹‚ with V piq x : K Ñ Mpiq b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We also generalise (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6) to such matrices: ~Vx~2 “ ´ }V p1q x }2, }V p2q x }2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , }V psq x }2¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 (Multi-objective unidirectional relative γ2 optimisation problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In notation of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 and the above assumptions on M and ∆, the multi-objective unidirectional relative γ2 optimisation problem Ð γ2pE|∆q “ Ð γ2pExy | ∆xyqx,yPD, is defined as follows: minimise p~Vx~2qxPD (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5a) subject to Exy “ V ˚ x p∆xy b IWqVy for all x, y P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5b) W is a vector space, Vx : K Ñ M b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5c) The bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) is equivalent to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) with the only difference in the objective, which justifies the use of the same notation Ð γ2pE|∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Later we will almost exclusively use the multi-objective version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the multiple-oracle case, we assume that the decomposition in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that it only changes the objective, and does not change the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Also, the constraint (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5b) in this case is equivalent to Exy “ sÿ i“1 ` V piq x ˘˚` ∆piq xy b IW ˘ V piq y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 21 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For a feasible solution Vx of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5), we call p~Vx~2qxPD the objective profile of the feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the single-oracle case, it is a vector in RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the multiple-oracle case, it is a vector in RD b Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The feasible objective space of the optimization problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) is the set of all objective profiles over all feasible solutions Vx of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If all Ex,y “ ex,y are one-dimensional, the feasible objective space of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) is a topologically closed subset of RD b Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is also true in general, but we only need the one-dimensional case, which we prove in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, for the multi-oracle case, we have the following variant of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, which binds the matrix Γ to the individual }V piq x }2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It can be used to prove trade-offs between input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For every feasible solution Vx to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) and every D ˆ D Hermitian matrix Γ, we have λmaxpΓ ˝ Eq ď sÿ i“1 λmaxpΓ ˝ ∆piqq max xPD ��V piq x ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Again, let V “ À xPD Vx and V piq “ À xPD V piq x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The proof follows the proof of Theo- rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 with the following change at the last step: λmaxpΓ ˝ Eq “ max v:}v}“1 v˚pΓ ˝ Eqv “ max v:}v}“1pV vq˚“ pΓ ˝ ∆q b IW ‰ V v “ max v:}v}“1 sÿ i“1 pV piqvq˚“ pΓ ˝ ∆piqq b IW ‰ V piqv ď sÿ i“1 λmaxpΓ ˝ ∆piqq max xPD ��V piq x ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 Properties Let us list some properties of the unidirectional relative γ2-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We are mostly interested in the case when the right-hand side ∆x,y is fixed, and the left-hand side ex,y is variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume that w and w1 are in the feasible objective spaces of optimization problems Ð γ2 ` ex,y|∆x,y ˘ x,yPD and Ð γ2 ` e1 x,y|∆x,y ˘ x,yPD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for all real c, c1 ě 0, the vector cw ` c1w1 is in the feasible objective space of Ð γ2 ` c1ep1q x,y ` c2ep2q x,y | ∆x,y ˘ x,yPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume that ` vx ˘ xPD is a feasible solution to Ð γ2 ` ex,y | ∆x,y ˘ x,yPD with objective profile w, and v1 x is defined similarly for w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, `?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='cvx‘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' c1v1 x ˘ xPD is a feasible solution to Ð γ2 ` cex,y` c1e1 x,y | ∆x,y ˘ x,yPD with objective profile cw ` c1w1 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8a) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will often have that ∆xx “ 0 for all x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In this case, it is easy to specify all families of ex,y that have a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume that ∆x,y in addition to ∆x,y “ ∆˚ y,x satisfy ∆x,x “ 0 for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let pex,yqx,yPD be any collection of complex numbers such that ex,y “ e˚ y,x for all x, y P D, and ex,y “ 0 whenever ∆x,y “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then the optimisation problem Ð γ2 ` ex,y | ∆x,y ˘ x,yPD has a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Due to Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9, it suffices to consider the case when there exist distinct x0, y0 P D such that ex0,y0 “ e˚ y0,x0 are the only non-zero ex,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By the assumption, ∆x0,y0 ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, there exist vectors u, v such that u˚∆x0,y0v “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Define the feasible solution as vx0 “ u, vy0 “ ex0,y0v, and vx “ 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Describing the set of ex,y that have feasible solution in the general case (when ∆x,x ‰ 0) is more complicated, and we do not do it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If ∆x,x “ 0 for all x, then the feasible objective space of Ð γ2pE|∆q is upwards closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=', if w P RD b Rs is in the feasible objective space, and w1 ě w (component-wise), then w1 is also in the feasible objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Vx be a feasible solution such that ~Vx~ “ wx for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Mx be pairwise orthogonal copies of M that are also orthogonal to MbW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' There exist V 1 x : K Ñ pMbWq‘Mx such that their projection to M b W agree to Vx and ~V 1 x~ “ w1 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' They also satisfy the constraints (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5b) with the properly enlarged W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, for x ‰ y this follows from the orthogonality of Mx and My, and for x “ y this follows from ∆x,x “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 7 Adversary Bound for State Conversion This is the central section of the paper, in which we define the adversary bound for state conver- sion with general input oracles, and prove that it equals Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, we restate the state conversion problem, define its Las Vegas complexity, and formulate the corresponding adversary optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, we explain the intuition behind the latter definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Sections 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 are devoted to the two main technical results: a lower bound for exact, and an upper bound for approximate state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' They are the corner- stones of what comes next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5, we prove an upper bound for exact state conversion, thus showing that the adversary bound is precisely equal to Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We finish the section with two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6, we consider a simple example of a state conversion problem with |D| “ 2, and in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7 we obtain the adversary bound of Boolean function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The main results in Sections 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 hold even for general linear input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5, we have to assume that the input oracles are unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 Definitions Our choice of problem for this section is state conversion with general input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The motivation for this initial choice is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, we want more control on the input oracle: we require that, for each x P D, we have only one input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Second, we would like to have larger flexibility on the side of the algorithm, that is why we choose the state conversion problem, where we have to map one state into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the beginning, we even do it approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Going to more specific tasks, like state generation, does not give us anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will extend the output and the input conditions to subspace conversion in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us give an explicit definition of state conversion, which follows from the general consid- eration of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 (State Conversion with General Input Oracles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let D be a set of labels, and M and K vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each x P D, let Ox : M Ñ M be a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A state conversion problem is given by a collection of tuples ξx ÞÑ τx where x ranges over D and ξx, τx P K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume that K is embedded in the space H of a quantum algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We 23 say that the algorithm A solves the state conversion problem ξx ÞÑ τx on input oracles Ox, if ApOxqξx “ τx for all x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Some of the results in this section hold even if we only assume that Ox are linear trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, we will usually assume that Ox are contractions or unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The lower bound result hold even for infinite D, but for the upper bounds it is crucial that D is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This definition also includes the case of multiple input oracles as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4), each Ox “ Op1q x ‘ Op2q x ‘ ¨ ¨ ¨ Opsq x with Opiq x acting in Mpiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We derive Las Vegas complexity of this problem from the general definition of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We study not only the worst-case complexity, but consider each input x P D individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 (Las Vegas complexity of State Conversion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume we have a state conversion problem is as in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, and an algorithm A that solves it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The Las Vegas complexity of the algorithm A on input x P D, is defined as LxpAq “ LpA, Ox, ξxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The worst-case Las Vegas complexity is defined as maxxPD LxpAq, in which case, we assume we have a single input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The complexity profile of the algorithm A is the vector in RD b Rs given by LDpAq “ pLxpAqqxPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The feasible complexity space of the problem is a subset of RD b Rs which is the set of all complexity profiles of the algorithms solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us now define the adversary optimisation problem corresponding to the state conversion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is a generalisation of the version of the adversary bound from [15] to the case of unidirectional input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 (Adversary Optimisation Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume ξx ÞÑ τx is a state conversion problem with unidirectional input oracles Ox : M Ñ M, as x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Its adversary optimisation problem is the following unidirectional γ2 optimisation problem: Ð γ2 ´ xξx, ξyy ´ xτx, τyy | IM ´ O˚ xOy ¯ x,yPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) Here IM stands for the identity on M, but we often omit this subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is easy to see that the constraints of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 are satisfied, and this is a legitimate unidirectional γ2- optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us write it down explicitly as we will be using it quite extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We consider it as a multi-objective optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' minimise ` ~vx~2˘ xPD (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2a) subject to xξx, ξyy ´ xτx, τyy “ @ vx, ppI ´ O˚ xOyq b IWqvy D for all x, y P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) W is a vector space, vx P M b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2c) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 Intuition Let us describe the intuition behind the bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For a collection of vectors pξxqxPD, let Gξ denote the corresponding Gram matrix: Gξrrx, yss “ xξx, ξyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Two collections of vectors can be transformed one into another by a unitary transformation if and only if they have the same Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since unitary transformations are free in quantum query algorithms, we may replace collections of vectors by the corresponding Gram matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, rather than saying that an algorithm solves state conversion ξx ÞÑ τx, we can say that it transforms Gξ into Gτ, or write Gτ ÞÑ Gξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, the left-hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) gives the difference of the corresponding Gram matrices Gξ ´ Gτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The right-hand side @ vx, ppI ´ O˚ xOyq b IWqvy D “ @ vx, vy D ´ @ pOx b IWqvx, pOy b IWqvy D (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) 24 gives the change in the Gram matrix when the state vx is processed by the oracle Ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The objective value ~vx~2 can be interpreted as the corresponding Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, the optimisation problem seeks for the best possible states vx to be processed by the oracle to get the required change in the Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The issue of how to get the states vx to the input oracle is ignored here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, one can see the adversary optimisation problem as a semi-definite relaxation of a quantum query algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To get the lower bound in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, we accumulate changes in the Gram matrix like in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) over all the queries made by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The proof closely follows the proof of Theorem 10 from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To get the algorithm in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4, we repeatedly apply a scaled down version of the query in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The result is that the Gram matrix slowly slides close to the line connecting Gξ to Gτ in the cone of D ˆ D semi-definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 Lower Bound (For Exact Version) Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume A is an algorithm that performs state conversion ξx ÞÑ τx with unidi- rectional access to general linear oracles Ox as x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, its complexity profile LDpAq is in the feasible objective space of the adversary optimization problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote for brevity ψt,x “ StpA, Oxqξx “ Ut´1 rOx Ut´2 rOx ¨ ¨ ¨ U1 rOxU0ξx, and let ψ1 t,x “ QtpA, Oxqξx be the state processed on step t by the input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The operators St and Qt are defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have xψ1,x, ψ1,yy “ xξx, ξyy, and ψT`1,x “ τx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This gives xξx, ξyy ´ xτx, τyy “ Tÿ t“1 ´ xψt,x, ψt,yy ´ xψt`1,x, ψt`1,yy ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Next, for the effect of one query rOx “ pOx b I‚q ‘ I˝: xψt,x, ψt,yy ´ xψt`1,x, ψt`1,yy “ xψt,x, ψt,yy ´ A rOxψt,x, rOyψt,y E “ @ ψ1 t,x, ψ1 t,y D ´ @ pOx b I‚qψ1 t,x, pOy b I‚qψ1 t,y D “ @ ψ1 t,x, ψ1 t,y D ´ @ ψ1 t,x, pO˚ xOy b I‚qψ1 t,y D “ @ ψ1 t,x, ppIM ´ O˚ xOyq b I‚qψ1 t,y D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) This means that we can take vx “ T à t“1 ψ1 t,x (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) as a feasible solution to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8b), ~vx~2 is equal to the Las Vegas complexity Lx, hence, this feasible solution has LDpAq as its objective profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The theorem is proven for exact and coherent state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, it can be used for approximate or non-coherent state conversion ξx ÞÑ τx as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, the latter is equivalent to exact coherent state conversion ξx ÞÑ τ 1 x for some τ 1 x satisfying the corresponding closeness requirements to τx as explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' See Section 10 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The map ξx ÞÑ vx is important enough, so that we introduce a special notation for it: VpA, Oq: ξ ÞÑ T à t“1 QtpA, Oqξ, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6) which is a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 Upper Bound (For Approximate Version) Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let ξx ÞÑ τx be a state conversion problem and Ox a general linear oracle, where x ranges over a finite set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume pvxqxPD is a feasible solution to the adversary optimization problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) with L “ maxxPD }vx}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for every ε ą 0, there exists an algorithm A with the following properties: it solves state conversion ξ` x ÞÑ τ ` x with unidirectional access to Ox, where ξ` x and τ ` x are some states (not necessarily in K) satisfying }ξ` x ´ ξx}, }τ ` x ´ τx} ď ε for all x P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' its Monte Carlo query complexity is T “ P L{ε2T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' for each x, its Las Vegas query complexity Lx is ~vx~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Specifically, the algorithm transforms ξ` x “ ξx ‘ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T vx ÞÝÑ τ ` x “ τx ‘ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T vx (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) in T queries, and the state processed by the input oracle on each query is vx{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote v1 x “ pOx b IWqvx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3), we obtain xξx, ξyy ´ xτx, τyy “ @ vx, ppI ´ O˚ xOyq b IWqvy D “ xvx, vyy ´ @ v1 x, v1 y D for all x, y P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is equivalent to @ v1 x, v1 y D ` xξx, ξyy “ xvx, vyy ` xτx, τyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This means that there exists a unitary transformation U satisfying Upv1 x ‘ ξxq “ vx ‘ τx for all x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us now describe the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It depends on an integer parameter T, which is also its query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Its space is of the form V ‘ K b J , where V is isomorphic to M b W and J is an T-qudit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Up to unitaries, the transformation in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) is equivalent to 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T |vx⟩V ` |ξx⟩K b ˆ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T Tÿ j“1 |j⟩J ˙ ÞÝÑ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T |vx⟩V ` |τx⟩K b ˆ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T Tÿ j“1 |j⟩J ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8) The algorithm performs this transformation by going through the states ψt,x “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T |vx⟩V ` |τx⟩K b ˆ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T t´1 ÿ j“1 |j⟩J ˙ ` |ξx⟩K b ˆ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T Tÿ j“t |j⟩J ˙ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9) just before the t-th query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that ψ1,x and ψT`1,x are the states on the left- and the right- hand sides of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the t-th query, apply the input oracle Ox b IW to the register V in ψt,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This results in the state 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T ˇˇv1 x � V ` |τx⟩K b ˆ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T t´1 ÿ j“1 |j⟩J ˙ ` |ξx⟩K b ˆ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T Tÿ j“t |j⟩J ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 26 Next, apply U to the space V ‘ K b |t⟩J , which gives ψt`1,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' After T iterations, we get the required transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The differences ξ` x ´ ξx and τ ` x ´ τx are both vx{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The norm of this vector is less than ε as long as T ě L{ε2, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, the Las Vegas complexity on input x is exactly T ¨ \u200c\u200c\u200c\u200c vx ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T \u200c\u200c\u200c\u200c 2 “ ~vx~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Figure 3 Gξ` Gξ Gτ ` Gτ Gτ 1 Visualisation of the algorithm A used in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm follows the straight line from Gξ` to Gτ `, as the Gram matrices of the states in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9) for various t are uniformly placed on this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The wiggly line indicates the application of A to Gξ in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm follows closely to the line connecting Gξ` and Gτ ` and terminates in a point Gτ 1 close to Gτ `, but not, generally, Gτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' An immediate corollary is that for contraction oracles we can replace ξ` x with the original ξx and get essentially the same bound on Monte Carlo complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume the premises of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5, where the input oracles Ox are con- tractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for every ε ą 0, there exists a quantum algorithm with Monte Carlo query complexity P 4L{ε2T that ε-approximately and coherently solves state conversion ξx ÞÑ τx with unidirectional access to Ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is a unidirectional version of the main technical result of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This version has slightly better dependence on ε, compared to [15], which had O ` ε´2 log 1 ε ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By the example due to Kothari [41], see [15], the dependence on ε is tight up to constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider the same algorithm A as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote by τ 1 x its final state when executed on the initial state ξx and the oracle Ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' See Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since Ox is a contraction, the whole algorithm ApOxq is a contraction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, }τ 1 x ´ τ ` x } “ }ApOxqξx ´ ApOxqξ` x } ď }ξx ´ ξ` x } ď ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since }τ ` x ´ τx} ď ε, the triangle inequality gives us }τx ´ τ 1 x} ď 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Dividing ε by 2, we get the required algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We also get a relation between Las Vegas and Monte Carlo complexities, which is a direct consequence of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume there is an algorithm that solves state conversion ξx ÞÑ τx with con- traction input oracles Ox exactly and has worst-case Las Vegas complexity L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, there exists an algorithm that ε-approximately and coherently solves the same problem and has Monte Carlo complexity OpL{ε2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 27 Since the distance }τx ´ τ 1 x} ď ε converts to error ε2 after measurement, it is reasonable to say that the complexity of the algorithm is inversely linear in the error, which is similar to the randomised case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is the result mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 Upper Bound For Exact Version The results of the previous two subsections are good enough for most purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5, we can take ξ` x and τ ` x as close to ξx and τx as we want and the Las Vegas complexity stays ~vx~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, in this section we improve this result and show how to perform exact state conversion ξx ÞÑ τx essentially in the same budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Together with the lower bound, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4, this shows that Las Vegas complexity is exactly equal to the adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is not only mathematically more satisfying and follows the convention of Las Vegas complexity to describe exact computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' One of the motivations behind this result is that a priori there is no good way to bind Las Vegas complexity on close initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If ξx and ξ` x are at a distance ε, then, from general principles, it can be deduced that their Las Vegas complexities differ by at most εT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' But this is useless because T generally is not bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This poses problems if, for example, we want to compose Las Vegas programs using Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 or 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 and we only have approximate versions of the subroutines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In this section, we extensively use the language of Gram matrices introduced in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Our first observation is that if both Gram matrices Gξ and Gτ are full rank, then state conversion can be performed exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume the premises of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If we additionally have that Gξ and Gτ are positive definite, then state conversion ξx ÞÑ τx can be solved exactly with Las Vegas complexity ~vx~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Moreover, the state processed by the oracle on each query is vx{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T, where T is the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since Gξ, Gτ ą 0, there exists an integer T such that Gξ, Gτ ě 1 T Gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Choose ξ´ x and τ ´ x so that their Gram matrices are Gξ´ “ Gξ ´ 1 T Gv and Gτ ´ “ Gτ ´ 1 T Gv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that, for all x, y P D: @ ξ´ x , ξ´ y D ´ @ τ ´ x , τ ´ y D “ xξx, ξyy ´ xτx, τyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, vx is a feasible solution to the adversary bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) for state conversion ξ´ x ÞÑ τ ´ x as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Applying Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 to the latter and using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7), we get an algorithm performing exact state conversion ξ1 x “ ξ´ x ‘ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T vx ÞÝÑ τ 1 x “ τ ´ x ‘ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T vx, where on each step the state vx{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' T is processed by the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' These collections of vectors have Gram matrices Gξ, and Gτ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, they can be turned into ξx and τx, respectively, by unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the illustration in Figure 3, the ξx and τx are equivalent to ξ` x and τ ` x , and the algorithm follows the straight line connecting Gξ` and Gτ `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The assumptions Gξ ą 0 and Gτ ą 0 are very strong, and almost never hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, the state generation problem has Gξ of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For (exact and coherent) Boolean function evaluation, the rank of Gτ is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, we will be able to apply this lemma by first “pushing” both Gξ and Gτ into the space of positive-definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' But for that we will need some additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, we assume that all Ox are unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Second, we get the point p}vx}2qxPD only as a limit of points in the feasible complexity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is because pushing Gξ and Gτ takes complexity, which can be made arbitrary small, 28 but cannot be made zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6, we will show that the above two assumptions are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, for simplicity we assume that all Ox are pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will lift this restriction in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us start with the question when a state conversion ξx ÞÑ τx is possible at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If there is an algorithm performing the transformation, we say that Gτ is achievable from Gξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote by Rξ the real affine space of D ˆ D Hermitian matrices A satisfying Arrx, xss “ }ξx}2 for all x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assuming that the input oracles Ox are unitary are pairwise distinct, we have: (a) If the state conversion ξx ÞÑ τx is possible, then Gτ P Rξ and Rτ “ Rξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (b) For any two M, M1 P Rξ, the optimisation problem Ð γ2 ` Mrrx, yss ´ M1rrx, yss | I ´ O˚ xOy ˘ x,yPD has a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The point (a) merely says that unitaries ApOxq do not change the norm of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The point (b) follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, in notation of the this proposition, ∆x,y “ I ´O˚ xOy “ 0 if and only if x “ y, and ex,x “ Mrrx, xss´M1rrx, xss “ 0 for all x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We can now describe our algorithm for exact state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let ξx ÞÑ τx be a state conversion problem with pairwise distinct unitary input oracles Ox, where x ranges over a finite set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume pvxqxPD is a feasible solution to the adversary optimization problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for every δ ą 0, there exists a quantum algorithm A with the following properties: A solves state conversion ξx ÞÑ τx exactly with unidirectional access to Ox;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' for each x P D, we have ��LxpAq ´ ~vx~2�� ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Together with Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 and Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7, this gives the following main result of the paper: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the assumptions of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10, the following two sets are equal: the topological closure of the feasible complexity space of the state conversion problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' and the feasible objective space of the corresponding adversary optimisation problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, we show that we can transform Gξ and Gτ into some M1, M2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As soon as we get into the space of full-rank Gram matrices, we can use Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Namely, we use it to perform the two middle steps in the following chain of transformations: Gξ ÞÑ p1 ´ εqGξ ` εM1 ÞÑ p1 ´ εqGξ ` εM2 ÞÑ p1 ´ εqGτ ` εM2 ÞÑ Gτ, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10) where ε ą 0 is some small number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' See Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The main work happens in the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We use Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 to show that the complexity of the other steps vanishes with ε Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The final idea is that we perform the last transformation in reverse using Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us proceed with the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We may assume that Gτ P Rξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Otherwise, neither the state conversion is possible, nor the optimisation problem has a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We may also assume that neither of ξx is zero, since then Gτ P Rξ implies τx “ 0 and any algorithm always transforms 0 ÞÑ 0, so we may drop this input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider the matrix M defined by Mrrx, yss “ # Gξrrx, xss “ Gτrrx, xss, if x “ y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11) 29 Clearly, M P Rξ and M ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9(b), the adversary bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) corresponding to the transformation Gξ ÞÑ M has a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Using Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 and continuity of the inner product, we can get Gram matrices achievable from Gξ that are arbitrarily close to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since M is positive definite, there exists a positive definite M1 among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let B be the algorithm that performs the transformation Gξ ÞÑ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Using the same argument but with ξx replaced by τx and Ox replaced by O˚ x, we get M2 ą 0 and an algorithm E that transforms Gτ ÞÑ M2 using the input oracles O˚ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9(a), both M1 and M2 are in Rξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' They are both positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By point (b) of the same lemma and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8, there exists a quantum algorithm C that transforms M1 ÞÑ M2 exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' These algorithms are depicted in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Figure 4 Gξ Gτ (a) M M1 M2 B C E Gξ Gτ (b) M M1 M2 Bε Cε Dε E´1 ε The algorithms used in the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The Gram matrices Gξ and Gτ are not of full rank here, therefore are depicted on the edge of the cone of positive semidefinite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The matrix M is positive definite, and so are all the matrices in a sufficiently small circle around M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In (b), the algorithms Bε and Cε are the scaled-down versions of B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm E´1 is additionally reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As ε Ñ 0, the algorithm Dε approaches the line connecting Gξ and Gτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For small enough ε ą 0, the following table lists the algorithms performing the transforma- tions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10) with their Las Vegas complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A graphical representation of the algorithms 30 is given in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Algorithm Transformation Complexity Bε Gξ ÞÑ p1 ´ εqGξ ` εM1 LxpBεq “ εLxpBq Cε p1 ´ εqGξ ` εM1 ÞÑ p1 ´ εqGξ ` εM2 LxpCεq “ εLxpCq Dε p1 ´ εqGξ ` εM2 ÞÑ p1 ´ εqGτ ` εM2 LxpDεq “ p1 ´ εq~vx~2 E´1 ε p1 ´ εqGτ ` εM2 ÞÑ Gτ LxpEεq “ εLxpEq The algorithm Bε is I ‘ B of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6, where B transforms εGξ ÞÑ εM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The complexity follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm Cε is analogous with C transform- ing εM1 ÞÑ εM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm E´1 ε is I ‘ E´1, where E´1 transforms εM2 ÞÑ εGτ due to Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To get Dε, note that both matrices are positive definite, and their difference is p1 ´ εqpGξ ´ Gτq, hence, we can use Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 with ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 ´ ε vx as a feasible solution to the corresponding adversary bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm A is the sequential composition of these subroutines, hence, by Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8, we have LxpAq “ εLxpBq ` εLxpCq ` p1 ´ εq~vx~2 ` εLxpEq Ñ ~vx~2 as ε Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 Example with Two Labels Here we consider an example when D “ t0, 1u and the input oracles O0 ‰ O1 are unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For normalized states, Gram matrices can be parametrized by a single off-diagonal parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We write Ga “ ˆ 1 a a˚ 1 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='12) Consider a transformation Ga ÞÑ Gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In other words, we have that xξ0, ξ1y “ a and xτ0, τ1y “ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The feasible objective space of the corresponding adversary optimisation prob- lem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) is the epigraph of a hyperbola: " pw0, w1q ˇˇˇ w0, w1 ě 0 and ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='w0w1 ě |a ´ b| }O0 ´ O1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, since O0 and O1 are unitaries, we get that Garrx, xss ´ Gbrrx, xss “ 0 “ xvx, ppI ´ O˚ i Oiq b IWqvxy for all vx and x “ 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' So we only have to analyse the off-diagonal term Garr0, 1ss ´ Gbrr0, 1ss “ a ´ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote d “ }I ´ O˚ 0O1} “ }O0 ´ O1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If v0, v1 is a feasible solution to the adversary optimisation problem, then from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b), we get |a ´ b| “ ��xv0, ppI ´ O˚ 0O1q b IWqv1y �� ď d }v0} ¨ }v1}, implying the lower bound in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 31 In the opposite direction, assume ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='w0w1 “ |a ´ b|{d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let u and v be the normalised left and right singular vectors of I ´ O˚ 0O1 with the singular value d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, we have a ´ b “ p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='w0uq˚pI ´ O˚ 0O1qpa ´ bq?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='w1v |a ´ b| , implying that pw0, w1q is in the feasible objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The claim follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11, the topological closure of the feasible complexity space of the correspond- ing state conversion problem equals (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We show that, in general, not all points in this set are attained as complexity profiles of the algorithms performing the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let O0 “ 1 and O1 “ ´1 be 1-dimensional unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider the transformation G1 ÞÑ Gi (where i is the imaginary unit) on these input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The point p1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2, 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2q is in the feasible objective space of the corresponding adversary optimisation problem, but not in the feasible complexity space of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The first statement follows from Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It remains to prove there is no algorithm solving the problem with this complexity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider an algorithm A that performs this transformation in T queries, and assume that it goes through the following Gram matrices during its execution: G1 ÞÑ Gc1 ÞÑ Gc2 ÞÑ ¨ ¨ ¨ ÞÑ GcT ´1 ÞÑ Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='14) First, we claim that only Gb with b P R are achievable from G1 in one query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, G1 corresponds to a state collection ξ0, ξ1 with ξ0 “ ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, the state processed by the input oracle is the same for x “ 0 and x “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote it ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For the states τ0 and τ1 after the query, we have xτ0, τ1y “ 1 ´ 2}ψ1}2 P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, among c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' , cT´1 there exists cj P Rzt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Write the algorithm as a sequential composition A “ C ˚ B, where B performs the transformation G1 ÞÑ Gcj in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='14), and C the transformation Gcj ÞÑ Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Using Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='12, we get that for any point pw0, w1q in the feasible objective space for transformation Ga ÞÑ Gb with our choice of input oracles w0 ` w1 ě 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='w0w1 ě |a ´ b|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4: L0pBq ` L1pBq ě |1 ´ cj| and L0pCq ` L1pCq ě |cj ´ i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Combining this with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 and the triangle inequality in C, we get L0pAq ` L1pAq “ L0pBq ` L0pCq ` L1pBq ` L1pCq ě |1 ´ cj| ` |cj ´ i| ą |1 ´ i| “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, p1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2, 1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2q is not in the feasible complexity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us now move to the case when O0 and O1 are contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Our goal is to show that Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11 are false in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For that, consider the transformation G1 ÞÑ G0 with the input oracles in C2 given by O0 “ ˆ 1 0 0 0 ˙ and O1 “ ˆ 0 ´1 0 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 32 Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For the above problem, the adversary optimisation problem has a feasible solution, but there is no algorithm performing the required transformation exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The feasible solution is v0 “ |0⟩ and v1 “ |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us prove there is no algorithm solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The initial Gram matrix G1 means that we have equal initial states ξ0 “ ξ1 of unit norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As G0 ‰ G1, the algorithm has to make at least one query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider the first query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since ξ0 “ ξ1, the state given to the oracle is the same for both inputs, denote it ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We may assume ψ1 ‰ 0, as otherwise this query can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let ψx be the state of the algorithm after the query on input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Without loss of generality, the algorithm is sliced, hence, ψ1 “ α|0⟩ `β|1⟩ for some α, β P C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As ψ1 ‰ 0, either α ‰ 0, or β ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If α ‰ 0, then ∥ψ1∥ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If β ‰ 0, then ∥ψ0∥ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In either case, it is impossible to get both terminal states τ0 and τ1 to have unit norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, there is no algorithm solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7 Boolean Function Evaluation Here, we derive the adversary bound for Boolean function evaluation as a simple special case of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us specify exactly what we mean by Boolean function evaluation in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let f : D Ñ t0, 1u with D Ď t0, 1un be a (partial) Boolean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We assume the input oracle Ox encodes x P D in the phase, and we consider the multi-oracle settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' That is, there are n unitary input oracles Opiq x : C Ñ C defined by Opiq x “ p´1qxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since Ox is Hermitian, there is no difference between unidirectional and bidirectional access to this oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We also assume that the function is evaluated in the phase, exactly and coherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' That is, the output space K “ C and the goal is to map |0⟩ ÞÑ p´1qfpxq|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since the space is one-dimensional, this can also be seen as an instance of unitary implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We get that xξx, ξyy ´ xτx, τyy “ 2 ¨ 1fpxq‰fpyq, where 1P is the indicator variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Similarly, I ´ O˚ xOy “ 2 Àn j“1 1xj‰yj, where À is a direct sum of n matrices, each of size 1 ˆ 1, resulting in an n ˆ n diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Dividing by 2, we get the adversary optimisation problem Ð γ2 ´ 1fpxq‰fpyq ˇˇ n à j“1 1xj‰yj ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='15) It is similar to the corresponding expression in [15], except that it is unidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Section 9, we will show that since Ox “ O˚ x, the unidirectional version is equal to the usual bidirectional relative γ2-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The multi-objective optimisation problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='15) exactly characterises the Las Vegas com- plexity of each of the n individual input symbols on every input x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is also possible to substitute the input oracle with the one that encodes xj in the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Namely, with the oracle rOpiq x : C2 Ñ C2 given by |b⟩ ÞÑ |b ‘ xj⟩, where ‘ is XOR here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, in the Fourier basis, rOpiq x “ I1 ‘ Opiq x , hence, the algorithm can just ignore the I1 part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The same holds for the output, if we require that the algorithm has to perform the transformation |b⟩ ÞÑ |b ‘ fpxq⟩ in the output space K “ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 8 Subspace Conversion In Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, we continue the settings of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 but without the assumption that input oracles are pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This leads to our investigation of the linear consistency of feasible 33 solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The corresponding problem can be formulated as subspace conversion, which we analyse in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, define the corresponding notion of complexity and extend the connection to the adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, we revisit the functional composition property of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The notion of complexity we introduced for the subspace conversion problem will allow us to formulate and prove a simpler estimate on the complexity of the composed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 Linear Consistency Throughout the section, we assume we have a state conversion problem ξx ÞÑ τx in K with input oracles Ox, where x ranges over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will be particularly interested in pairs of inputs x, y with Ox “ Oy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For O P tOx | x P Du, let DO “ tx P D | Ox “ Ou and KO “ spantξx | x P DOu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The important point is that the algorithm performs the same linear transformation ApOq on all x P DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, the pairs ξx ÞÑ τx for x P DO should be linearly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The adversary optimisation problem is in accord with this requirement as shown in the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume that the adversary optimisation problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) for a state conversion problem ξx ÞÑ τx with contraction oracles Ox has a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for each O, there exists a linear transformation TO : KO Ñ K such that τx “ TOξx for all x P DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Moreover, if O is unitary, TO is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Fix O, and restrict the optimisation problem to DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since O is a contraction, I ´ O˚O exists semi-definite, and S “ ppI ´ O˚Oq b IWq1{2 is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b), we have xξx, ξyy ´ xτx, τyy “ @ vx, ppI ´ O˚Oq b IWqvy D “ @ Svx, Svy D for all x, y P DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, there exists a unitary that maps ξx ÞÑ τx ‘ Svx for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, the mapping TO : ξx ÞÑ τx is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If O is unitary, then S “ 0, and TO is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that the latter result is false for general linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, if O “ 2I it is easy to construct a feasible solution for a non-linear state conversion 0 ÞÑ |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This does not contradict Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 though, because there the initial state is perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Effects like this is the main reason why we focus on contraction oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us also consider linear consistency of feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 (Linear consistency of feasible solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We say that a feasible solution vx to the adversary optimisation problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) is linearly consistent if, for each O, there exists a linear transformation VO : KO Ñ M b W such that vx “ VOξx for all x P Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' One way to ensure this condition is to impose the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 (Linear independence assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We say that a state conversion problem satisfies the linear independence assumption if, for each O, the vectors in tξx | x P DOu are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Under this assumption, we are losing nothing in relation to the transformation performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each O, we can uniquely extend this state conversion to all ξ P KO by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will call the latter the linearly extended state conversion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have (a) Any feasible solution obtained via Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 is linearly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 34 (b) Linear independence assumption implies linear consistency for all feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (c) Moreover, under linear independence assumption, any feasible solution can be uniquely extended to a linearly consistent feasible solution to the linearly extended state conversion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For Point (a), use VO “ VpA, Oq from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Point (b) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Point (c) follows by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let xi range over DO and yj over DO1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If @ ξxi, ξyj D ´ @ τxi, τyj D “ @ vxi, ppI ´ O˚O1q b IWqvyj D for all xi and yj, then Aÿ i aiξxi, ÿ j bjξyj E ´ Cÿ i aiτxi, ÿ j bjτyj G “ Aÿ i aivxi, ppI ´ O˚O1q b IWq ÿ j bjvyj E for all complex ai and bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Contrary to Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4(a), feasible solutions to the adversary optimisation problem need not satisfy linear consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For example, let D “ t0, 1, `, ´u, K be a qubit, ξ0 “ |0⟩, ξ1 “ |1⟩, ξ` “ p|0⟩ ` |1⟩q{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2, and ξ´ “ p|0⟩ ´ |1⟩q{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We let Ox “ I and τx “ ξx for all x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This problem in trivially solvable in 0 queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' But the following is a feasible solution to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1), which is not linearly consistent: v0 “ v1 “ 0, and v` “ v´ “ |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This poses a problem for strengthening Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The feasible solution above gives us a point p0, 0, 1, 1q in the feasible objective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the other hand, by the parallelogram identity, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, we have that for every algorithm A: L`pAq ` L´pAq “ L0pAq ` L1pAq, implying that the point p0, 0, 1, 1q is not in the topological closure of the feasible complexity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' One can say that this example is artificial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' There is no need to deteriorate the solution v0 “ v1 “ v` “ v´ “ 0 by increasing v` and v´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The following result states that this is a general observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Recall that a solution to a multi-objective optimisation problem is called Pareto optimal if it is not strictly dominated by any other solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In our case that means that there is no other feasible solution v1 x to the same optimisation problem such that ~v1 x~2 ď ~vx~2 for all x and ~v1 x~2 ă ~vx~2 for some x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Any Pareto optimal solution to the adversary optimisation problem with contraction input oracles is linearly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let vx be a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Take any O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will show that either vx is linear consistent on DO, or there is a feasible solution that strictly dominates vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By an argument like in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4(b) and (c), there exists a feasible solution v1 x that is linearly consistent on DO and equal to vx outside of DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' That is, there exists a linear map V 1 : KO Ñ M b W such that v1 x “ V 1ξx for all x P DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Recall the conditions (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b): xξx, ξyy ´ xτx, τyy “ @ vx, ppI ´ O˚ xOyq b IWqvy D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, let us consider these constraints for x P DO and y R DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (The constraints with x R DO and y P DO are equivalent to these ones due to the symmetry imposed on a unidirectional relative γ2-optimisation problem, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Π1 denote the projector onto the span of 35 ppI ´ OOyq b IWqvy as y ranges over DzDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The constraints (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) are linear in vx and define Π1vx uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, Π1vx “ Π1v1 x, and the mapping ξx ÞÑ Π1vx is linear for x P DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Next, consider the constraints (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) for x, y P DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Similarly to the proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, the operator pI ´ O˚Oq b IW is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let S “ ppI ´ O˚Oq b IW q1{2 and Π2 denote the projector onto its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We claim that the mapping ξx ÞÑ Π2vx is linear on DO as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Indeed, for x, y P DO, we have: @ Sv1 x, Sv1 y D “ xξx, ξyy ´ xτx, τyy “ xSΠ2vx, SΠ2vyy Hence, there is a unitary U that maps Sv1 x ÞÑ SΠ2vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, Π2vx “ S`USV 1ξx, where S` is the Moore-Penrose pseudoinverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Π denote the projector onto the span of Π1 and Π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since both ξx ÞÑ Π1vx and ξx ÞÑ Π2vx are linear on DO, the same holds for ξx ÞÑ Πvx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If we replace vx by Πvx for each x P DO, we get a feasible solution that is linearly consistent on DO and dominates vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, by imposing the linear consistency condition of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 we are only losing solutions where some of the objectives ~vx~2 are artificially inflated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the following, we will only consider linearly consistent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4(c), we may assume the problem satisfies the linear independence condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have the following generalisation of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let ξx ÞÑ τx be a state conversion problem with unitary oracles Ox, where x ranges over a finite set D, and which satisfies the linear independence condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume pvxqxPD is a feasible solution to the adversary optimization problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1), and let TO and VO be like in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 and Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for every δ ą 0, there exists a quantum algorithm A with the following properties: for every O, and ξ P KO, A transforms ξ ÞÑ TOξ on input oracle O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' moreover, ��LpA, O, ξq ´ ~VOξ~2�� ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The proof is a modification of the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Redefine Rξ as the real affine space of DˆD Hermitian matrices A satisfying Arrx, yss “ xξx, ξyy for all x, y P D with Ox “ Oy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The two points of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9 still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The matrix M is defined by Mrrx, yss “ # Gξrrx, yss “ Gτrrx, yss, if Ox “ Oy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) Again, M P Rξ, as well as M ą 0, since it is a block-diagonal matrix whose blocks are Gram matrices of linearly independent collections of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Other than that, the algorithm is exactly the same as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The algorithm transforms ξx ÞÑ τx on the input oracle O for all x P DO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By linearity, it transforms ξ ÞÑ TOξ for all ξ P KO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The same linearity property holds for all queries made by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, the Las Vegas query complexity of the Dε subroutine of the algorithm when the A is executed on the initial state ξ P KO is p1 ´ εq~VOξ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The complexities of other subroutines tend to zero as ε Ñ 0, which gives the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 Subspace Conversion Problem As one can see, what Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 actually solves is the subspace conversion problem from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us restate the problem assuming general input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 36 Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7 (Subspace Conversion, Restated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let D be a set of labels, and M and K be vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each x P D, let Ox : M Ñ M be a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A subspace conversion problem is given by a collection of linear maps Tx : Kx Ñ K with Kx Ď K, where x ranges over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume that K is embedded in the space H of a quantum algorithm A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We say that the algorithm A solves the subspace conversion problem Tx on input oracles Ox, if, for every x P D, the map ApOxq agrees with Tx on Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We define the complexity LxpAq on the input x as the supremum of LpA, O, ξq as ξ ranges over the unit vectors in Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The remaining complexity-related definitions are as in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the case of a single input oracle, the supremum in the definition of LxpAq is just the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the case of multiple input oracles of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2, the supremum is understood with respect to the dominance relation u ď v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In other words, it is the entry-wise maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that it is not true, in general, that there exists a unit ξ P Kx such that LxpAq “ LpA, Ox, ξq, since different input oracles can attain their maxima at different ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The corresponding adversary optimisation problem is as follows: Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8 (Adversary for Subspace Conversion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider a subspace conversion problem as defined above with input oracles Ox, and let Kx be the projector onto Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The corresponding adversary optimisation problem is given by Ð γ2 ` K˚ xKy ´ T ˚ x Ty | I ´ O˚ xOy ˘ x,yPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) Note that while (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) states that Vx : K Ñ MbW, in this optimisation problem we actually have Vx : Kx Ñ M b W, as Tx is defined on Kx, and the coimage of Kx is Kx as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, for the state conversion problem, where each Kx is one-dimensional, we get back the definition from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the other extreme, for the unitary implementation problem, where Kx “ K for all x, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) reads as Ð γ2 ` I ´ T ˚ x Ty | I ´ O˚ xOy ˘ x,yPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 can be reformulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Tx : Kx Ñ K be an isometric subspace conversion problem with unitary input oracles Ox and finite set of labels D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, the topological closure of the feasible complexity space of this problem coincides with the feasible objective space of the corresponding adversary optimisation problem (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If Vx is a feasible solution, then by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5b): K˚ xKy ´ T ˚ x Ty “ V ˚ x ppI ´ O˚ xOyq b IWqVy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Multiplying by ξ˚ on the left and ξ1 on the right gives @ Kxξ, Kyξ1D ´ @ Txξ, Txξ1D “ @ Vxξ, ppI ´ O˚ xOyq b IWqVyξ1D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) Hence, Vxξ is a feasible solution to the adversary optimisation problem of state conversion ξ ÞÑ Txξ with oracles Ox as x ranges over D and ξ over Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The theorem follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In other words, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) is just a way to write down a linearly consistent feasible solution to an adversary optimisation problem, where Vx acts like VO in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The objective ~Vx~2 is then the largest (entry-wise) complexity on the oracle Ox as ξ ranges over unit vectors in Kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 37 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 Composition, Revisited Here we revisit the composition properties from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 using the notions from Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Tx : Nx Ñ N be a subspace conversion problem as x ranges over D, and B be an algorithm solving this problem on input oracles Ox : M Ñ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each x P D, let O1 x : N Ñ N agree with Tx on Nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let ξx ÞÑ τx be a state conversion problem with the input oracles O1 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume that a sliced algorithm A solves this problem and has the following property: @x P D @t: QtpA, O1 xqξx P Nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) We can estimate the complexity of the composed algorithm as the product of complexities of its constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Under the above assumption, the composed algorithm A ˝ B defined in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10 solves the state conversion problem ξx ÞÑ τx with input oracles Ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Moreover, LxpA ˝ Bq ď LxpAqLxpBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) and the fact that B solves the subspace conversion problem, we have that O1 x and BpOxq satisfy the condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, we can replace the input oracle O1 x of A by BpOxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The first statement then follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Concerning complexity, we have: LpA ˝ B, Ox, ξxq “ ÿ t L ´ B, Ox, Qt ` A, BpOxq ˘ ξx ¯ “ ÿ t L ´ B, Ox, Qt ` A, O1 x ˘ ξx ¯ ď LxpBq ÿ t ��Qt ` A, O1 x ˘��2 “ LxpBqLxpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Here, we used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) on the first step, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7 on the second step, and the definition of LxpBq and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 on the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the above proposition it is assumed that the algorithm A has a single input oracle (while B can have multiple input oracles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Similarly, it is possible to get an analogue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Again, assume A has multiple input oracles Opiq : N piq Ñ N piq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each i, let T piq x : N piq x Ñ N piq be a subspace conversion problem with x ranging over D, and Bpiq be an algorithm that solves the above problem with input oracle Ox : M Ñ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9, the algorithm B “ À i Bpiq solves state conversion Tx “ À i T piq x : Nx Ñ N, where Nx “ À i N piq x and N “ À i N piq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let, for each x P D and i, O1 x piq be a linear map on N piq that agrees with T piq x on N piq x , and denote O1 x “ À i O1 x piq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, using a similar estimate as in the proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10, but with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6) instead of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5), we get: LxpA ˝ Bq ď ÿ i LxpAqrriss ¨ LxpBpiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Above we assumed for simplicity that all the subspace conversion problems T piq x have the same set of labels D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If the i-th problem has the set of labels Dpiq, it is possible to take D as the Cartesian product D “ ś i Dpiq or some subset thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 38 9 Bidirectionality In this section, we consider aspects specific to bidirectional access to the input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, we show how one can obtain the main results from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As mentioned in the introduction, bidirectional case is just a special case of the unidirectional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each state conversion problem with unitary input oracle pOxqxPD, the feasible complexity space assuming bidirectional access to the oracle Ox coincides to the fea- sible complexity space assuming unidirectional access to the oracle Ox ‘ O˚ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Moreover, the corresponding Monte Carlo complexities differ at most by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Each algorithm A with bidirectional access to Ox can be simulated with unidirectional access to Ox ‘ O˚ x by using the parts Ox and O˚ x of the oracle to process direct and reverse queries of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Both Monte Carlo and Las Vegas complexities do not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the other hand, if A has unidirectional access to Ox ‘ O˚ x, it can be simulated with bidirectional access to Ox by first processing the Ox-part with the direct query, and then the O˚ x-part with the reverse query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Las Vegas complexity does not change, and the Monte Carlo complexity grows by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us define the (bidirectional) relative γ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We start with the single-objective version, which is the version used in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 (Bidirectional relative γ2-bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let K, and M be vector spaces, and D be a set of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let E “ tExyu and ∆ “ t∆xyu, where x, y P D be two families of linear operators: Axy : K Ñ K and ∆xy : M Ñ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The relative γ2-norm Ø γ2pE|∆q “ Ø γ2pExy | ∆xyqx,yPD, is defined as the optimal value of the following optimisation problem, where Ux and Vx are linear operators, minimise maxxPD maxt∥Ux∥2, ∥Vx∥2u (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1a) subject to Exy “ U ˚ x p∆xy b IWqVy for all x, y P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1b) W is a vector space, Ux, Vx : K Ñ M b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1c) The one-dimensional version is : minimise maxxPD maxt∥ux∥2, ∥vx∥2u (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2a) subject to exy “ @ ux, p∆xy b IWqvy D for all x, y P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) W is a vector space, ux, vx P M b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2c) The relative γ2-norm can be also defined in terms of the unidirectional γ2-bound as Ø γ2pExy | ∆xyqx,yPD “ Ð γ2p rExy | r∆xyqx,yPDYD1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) Here D1 “ tx1 | x P Du is a disjoint copy of D, and rE and r∆ are defined as rEx,y1 “ rE˚ x1,y “ Ex,y, rEx,y “ rEx1,y1 “ 0, r∆x,y1 “ r∆˚ x1,y “ ∆x,y, r∆x,y “ r∆x1,y1 “ 0 for all x, y P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This instantly gives a dual for the the one-dimensional version of the bound, which was already proven in [15]: 39 Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The optimal value of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) is equal to the optimal value of the following opti- mization problem: maximise }Γ ˝ E} (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4a) subject to }Γ ˝ ∆} ď 1, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4b) where Γ ranges over D ˆ D matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Use the above representation, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4, and the fact that }A} “ λmax ˆ 0 A A˚ 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Now let us move to the connection between unidirectional and bidirectional oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1, unidirectional access to Ox is equivalent to bidirectional access to Ox ‘ O˚ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The following proposition shows that we can substitute unidirectional γ2 bound with oracle Ox ‘ O˚ x with bidirectional γ2-norm with oracle Ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let Ox “ Op1q x ‘¨ ¨ ¨‘Opsq x be unitary oracles as x ranges over D, and assume that ey,x “ e˚ x,y and ex,x “ 0 are complex numbers for all x, y P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider the following two optimization problems Ø γ2pex,y | I ´ O˚ xOyqx,yPD and Ð γ2 ` ex,y | I ´ pO˚ xOy ‘ OxO˚ yq ˘ x,yPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, for every collection pLxqxPD, with Lx P Rs, the following statements are equivalent: (a) there exists a feasible solution ux, vx to the first optimization problem with Lx “ p~ux~2 ` ~vx~2q{2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (b) there exists a feasible solution ux, vx to the first optimization problem with Lx “ ~ux~2 “ ~vx~2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (c) there exists a feasible solution ˜vx to the second optimization problem with Lx “ ~˜vx~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First, let us prove paq ñ pcq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume we have a feasible solution ux, vx to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) with ∆x,y “ I ´ O˚ xOy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us denote rOx “ Ox b IW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, we have ex,y “ A ux, pI ´ rO˚ x rOyqvy E ùñ ex,y “ xux, vyy ´ A rOxux, rOyvy E , and ey,x “ A uy, pI ´ rO˚ y rOxqvx E ùñ ex,y “ xvx, uyy ´ A rOxvx, rOyuy E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Consider the following two equalities � ux ` vx, ` I ´ rO˚ x rOy ˘ puy ` vyq � “ xux, uyy ` xux, vyy ` xvx, uyy ` xvx, vyy ´ A rOxux, rOyuy E ´ A rOxux, rOyvy E ´ A rOxvx, rOyuy E ´ A rOxvx, rOyvy E , and � rOxux ´ rOxvx, ` I ´ rOx rO˚ y ˘ p rOyuy ´ rOyvyq � “ A rOxux, rOyuy E ´ A rOxux, rOyvy E ´ A rOxvx, rOyuy E ` A rOxvx, rOyvy E ´ xux, uyy ` xux, vyy ` xvx, uyy ´ xvx, vyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 40 Hence, ˜vx “ pux ` vxq ‘ p rOxux ´ rOxvxq 2 is a feasible solution to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) with ∆x,y “ I ´ pO˚ xOy ‘ OxO˚ yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have ~˜vx~2 “ \u200c\u200cux ` vx \u200c\u200c2 ` \u200c\u200c rOxux ´ rOxvx \u200c\u200c2 4 “ \u200c\u200cux ` vx \u200c\u200c2 ` \u200c\u200cux ´ vx \u200c\u200c2 4 “ \u200c\u200cux \u200c\u200c2 ` \u200c\u200cvx \u200c\u200c2 2 , where we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8a), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8b), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8c) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='9), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This proves that paq ñ pcq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Now let us prove pcq ñ pbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume ˜vx is a feasible solution to the second optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let ˜v1 x and ˜v2 x be the parts of ˜vx processed by I ´ O˚ xOy and I ´ OxO˚ y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Set ux “ ˜v1 x ‘ rO˚ x˜v2 x and vx “ ˜v1 x ‘ r´ rO˚ x˜v2 xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, xux, ppI ´ O˚ xOyq b pIW ‘ IWqqvyy “ A ˜v1 x, pI ´ rO˚ x rOyq˜v1 y E ` A rO˚ x˜v2 x, p rO˚ x rOy ´ Iq rO˚ y ˜v2 y E “ A ˜v1 x, pI ´ rO˚ x rOyq˜v1 y E ` A ˜v2 x, pI ´ rOx rO˚ yq˜v2 y E “ ex,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Again, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8b) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='8c), ~ux~ “ ~vx~ “ ~˜vx~.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This proves pcq ñ pbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The remaining implication pbq ñ paq is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, we can make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The multi-objective bidirectional relative γ2-optimisation problem Ø γ2 ` ex,y | ∆x,y ˘ x,yPD is defined as minimise ´~ux~2 ` ~vx~2 2 ¯ xPD (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5a) subject to exy “ @ ux, p∆xy b IWqvy D for all x, y P D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5b) W is a vector space, ux, vx P M b W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5c) Alternatively, one may substitute (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5a) with minimise ´ max ␣ ~ux~2, ~vx~2(¯ xPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 (Bidirectional Adversary Optimisation Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume ξx ÞÑ τx is a state conversion problem with bidirectional input oracles Ox : M Ñ M, as x P D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Its adversary optimisation problem is Ø γ2 ´ xξx, ξyy ´ xτx, τyy | IM ´ O˚ xOy ¯ x,yPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6) An important corollary is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assuming bidirectional access, the adversary bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) can be replaced with the corresponding bidirectional version (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6), and the results of the corresponding Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='10, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='11, and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 41 For instance, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6 after this transformation is the main technical result from [15] with slightly better dependence on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' And the adversary bound for Boolean function evalua- tion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='15) equals Ø γ2 ´ 1fpxq‰fpyq ˇˇ n à j“1 1xj‰yj ¯ , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) which is equivalent to the known bound from [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The corresponding dual (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) is the lower bound from [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 10 Unitary Permutation Inversion The goal of this section is to prove a separation between unidirectional and bidirectional access to an oracle on a natural problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will achieve this using the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1 (Unitary Permutation Inversion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The set of labels is the set of permutations on n elements D “ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For each π P Sn, let Oπ : Cn Ñ Cn be the input oracle defined by Oπ|i⟩ “ |πpiq⟩ for all i P rns The task is to find π´1p1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' First note that this problem is different from the usual permutation inversion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In the latter, the permutation π is encoded using the standard input oracle |i⟩|b⟩ ÞÑ |i⟩|b ‘ πpiq⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The latter is a well-known problem, first defined in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is similar to Grover’s search, but different enough to complicate direct reductions from the lower bound for unstructured search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis [1] gave a tight lower bound of Ωp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Nayak [49] gave a direct reduction from unstructured search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' See also a recent paper by Rosmanis [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since the unidirectional and bidirectional access are equivalent for standard oracle, we resort to the unitary oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The reason of requiring π to be a permutation is solely to ensure that Oπ is a unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The problem can be trivially solved in one query with bidirectional access: apply O˚ π to |1⟩ and read out the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Since unitary inversion using the standard oracle requires Ωp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='nq queries, this means that the unitary permutation oracle |i⟩ ÞÑ |πpiq⟩ cannot be simulated by the standard oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Intuitively, it seems the problem should be hard for unidirectional input oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We show that this is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Any quantum query algorithm solving the unitary inversion problem (with bounded error and non-coherently) with unidirectional access to the input oracles has to make Ωp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='nq queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note, however, that there is no matching upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Grover’s search cannot be directly applied here because of the unidirectional access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The remaining part of this section is devoted to the proof of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The proof relies on Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, and we have to find the adversary matrix Γ from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Interestingly, the analysis is a variant of the usual positive-weighted adversary, but it is different from the one used by Ambainis in the lower bound proof of the usual permutation inversion problem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We need the following technical result, which was used [58] to reduce the combinatorial formulation of the positive-weighted adversary like in [1] to the spectral formulation as in [9, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We give a slightly modified version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let A be a matrix with entries 0, ˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, }A} ď max i,j : Arri,jss‰0 a RiCj, where Ri and Cj is the number of non-zero elements in the i-th row and j-column, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 42 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Taking the absolute value of each entry can only increase the norm, hence, we can assume the matrix A only has 0,1 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, this is a special case of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2 of [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Assume |0⟩ ÞÑ |τπ⟩ is a state-generating problem such that measuring τπ gives π´1p1q with probability at least 2{3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We use this property to ensure that Rexτπ, τσy ď 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2 3 for π, σ P Sn such that π´1p1q ‰ σ´1p1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) Define the corresponding output object, which is an Sn ˆ Sn-matrix E with Errπ, σss “ 1 ´ xτπ, τσy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us define the adversary matrix Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Denote by Cn the subset of Sn formed by permutations having a single cycle of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We will only consider permutations in Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We say that π, σ P Cn are in relation, denoted π ú σ, if π and σ have cyclic structures of the following form: π: 1 ÞÑ ¨ ¨ ¨ ÞÑ pk ÞÑ pk`1 ÞÑ ¨ ¨ ¨ pℓ ÞÑ pℓ`1 ÞÑ ¨ ¨ ¨ pn ÞÑ 1, σ: 1 ÞÑ ¨ ¨ ¨ ÞÑ pk ÞÑ pℓ`1 ÞÑ ¨ ¨ ¨ pn ÞÑ pk`1 ÞÑ ¨ ¨ ¨ pℓ ÞÑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) for some 1 ď k ă ℓ ă n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In other words, the interval pk`1 ÞÑ ¨ ¨ ¨ ÞÑ pℓ is taken out and put at the end of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Alternatively, one can say that the suffix pk`1 ÞÑ ¨ ¨ ¨ ÞÑ pn is cyclically shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This is a symmetric relation, but neither reflexive, nor transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As usual for the positive-weighted adversary, define an CnˆCn matrix Γ by Γrrπ, σss “ 1πúσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have the following properties of the matrix Γ: Γ is a Hermitian matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Γrrπ, σss “ 0 if π´1p1q “ σ´1p1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' λmaxpΓq “ Ωpn2q with the principal eigenvector given by the all-1 vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' λmaxp´Γq ď n ´ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The first two properties follow from the definition of the relation: If π ú σ, then σ ú π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Also, in this case, π´1p1q ‰ σ´1p1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The third property follows from the fact that each row has exactly pn ´ 1qpn ´ 2q{2 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Now, let us prove the fourth property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is equivalent to pn ´ 2qI ` Γ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us prove the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Fix 1 ď k ď n ´ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Say that π „k σ if π “ σ or π and σ are in relation like in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) with this fixed value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Note that „k is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Define the matrix Γk by Γkrrπ, σss “ 1π„kσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is a block-diagonal matrix with all-1 blocks on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, Γk ě 0, which gives n´2 ÿ k“1 Γk “ pn ´ 2qI ` Γ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let u be the normalised all-1 vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, λmaxpΓ ˝ Eq ě u˚pΓ ˝ Equ ě ˆ 1 ´ 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 2 3 ˙ u˚Γu “ Ωpn2q, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) where we used the second point of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 and (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) on the second step, and the third point of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 on the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 43 It remains to estimate Γ ˝ ∆, where ∆π,σ “ I ´ O˚ πOσ “ I ´ Oπ´1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By the definition of Γ, we can restrict our attention to the pairs π, σ, which are in relation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In notation of (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2), we have that π´1σ is a single cycle of length 3 π´1σ: pk ÞÑ pℓ ÞÑ pn ÞÑ pk and identity elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, ∆π,σ “ ¨ ˝ 1 0 ´1 ´1 1 0 0 ´1 1 ˛ ‚ (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) where the rows and columns are labelled by pk, pℓ, pn in this order, and the matrix has zeroes everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The matrix Γ ˝ ∆ is labelled by the elements pπ, iq P Cn ˆ rns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The block (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) when embedded in the latter has rows pπ, pkq, pπ, pℓq, pπ, pnq and columns pσ, pkq, pσ, pℓq, pσ, pnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We would like to apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 to Γ ˝ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For instance, we see that there are at most n choices of ρ P Cn such that ∆π,ρ has non-zero elements in row pπ, pkq, since pk has to be one of the two elements used to define the relation π ú ρ for this to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Similarly, in notation of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3, we get the following estimates: Rπ,pk, Rπ,pℓ, Cσ,pk, Cσ,pn ď 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5a) For one row and one column we get a worse estimate, where we count the total number of ρ in relation with π (or σ, respectively): Rπ,pn, Cσ,pℓ ď 2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5b) Therefore, we should treat the element on the intersection of the latter row and the latter column separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Rewrite (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4): ∆π,σ “ ¨ ˝ 1 0 ´1 ´1 1 0 0 ´1 1 ˛ ‚“ ¨ ˝ 1 0 ´1 ´1 1 0 0 0 1 ˛ ‚` ¨ ˝ 0 0 0 0 0 0 0 ´1 0 ˛ ‚, Let us denote the first and the second matrices in the last sum by ∆1 π,σ and ∆2 π,σ, respectively, and the corresponding families by ∆1 and ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' }Γ ˝ ∆1} “ Opn3{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This follows from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3 using the estimates in (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' λmaxpΓ ˝ ∆2q “ Opnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The matrix Γ ˝∆2 is just the matrix ´Γ where the row and the column label π becomes ` π, π´1p1q ˘ and the matrix is extended by zeroes elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hence, the claim follows from the fourth point of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Combining Claims 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6, we get that λmaxpΓ ˝ ∆q “ Opn3{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Together with (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3), this gives the required lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 44 11 Discussion and Future Work In this paper, we defined a natural notion of Las Vegas complexity, and demonstrated its versatility for various composition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We proved that a Las Vegas algorithm can be turned into an approximate Monte Carlo algorithm with a slight increase in complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have shown that Las Vegas complexity is equal to the adversary bound for exact state and subspace conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The latter is exciting as the same object is shown to have two different facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For some problems, intuition gathered from quantum algorithms might be helpful in coming up with good Las Vegas algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For other problems, it might be easier to forget about limitations of quantum algorithms and work directly with optimisation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Our algorithm of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4 can be seen as a way to guess arbitrarily large states to process by the input oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' It is interesting to understand consequences of such a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Due to its exactness, Las Vegas complexity results in “cleaner” algorithms without necessity to worry about error reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Similar results have been already obtained for function evalu- ation using compositional properties of the adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' But having “clean” subroutines for various state-generating and state-converting problems might be helpful as well, especially, given that they not always have built-in tools for error reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This paper should be seen as a prequel to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [15] since it gives a more general and simple exposition of the first half of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [15], but mostly ignores the second half, which deals with applications to function and relation evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Complete reconciliation of the results from [15] with the current paper is left as important future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us just mention two results that can be easily obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Purifiers of [15] imply that, assuming bidirectional access, a Monte Carlo algorithm for approximate and non-coherent function evaluation can be turned into an exact coherent Las Vegas algorithm for the same function with constant increase in complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As mentioned in the introduction, this is in contrast to randomised Las Vegas complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The bound (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7) is equal to Las Vegas complexity of bidirectional function evaluation also for non-Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, we only get this result up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Understanding the exact relation between the two is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We list just a few other open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' What is Las Vegas complexity of various important subroutines, for instance, amplitude amplification?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Can the adversary bound for contraction oracles be applied for some problems like faulty oracles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Is there an nice formulation of the adversary bound for (approximate and non-coherent) function evaluation with unidirectional input oracles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, what is the true complexity of the unitary permutation inversion problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The purifiers mentioned above seem to crucially depend on bidirectionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Finally, an interesting research direction is to obtain analogues of some of the results in this paper for time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Acknowledgements We thank anonymous reviewers for their comments on an earlier draft of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' is supported by the ERDF project number 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5/18/A/020 “Quantum algorithms: from complexity theory to experiment”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum lower bounds by quantum arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Journal of Computer and System Sciences, 64(4):750–767, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: STOC’00, arXiv:quant-ph/0002066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 45 [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A new quantum lower bound method, with an application to a strong direct product theorem for quantum search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theory of Computing, 6(1):1–25, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/0508200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum search with variable times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theory of Computing Systems, 47(3):786–807, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: STACS’08, arXiv:quant-ph/0609188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Variable time amplitude amplification and quantum algorithms for linear alge- bra problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 29th STACS, volume 14 of LIPIcs, pages 636–647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Dagstuhl, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Balodis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Santha, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Smotrovs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Separations in query complexity based on pointer functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Journal of the ACM, 64(5):32, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: STOC’16, arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='04719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Childs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Any AND-OR formula of size N can be evaluated in time N 1{2`op1q on a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' SIAM Journal on Computing, 39(6):2513–2530, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ambainis, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Magnin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' R¨otteler, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Roland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Symmetry-assisted adversaries for quantum state generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 26th IEEE CCC, pages 167–177, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Barnum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Algorithms and Complexity for Continuous Prob- lems, volume 6391 of Dagstuhl Seminar Proceedings, pages 1–25, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/0703141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Barnum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Saks, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum decision trees and semi-definite programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 18th IEEE CCC, pages 179–193, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Beals, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Buhrman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Cleve, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Mosca, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum lower bounds by polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Journal of the ACM, 48(4):778–797, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: FOCS’98, arXiv:quant-ph/9802049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Learning-graph-based quantum algorithm for k-distinctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 53rd IEEE FOCS, pages 207–216, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Span programs for functions with constant-sized 1-certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 44th ACM STOC, pages 77–84, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum walks and electric networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3143, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum algorithms for learning symmetric juntas via the adversary bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Computa- tional Complexity, 24(2):255–293, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: CCC’14, arXiv:1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Variations on quantum adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='06943, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Blais.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum algorithm for monotonicity testing on the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theory of Computing, 11(16):403–412, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='02868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Childs, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Jeffery, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Kothari, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Magniez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Time-efficient quantum walks for 3-distinctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 40th ICALP, Part I, volume 7965 of LNCS, pages 105–122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Springer, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Span programs and quantum algorithms for st-connectivity and claw detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 20th ESA, volume 7501 of LNCS, pages 193–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Rosmanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the power of non-adaptive learning graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Computational Com- plexity, 23(2):323–354, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: CCC’13, arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Rosmanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Adversary lower bounds for the collision and the set equality problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum Information & Computation, 18(3-4):200–224, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Rosmanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Tight quantum lower bound for approximate counting with quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='06879, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Belovs and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Adversary lower bound for the k-sum problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 4th ACM ITCS, pages 323–328, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='6528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 46 [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Ben-David and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Kothari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum distinguishing complexity, zero-error algorithms, and statistical zero knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 14th TQC, volume 135 of LIPIcs, pages 2:1–2:23, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='03660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Bennett, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Bernstein, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Brassard, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Vazirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Strengths and weaknesses of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' SIAM Journal on Computing, 26(5):1510–1523, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/9701001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Berzina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Dubrovsky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Freivalds, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lace, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Scegulnaja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum query complexity for some graph problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 30th SOFSEM, volume 2932 of LNCS, pages 140–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Springer, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Bhatia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Princeton University Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Brandeho and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Roland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A Universal Adiabatic Quantum Query Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 10th TQC, volume 44 of LIPIcs, pages 163–179, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [28] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Buhrman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Cleve, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' de Wolf, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Zalka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Bounds for small-error and zero-error quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 40th IEEE FOCS, pages 358–368, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:cs/9904019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Buhrman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' De Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum zero-error algorithms cannot be composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Information processing letters, 87(2):79–84, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/0211029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [30] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Buhrman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum verification of matrix products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 17th ACM-SIAM SODA, pages 880–889, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/0409035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [31] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Buhrman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' de Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Complexity measures and decision tree complexity: a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theo- retical Computer Science, 288:21–43, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Cleve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' An introduction to quantum complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/9906111, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' D¨orn and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thierauf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The quantum query complexity of algebraic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 16th FCT, volume 4639 of LNCS, pages 250–260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Springer, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:0705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' D¨urr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Heiligman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Høyer, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Mhalla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum query complexity of some graph problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 31st ICALP, volume 3142 of LNCS, pages 481–493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Springer, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/0401091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Elitzur and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Vaidman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum mechanical interaction-free measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Foundations of Physics, 23(7):987–997, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:hep-th/9305002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [36] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Farhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Goldstone, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Gutmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A quantum algorithm for the Hamiltonian NAND tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theory of Computing, 4:169–190, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Høyer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lee, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Negative weights make adversaries stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 39th ACM STOC, pages 526–535, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/0611054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Jeffery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum subroutine composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='14146, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Kimmel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Oracles with Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 10th TQC, volume 44 of LIPIcs, pages 1–26, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='02174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Kondacs and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Watrous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' On the power of quantum finite state automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 38th IEEE FOCS, pages 66–75, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:cs/9904019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [41] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Kothari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Semidefinite programming characterization of bounded-error quantum query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [42] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Le Gall, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Nishimura, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Tani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum algorithms for finding constant-sized sub- hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 20th COCOON, volume 8591 of LNCS, pages 429–440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [43] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lee, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Magniez, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Santha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Improved quantum query algorithms for triangle finding and associativity testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 24th ACM-SIAM SODA, pages 1486–1502, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [44] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Mittal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum query complexity of state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 52nd IEEE FOCS, pages 344–353, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 47 [45] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Shraibman, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A direct product theorem for discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 23rd IEEE CCC, pages 71–80, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [46] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lin and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Upper Bounds on Quantum Query Complexity Inspired by the Elitzur-Vaidman Bomb Tester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 30th CCC, volume 33 of LIPIcs, pages 537–566, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='0932.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [47] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Linial, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Mendelson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Schechtman, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Shraibman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Complexity measures of sign matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Combinatorica, 27(4):439–463, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [48] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Linial and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Shraibman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Lower bounds in communication complexity based on factorization norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Random Structures & Algorithms, 34:368–394, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: STOC’07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Nayak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Inverting a permutation is as hard as unordered search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theory of Computing, 7(1):19–25, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:arXiv:1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [50] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Nielsen and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Chuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Quantum computation and quantum information: 10th anniversary edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Cambridge University Press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [51] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Regev and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Schiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Impossibility of a quantum speed-up with a faulty oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 35th ICALP, volume 5125 of LNCS, pages 773–781, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [52] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 50th IEEE FOCS, pages 544–551, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:0904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [53] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reflections for quantum query algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 22nd ACM-SIAM SODA, pages 560–569, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [54] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Span-program-based quantum algorithm for evaluating unbalanced formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 6th TQC, volume 6745 of LNCS, pages 73–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [55] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Reichardt and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Span-program-based quantum algorithm for evaluating formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theory of Computing, 8:291–319, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: STOC’08, arXiv:0710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [56] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Rosmanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Tight bounds for inverting permutations via compressed oracle arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='08975, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [57] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The multiplicative quantum adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 23rd IEEE CCC, pages 237–248, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:quant-ph/0703237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [58] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ˇSpalek and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' All quantum adversary methods are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Theory of Computing, 2:1–18, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Earlier: ICALP’05, arXiv:quant-ph/0409116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [59] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Tomczak-Jaegermann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Longman Scientific & Tech- nical, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [60] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Yolcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The adversary bound revisited: From optimal query algorithms to optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='16293, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' [61] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Zhan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Kimmel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Hassidim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Super-polynomial quantum speed-ups for Boolean evaluation trees with hidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' of 3rd ACM ITCS, pages 249–265, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' arXiv:1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='0796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' A Duality We use semi-definite duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The dual is constructed by explicitly writing down the Lagrangian and transforming it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, weak duality (the maximisation problem bounds the minimisation problem from below) is apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' To prove strong duality (their optimal values are equal), we rely on Slater’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The latter says that strong duality holds if one of the optimisation problems is convex and strictly feasible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' there exists a feasible solution making all the inequalities in the problem strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 48 It turns out that the calculations are concise using multidimensional tensors with con- tractions given by the inner product formula between Hermitian matrices: xA, By “ tr A˚B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' However, given that matrices are tensors themselves, this notation might be confusing, so we opted to use the following one, that we find more intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We assume the matrices are square and are labelled by elements of direct products of some sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' If A is a matrix labelled by X ˆ Y , and B is a matrix labelled by X ˆ Z, then A ˝ B is a matrix labelled by X ˆ Y ˆ Z given by A ˝ Brrpx, y, zq, px1, y1, z1qss “ Arrpx, yq, px1, y1qss Brrpx, zq, px1, z1qss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' This includes the usual Hadamard product (when |Y | “ |Z| “ 1), the tensor product (when |X| “ 1) and the version of the Hadamard product used in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3b) (when |Y | “ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' For the matrix A as above, let ř Y A be the X ˆ X matrix given by `ř Y A ˘ rrx, x1ss “ ÿ y,y1PY Arrpx, yq, px1, y1qss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' ř without the subindex stands for the total sum of all entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' In particular, we have xA, A1y “ řpA ˝ A1q and the partial trace is trY pAq “ ř Y pA ˝ IY q, where A is complex conjugate and IY is the Y ˆ Y identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' We have three sets of labels: D, and the bases of M and W, for which we use letters M and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b): exy “ tr “ v˚ xp∆xybIW qvy ‰ “ tr “ vyv˚ xp∆xy bIWq ‰ “ tr “ pvxv˚ yq˚p∆xy bIWq ‰ “ ř` vxv˚y ˝∆xy ˝IW ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let us merge all these conditions into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let E be the D ˆD matrix given by pexyq, and ∆ be the pD ˆ Mq ˆ pD ˆ Mq matrix with the blocks ∆x,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Both these matrices are Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let also v be the vector in CDˆMˆW obtained by joining all vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, all the constraints in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) can be concisely written as E “ ř M,Wpvv˚ ˝ ∆ ˝ IWq “ ř M `ř Wpvv˚ ˝ IW q ˝ ∆ ˘ “ ř M ` X ˝ ∆ ˘ , where X is a positive semi-definite pD ˆ Mq ˆ pD ˆ Mq-matrix given by X “ trW pvv˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Conversely, any positive semi-definite matrix can be written in this way for a large enough W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Also, the matrix ř MpX ˝ ID,Mq is the diagonal matrix with }vx}2 on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, we get the following equivalent formulation of the optimisation problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2): minimise t (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1a) subject to tID ě ř MpX ˝ ID,Mq (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1b) E “ ř MpX ˝ ∆q (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1c) X ě 0, t P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1d) We introduce two Lagrangian multipliers Y ě 0 and Λ which are DˆD Hermitian matrices, resulting in the following Lagrangian: t ` ř D ” Y ˝ `ř MpX ˝ ID,Mq ´ tID ˘ı ` ř D ” Λ ˝ ` E ´ ř MpX ˝ ∆q ˘ı After rearrangement: ř DpΛ ˝ Eq ` t “ 1 ´ tr Y ‰ ` ř D,M “ X ˝ pY ˝ ID,M ´ Λ ˝ ∆q ‰ 49 This gives the following dual: maximise ř DpΛ ˝ Eq (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2a) subject to tr Y “ 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b) Λ ˝ ∆ ď Y ˝ ID,M (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2c) Y ě 0, Λ Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2d) Note that this optimisation problem is strictly feasible as it suffices to take Λ “ 0 and Y a multiple of the identity matrix satisfying tr Y “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Therefore, by Slater’s condition, the optimal values of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' By studying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2c), we see that we can assume that Y is rank-1 (by extending the diagonal matrix Y ˝ID,M), and we can write Λ as Γ˝Y for some Hermitian DˆD-matrix Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2c) becomes Y ˝ Γ ˝ ∆ ď Y ˝ ID,M, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) and the objective (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2a) becomes ř DpY ˝ Γ ˝ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) This is clearly continuous in Y for fixed Γ and E, thus, we can additionally assume that Y has non-zero diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Then, the Hadamard inverse of Y is defined and positive semi-definite, hence, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='3) is equivalent to Γ ˝ ∆ ď ID,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) Altogether, the objective (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4) with conditions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='2b), and Y is positive semi-definite rank-1 gives us the dual maximise λmaxpΓ ˝ Eq subject to λmaxpΓ ˝ ∆q ď 1 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Proof of Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The set of feasible solutions of the optimisation problems (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='5) is the same so we can use the same characterisation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Let W denote the feasible objective space of the optimisation problem, and BR denote the set of vectors in RD b Rs with the sum of entries bounded by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The objective profile w “ p~vx~qxPD can be obtained by summing the diagonal entries of the corresponding matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' Thus, W X BR is the image under a continuous map of the set of feasible solutions X to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content='1) with tr X ď R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' The latter set is easily seen to be compact, hence, W X BR is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' As R is arbitrary, W is closed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} +page_content=' 50' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf'} diff --git a/T9FIT4oBgHgl3EQffis3/content/tmp_files/2301.11279v1.pdf.txt b/T9FIT4oBgHgl3EQffis3/content/tmp_files/2301.11279v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..73bbc07ffd21f604f39280063b5a20f15d3a9406 --- /dev/null +++ b/T9FIT4oBgHgl3EQffis3/content/tmp_files/2301.11279v1.pdf.txt @@ -0,0 +1,1305 @@ +Highlights +Gaussian process regression and conditional Karhunen-Lo´eve mod- +els for data assimilation in inverse problems⋆ +Yu-Hong Yeung, David A. Barajas-Solano, Alexandre M. Tartakovsky +• We propose CKLEMAP as an efficient alternative to the maximum +a posteriori probability (MAP) method of parameter estimation for +partial differential equations. +• The efficiency is due to the use of a conditional Karhunen-Lo´eve repre- +sentation of the parameter field and an acceleration scheme for Jacobian +computations. +• CKLEMAP and MAP scale as N 1.3 and N 3, where N is the number +of nodes of degrees of freedom in the discretization of the governing +partial differential equation. +• CKLEMAP is as accurate as MAP but significantly faster for large- +scale parameter estimation problems. +arXiv:2301.11279v1 [cs.LG] 26 Jan 2023 + +Gaussian process regression and conditional +Karhunen-Lo´eve models for data assimilation in inverse +problems +Yu-Hong Yeunga, David A. Barajas-Solanoa, Alexandre M. Tartakovskya,b,∗ +aPhysical and Computational Sciences Directorate, Pacific Northwest National +Laboratory, Richland, 99354, WA, USA +bDepartment of Civil and Environmental Engineering, University of Illinois +Urbana-Champaign, Urbana, 61801, IL, USA +Abstract +We present a model inversion algorithm, CKLEMAP, for data assimilation +and parameter estimation in partial differential equation models of physi- +cal systems with spatially heterogeneous parameter fields. These fields are +approximated using low-dimensional conditional Karhunen-Lo´eve expansions +(CKLEs), which are constructed using Gaussian process regression (GPR) +models of these fields trained on the parameters’ measurements. We then +assimilate measurements of the state of the system and compute the max- +imum a posteriori (MAP) estimate of the CKLE coefficients by solving a +nonlinear least-squares problem. When solving this optimization problem, +we efficiently compute the Jacobian of the vector objective by exploiting +the sparsity structure of the linear system of equations associated with the +forward solution of the physics problem. +The CKLEMAP method provides better scalability compared to the stan- +dard MAP method. In the MAP method, the number of unknowns to be +estimated is equal to the number of elements in the numerical forward model. +⋆This research was partially supported by the U.S. Department of Energy (DOE) Ad- +vanced Scientific Computing program. Pacific Northwest National Laboratory is operated +by Battelle for the DOE under Contract DE-AC05-76RL01830. +∗Corresponding author +Email addresses: Yu-Hong.Yeung@pnnl.gov (Yu-Hong Yeung), +David.Barajas-Solano@pnnl.gov (David A. Barajas-Solano), amt1998@illinois.edu +(Alexandre M. Tartakovsky) +Preprint submitted to Journal of Computational Physics +January 27, 2023 + +On the other hand, in CKLEMAP, the number of unknowns (CKLE coeffi- +cients) is controlled by the smoothness of the parameter field and the num- +ber of measurements, and is in general much smaller than the number of +discretization nodes, which leads to a significant reduction of computational +cost with respect to the standard MAP method. To show this advantage +in scalability, we apply CKLEMAP to estimate the transmissivity field in a +two-dimensional steady-state subsurface flow model of the Hanford Site by +assimilating synthetic measurements of transmissivity and hydraulic head. +We find that the execution time of CKLEMAP scales nearly linearly as N 1.33, +where N is the number of discretization nodes, while the execution time of +standard MAP scales as N 2.91. The CKLEMAP method improved execu- +tion time without sacrificing accuracy when compared to the standard MAP +method. +Keywords: +Model inversion, Gaussian process regression, conditional +Karhunen-Lo´eve expansion, maximum a posteriori (MAP) +1. Introduction +Parameter estimation is a critical part of developing partial differential +equation (PDE) models of natural or engineered systems. In heterogeneous +systems, parameters vary in space (and, possibly, time), and the destructive +nature and high cost of collecting measurements limit the number of direct +parameter measurements that can be gathered. As a consequence, modelers +are tasked with solving the inverse problem, i.e., estimating parameters from +a limited number of direct measurements and, usually, a larger number of +indirect measurements, e.g., measurements of the states in the PDE model. +In the context of subsurface flow and transport, such observables include +hydraulic head and tracer breakthrough measurements at observation wells, +among others. +The heterogeneity of parameters gives rise to two challenges: (1) spa- +tial heterogeneity must be parameterized, either naively, using the grid dis- +cretization of the PDE’s domain, or through some other scheme; and (2) +sparse-in-space measurements are often not enough to fully characterize spa- +tial heterogeneity, thus it is necessary to introduce assumptions about spatial +heterogeneity that regularize the inverse problem. +Once parameterization and regularization schemes have been selected, +one can compute the maximum a posteriori (MAP) estimate of the model +2 + +parameters. The MAP estimate is computed by solving a PDE-constrained +optimization problem consisting of minimizing a certain norm of the differ- +ence between predicted and measured observables (data misfit term) plus a +regularizing penalty. Assuming that the solution is obtained at a global min- +imum, the MAP estimate is equivalent to the largest mode of the Bayesian +posterior with the data misfit term corresponding to the (negative) Bayesian +log-likelihood and the regularizing penalty corresponding to the (negative) +Bayesian log-prior [1, 2, 3]. One can drop the PDE constraint by modeling +the predicted observables via a “surrogate” model, at the cost of constructing +said model either on the fly (e.g., [4]) or ahead of tackling the inverse prob- +lem (e.g., [5, 6, 7]). Alternatives to MAP estimation for nonlinear problems +include iterative linear filtering and smoothing [8, 9]. In this work, by “MAP +method” we will refer to MAP estimation via nonlinear least-squares using +the parameterization in terms of the degrees of freedom of the spatial grid +discretization of the forward solver scheme. +The pilot point method (PPM) [10, 11, 12] provides parameterization +and regularization by modeling parameter fields as a regressor computed +from a set of spatially discrete values (“pilot points”) of the parameter fields. +These pilot points then become the parameters to be estimated via PDE- +constrained optimization. The choice of the number and locations of pilot +points is not trivial and significantly affects the quality and time-to-solution +of the inverse problems. To address these challenges, [12] proposed to use +the singular value decomposition of the sensitivities of observables with re- +spect to the pilot points to reduce the effective dimension of the pilot point +parameterization. Beyond PPM, other parameterizations and regularization +schemes have been proposed. For example, [13] represented the parameter +field with a deep neural network and [14, 5] used the latent space represen- +tation of the parameter fields defined by a variational autoencoder and a +convolutional adversarial autoencoder, respectively. +Scientific machine learning (SciML) algorithms provide both an alter- +native and a complement to the PDE-constrained optimization and linear +filtering-based approaches to inverse problems described above. SciML ap- +proaches for inverse problems can be roughly classified into two families: +physics-informed deep learning (DL) and DL for constructing surrogate mod- +els. In physics-informed DL methods [15, 16, 17], the parameters and states +of PDE models are represented by DL models such as feed-forward or con- +volutional neural networks; then, the parameters of these DL models are +estimated by minimizing an objective consisting of the data misfit term plus +3 + +a weighted penalty on the PDE model residuals evaluated at certain points in +the simulation domain. This objective corresponds to the so-called “penalty” +approximation of the corresponding constrained minimization problem with +a fixed penalty weight [18]. The physics-informed DL approaches rely on +the expressive capacity of DL models to accurately represent parameters and +states. On the other hand, DL surrogate modeling approaches use DL models +to approximate the map from parameters to observables [5, 6, 7, 19]. These +approaches rely on the capacity of DL models to approximate functions of +high-dimensional inputs. Other recent developments include “neural opera- +tor” methods, which aim to learn the PDE solution as an explicit function +of the model parameters [20]. +Karhunen-Lo`eve expansions (KLEs) are extensively employed to param- +eterize spatially heterogeneous fields for both uncertainty quantification and +model inversion tasks. In [21], the conditional KLE of the parameter field +was conditioned on the direct field’s measurements, leading to conditional +KL expansions (CKLEs). It was demonstrated that using CKLEs instead of +KLEs reduces the variance of the stochastic model of the parameter field and +reduces uncertainty in the forward models. In [22, 23], CKLEs were used to +represent both parameter and state fields for solving inverse problems. The +CKLE parameters were estimated by minimizing the residuals of the govern- +ing equations. The resulting “physics-informed CKLE” algorithm (PICKLE) +was shown to provide approximate solutions to the inverse problem of accu- +racy comparable to PDE-constrained optimization-based methods but at a +significantly lower computational cost. +Here, we propose solving inverse problems in PDE models by representing +the parameter fields using CKLEs conditioned on available direct measure- +ments of these fields and then estimating the CKLE coefficients via nonlinear +least-squares. We refer to this combination of MAP estimation and CKLEs +as “CKLEMAP.” Compared to PICKLE, CKLEMAP is free of the errors +introduced by the approximation of the state with the CKLE expansions +and the penalty approximation of the PDE constraint, which leads to more +accurate solutions to the inverse problem at the cost of having to solve the +forward problem during the nonlinear least-squares minimization procedure. +Nevertheless, we significantly reduce the execution time of model inversion +with respect to the MAP method by drastically reducing the number of pa- +rameters to be estimated. We note that while KLEs, and more generally +the spectrum of Gaussian process covariance models, have been extensively +used to parameterize heterogeneous fields in Bayesian parameter estimation +4 + +(e.g., [24, 25, 26, 6]), the application of KLE in deterministic inverse meth- +ods has not been explored and is the subject of our work. Furthermore, we +demonstrate the advantage of using the CKLE representation as opposed to +the one based on KLE. +We apply CKLEMAP to a high-dimensional (approximately 1000 param- +eters in the CKLE are needed to accurately represent the transmissivity field) +stationary groundwater flow model of the Hanford Site, a former nuclear pro- +duction complex on the west shore of the Columbia River in the Columbia +Basin in the southeast part of the state of Washington in the United States +and currently operated by the United States Department of Energy. +We +use CKLEMAP to estimate the transmissivity field from synthetic measure- +ments of the transmissivity and hydraulic head fields. These measurements +are generated using the hydraulic conductivity measurements and boundary +conditions obtained in the Hanford Site calibration study [27]. +We compare the CKLEMAP and MAP methods and find that both meth- +ods are very close in accuracy with respect to the reference field. On the other +hand, we find that the computational cost of MAP increases with the prob- +lem size (the number N of finite volume cells) as N 2.91, while the cost of +CKLEMAP increases as N 1.33. We also observe that for N = 5900, the ex- +ecution time of CKLEMAP is one order of magnitude smaller than that of +MAP, and for NFV = 23600, we estimate that CKLEMAP would be more +than two orders of magnitude faster than MAP (the execution time of CK- +LEMAP is found to be ≈ 8 × 102 s, and the execution time of MAP of +approximately 2 × 105 s is estimated from the scaling relationship). The +choice of synthetic (as opposed to the field) measurements of the hydraulic +head allows us to have a reference transmissivity field for comparing the accu- +racy of the MAP and CKLEMAP methods while preserving the complexity +of boundary conditions and the transmissivity field of the Hanford Site. +2. Groundwater flow model +We consider two-dimensional flow in a heterogeneous porous medium in +the domain D ⊂ R2. Given some sparse measurements of the transmissivity +T(x): D → R+ and the hydraulic head u(x): D → R, our goal is to estimate +the spatial distribution of transmissivity. Flow in porous media is described +5 + +by the boundary value problem (BVP) +∇ · [T(x)∇u(x)] = 0, +x ∈ D, +(1) +T(x)∇u(x) · ⃗n(x) = −qN(x), +x ∈ ΓN, +(2) +u(x) = uD(x), +x ∈ ΓD, +(3) +where ΓN and ΓD are the disjoint subsets of the boundary of the domain D, +where the Neumann and Dirichlet boundary conditions (BCs) are prescribed, +respectively. +The flux qN ∈ R at the Neumann boundary ΓN is in the +direction of the outward-pointing unit vector ⃗n ∈ R2 normal to ΓN. The +prescribed hydraulic head at ΓD is denoted as uD ∈ R. +In groundwater models, Dirichlet BCs describe water levels in the lakes +and rivers connected to the aquifer. +Since it is possible to measure the +water levels relatively accurately, we treat the Dirichlet boundary conditions +as deterministic. Furthermore, we assume that the homogeneous Neumann +boundary condition (qN = 0) is imposed over the subset of ΓN formed by +the impermeable boundaries of the aquifer. The rest of ΓN is assumed to be +formed by recharge areas where the values of qN > 0. The boundary fluxes +from recharge areas are difficult to measure; therefore, we treat the non-zero +fluxes as random variables and estimate them along with the transmissivity +field T as part of the inverse solution. +The MAP method (described in detail in Section 3) requires solving the +governing equation for different BCs and realizations of T, which in general +must be done numerically. In this study, we solve the governing equation +using a cell-centered finite volume (FV) scheme with N quadrilateral cells, +and the fluxes across cell faces are approximated using the two-point flux ap- +proximation (TPFA). For simplicity, we assume that ΓN and ΓD are entirely +composed of cell faces. Let ˆxi denote the ith cell center, with i ∈ [1, N]. +We denote by ui ≡ u(ˆxi) and yi ≡ y(ˆxi) the discrete values of the hydraulic +head field u and log-transmissivity y ≡ ln T field evaluated at the ith FV +cell centers. +These discrete values are organized into the column vectors +u ≡ [u1, . . . , uN]⊤ ∈ RN and y ≡ [y1, . . . , yN]⊤ ∈ RN, respectively. +Then, the FV-TPFA discretization of the BVP (1)–(3) yields the system +of equations linear in u, +l(u, y) ≡ A(y)u − b(y) = 0, +(4) +with stiffness matrix A: RN → RN×N and right-hand vector side b: RN → +RN. Here, l: RN ×RN → RN denotes the vector of discretized BVP residuals +6 + +whose entries correspond to the mass balance for each FV cell. The set of FV +cells C can be partitioned into three subsets: N, the set NN of cells adjacent +to ΓN, D, the set ND of cells adjacent to ΓD, and the set of “interior” +cells I = C \ (D ∪ N) (that is, the cells to which boundary conditions do +not contribute directly to their mass balance). The set I has cardinality +NI = N − NN − ND. +3. MAP formulation +We assume that Nus and Nys measurements of u and y, denoted by us +and ys, respectively, are collected at the cell centers indicated by the vectors +of observation indices Iu and Iy, respectively. That is, +[us]i ≡ u(ˆx[Iu]i), +[ys]i ≡ y(ˆx[Iu]j), +i ∈ [1, Nus], j ∈ [1, Nys]. +Using these measurements, we aim to estimate y. +The MAP estimator [1] of y is computed by minimizing the sum of the +ℓ2-norm of the discrepancy between measurements and model predictions, +plus a regularization penalty on y, that is, by solving the PDE-constrained +minimization problem +min +u,y +1 +2∥us − Huu∥2 +2 + 1 +2∥ys − Hyy∥2 +2 + γR(y), +s.t. +l(u, y) = 0, +(5) +where R(y) is the regularization penalty, γ > 0 is a regularization weight, and +Hu : RNus×N and Hy : RNys×N are observation matrices, which downsample +u and y using the observation indices Iu and Iy, respectively. Specifically, +Hu ≡ IN[Iu], and Hy ≡ IN[Iy] are submatrices of the N ×N identity matrix +IN corresponding to the rows of indices Iu and Iy, respectively. +For y, we employ the so-called “H1 regularization,” which penalizes the +H1 seminorm of y (the ℓ2-norm of the gradient of y). In the discrete case, +the H1 seminorm penalty is of the form ∥Dy∥2 +2, where D is the TPFA dis- +cretization of the gradient operator such that Dy is equal to the gradients +of y across the interior faces of the FV discretization. The resulting PDE- +constrained minimization reads +min +u,y +1 +2∥us − Huu∥2 +2 + 1 +2∥ys − Hyy∥2 +2 + γ +2∥Dy∥2 +2, +s.t. +l(u, y) = 0, +(6) +7 + +The MAP estimates ˆu and ˆy obtained from Eq. (6) are equivalent to the +largest mode (ˆu, ˆy) of the joint posterior distribution of (u, y) in a Bayesian +interpretation of the inverse problem, in which the data misfit terms corre- +spond to a Gaussian negative log-likelihood and the regularization penalty +to a Gaussian negative log-prior. +4. CKLEMAP method for inverse problems +4.1. Parameterizing y(x) via conditional Karhunen-Lo´eve expansions +As in the PICKLE method [22, 23], we represent the unknown parameter +field y(x) using the truncated CKLE +yc(x, ξ) ≡ ¯yc(x) + +Ny +� +i=1 +φy +i (x) +� +λy +i ξi, +(7) +where ξ ≡ (ξ1, ξ2, . . . , ξNy)⊤ is the vector of CKLE coefficients and the eigen- +pairs {φy +i (x), λy +i }Ny +i=1 are the solutions of the eigenvalue problem +� +D +Cc +y(x, x′)φy(x′) dx′ = λyφy(x). +(8) +Here, ¯yc(x) and Cc +y(x, x′) denote the mean and covariance of y(x) conditioned +on the measurements yc. +The CKLE is truncated (i.e., Ny is selected) such as to achieve a desired +relative tolerance +rtoly ≡ +N +� +i=Ny+1 +λy +i / +N +� +i=1 +λy +i , +(9) +where N is the number of FV cells. +The GPR (or Kriging) equations are used to compute yc(x) and Cc +y(x, y): +¯yc(x) = C(x)C−1 +s ys, +(10) +Cc +y(x, x′) = Cy(x, x′) − C(x)C−1 +s C(x′), +(11) +where Cs is the Nys × Nys observation covariance matrix with elements +[Cs]ij = Cy(ˆx[Iy]i, ˆx[Iy]j) and C(x) is the Nys-dimensional vector function +with components [C(x)]i = Cy(x, ˆx[Iy]i). +8 + +The prior covariance kernel Cy(x, y) is estimated as in the GPR method +by choosing a parameterized covariance model and computing its hyperpa- +rameters by minimizing the marginal log-likelihood of the data ys [28]. In +this work, we employ the 5/2-Mat´ern kernel as the prior covariance model, +Cy(x, y) = σ2 +� +1 + +√ +5|x − y| +l ++ 5 +3 +|x − y|2 +l2 +� +exp +� +− +√ +5|x − y| +l +� +, +with hyperparameters σ and λ, which correspond to the standard deviation +and the correlation length, respectively. +By representing y(x) via the CKLE (7), we replace the discrete vector +y as the unknown of the inverse problem with the CKLE coefficients ξ. +Specifically, we propose parameterizing y in the MAP problem (6) via the +discrete CKLE +yc(ξ) ≡ ¯yc + Ψyξ, +(12) +where +[¯yc]i ≡ ¯yc(ˆxi), +[Ψy]ij ≡ +� +λy +jφy +j(ˆxi). +We refer to this approach as the “CKLEMAP” method. +Given that, for +sufficiently smooth log-transmissivity fields, the number of CKLE coefficients +required to accurately represent yc is much smaller than the number of FV +cells, i.e., Ny ≪ N, the CKLEMAP method is less computationally expensive +than the MAP method. +4.2. CKLEMAP minimization problem formulation +By solving Eq. (4) with y = yc(ξ), it can be seen that u can be expressed +as a function of ξ; specifically, +u(ξ) = [A(ξ)]−1 b(ξ), +(13) +where A (ξ) = A (yc(ξ)) and b (ξ) = b (yc(ξ)). By expressing u as a func- +tion of ξ, we can remove the PDE constraint from Eq. (6), leading to the +CKLEMAP unconstrained minimization problem +min +ξ +1 +2∥us − Huu(ξ)∥2 +2 + 1 +2∥ys − Hyyc(ξ)∥2 +2 + γ +2∥Dyc(ξ)∥2 +2. +(14) +To solve the CKLEMAP problem Eq. (14), we recast it as the nonlinear +least-squares minimization problem +min +ξ +1 +2 ∥f(ξ)∥2 +2 , +f(ξ) = +� +� +us − Huu(ξ) +ys − Hyyc(ξ) +√γ Dyc(ξ) +� +� , +9 + +which we solve using the Trust Region Reflective algorithm [29]. The least- +square minimization algorithm requires the evaluation of the Jacobian Jξ +of the objective vector of the least-squares problem, f, which is also the +most computationally demanding part of the least-square minimization. This +Jacobian evaluation is done in two steps. First, we evaluate the Jacobian of +the objective vector with respect to yc, which reads +Jξ = Jyc +� ∂yc +∂ξ +I +� += +� +� +−Hu +∂u(yc) +∂yc +−Hy +√γ D +� +� +�Ψy +I +� +. +(15) +The partial derivative ∂u/∂yc is evaluated via the chain rule [3, 23] as de- +scribed in Section 4.3. We note that most elements of Jyc are constant over +iterations except the partial derivatives in the first block row. These con- +stant values are computed once before the least-square minimization and +reused in each iteration. With Jyc computed, Jξ can then be evaluated by +postmultiplying the first block column by Ψy. +4.3. Computations of partial derivatives in the evaluation of Jacobian +In this section we describe how the partial derivative ∂u/∂yc, required to +evaluate the Jacobian of Eq. (15), are evalauted. Let p denote yc +i. Differen- +tiating Eq. (4) with respect to p yields +dl +dp = ∂l +∂u +∂u +∂p + ∂l +∂p = A∂u +∂p + +�∂A +∂p u − ∂b +∂p +� += 0, +(16) +which can be readily solved for ∂u/∂p, leading to the expression +∂u +∂p = −A−1 +�∂A +∂p u − ∂b +∂p +� += −A−1 ∂l +∂p +���� +u +. +(17) +It can be seen that evaluating ∂u/∂yc requires evaluating the sensitivities of +the TPFA stiffness matrix A and right-hand side vector b with respect to +yc. Substituting Eq. (17) into the first row block of Eq. (15) and taking the +transpose yields +� ∂l +∂yc +���� +u +�⊤ +A−1H⊤ +u, +(18) +by the fact that A is symmetric. +10 + +Note that in the MAP method, the Jacobian is given as +Jy = +� +� +−Hu +∂u(y) +∂y +−Hy +√γ D +� +� , +(19) +and the partial derivatives are computed as in the CKLEMAP method, with +y being treated the same way as yc. +4.4. Accelerated CKLEMAP method +In the “accelerated” CKLEMAP method, we compute A−1H⊤ +u efficiently +by exploiting the sparsity structure of the Cholesky factor of A. Recall that +each column of H⊤ +u = (IN[Iu])⊤ has only one non-zero entry. Therefore, if +the sparsity structure of the Cholesky factor L of A is known, the sparsity +structure of each column of Z = L−1H⊤ +u is {closureL(i) | i ∈ Iu}, that is, +the subset of vertices in the graph G(L) that have a path from each vertex +i ∈ Iu [30]. +Figure 1 shows an example of a closure. +Furthermore, the +graph of a Cholesky factor L is a directed tree, and any closure induced by a +vertex i is all the vertices along the path from i to the root of the tree [31]. +This enables a simple algorithm to find the sparsity structure of the solution +of LZ = H⊤ +u. Figure 2 illustrates this algorithm together with a graphical +example. Once we have the sparsity structure Zi of zi, the column i of Z, we +only need the submatrix L[Zi, Zi] instead of the whole matrix L to solve for +zi. Such submatrix is highlighted in blue dots in the lower triangular matrix +L in Figure 2b. This eliminates the unnecessary computations involving the +part of L that does not contribute to the final solutions, thus accelerating +the computations. Furthermore, since the topology of the FV discretization +is static, the sparsity structure of the Cholesky factor L is fixed throughout +the entire least-square minimization procedure. Given this, together with +the fact that Hu is constant, it follows that Zi is also fixed and only needs to +be computed once. Figure 3 shows the closures of two observation locations +in Iu on the Hanford Site experiment to be discussed in detail in Section 5. +The gray lines indicate the cells that do not contribute to the columns of the +Jacobian corresponding to either of these two locations. +We note that, although the computations of the Jacobian can be acceler- +ated by 3–4 times using the procedure described above, the overall execution +time reduction in solving the minimization problems exhibited by the nu- +merical experiments of Section 5 is 10–20%. This is because the nonlinear +11 + +1 +2 +3 +4 +5 +6 +7 +8 +Figure 1: Closure of a unit column vector e3 ≡ [0, 0, 1, 0, . . .]⊤ in a graph G(A). The +nonzero entries of A−1e3 are those nodes in the closure, i.e., {3, 4, 6, 7, 8}. +1: procedure FindSparsity(L, x) +2: +j ← x +3: +S ← {j} +4: +while j ̸= N do +5: +j ← argmini>jL[i, j] ̸= 0 +6: +S ← S ∪ {j} +7: +end while +8: +return S +9: end procedure +(a) Algorithm +L +× z = ex +(b) Graphical Example +Figure 2: Algorithm for finding the sparsity structure S of z = L−1ex. +least-squares minimization algorithm, the Trust Region algorithm, dominates +most of the execution time. The execution times can be further reduced by +optimizing the implementation of the Trust Region algorithm. +5. Numerical experiments +5.1. Case study +We evaluate the performance of the proposed CKLEMAP formulation +against MAP with a case study of parameter estimation in a steady-state +two-dimensional groundwater model of the Hanford Site. The reference log- +transmissivity field ˜y and boundary conditions uD and qN are based on the +data obtained from a three-dimensional Hanford Site calibration study [27] +and are shown in Figure 4. The details of the reference transmissivity field +generation are given in [23]. +To study the scalability of the CKLEMAP +and MAP methods with the problem size (i.e., the number of cells in the FV +model), we generate the reference field at two additional resolutions with four +times and 16 times the number of cells in the base FV model, respectively. +12 + +directed tree G(L) and its root +closure 1 +closure 2 +common closure +Figure 3: The directed tree G(L) structure on with two closures from different cells. +13 + +The numbers of cells in the low, medium, and high-resolution models +are 1475, 5900, and 23600, respectively. For a higher resolution mesh, we +divide each cell in a lower resolution model into four equiareal subcells and +interpolate ˜y at the centers of each subcell, as well as uD and qN at the +midpoints of each boundary edge of the boundary subcells. +There are 558 wells at the Hanford Site where u can be potentially mea- +sured [27]. Some of these wells are located in the same coarse or fine cells. +Figure 4 shows the locations of the cells in the low-resolution FV model that +contain at least one well. +Since our model uses exclusively cells but not +points to specify spatial locations, multiple wells are treated as a single well +if they are located in the same cell. As a result, there are 323 wells in the +low-resolution FV model, while the medium-resolution model has 408 wells. +The aforementioned Hanford Site calibration study defined the Dirichlet +and Neumann boundaries ΓD and ΓN as shown in Figure 4, and provides the +estimates of the heads uD and the fluxes qN at these boundaries. In setting +boundary conditions for our comparison study, we assume that uD and qN +are both known and are given by the estimate. +For each reference log-transmissivity field ˜y, we generate the hydraulic +head field ˜u by solving the Darcy flow equation on the corresponding FV mesh +with the Dirichlet and (deterministic) Neumann boundary conditions that are +set as described above. The values of the reference y and u fields at all cell +locations ˆxi are organized into the vectors ˜y and ˜u, respectively. Then, we +randomly pick Nys well locations and treat the values of ˜y at these locations +as y measurements to form ys. +Similarly, we draw Nus measurements of +the hydraulic head u from ˜u to form us. These measurements are treated +as synthetic data sets and used in the CKLEMAP and MAP methods to +estimate the entire y and u fields. +We note that the aquifer at the Hanford Site is unconfined, and the +use of Eq. (1) to describe flow at the Hanford Site relies on a conceptual +simplification. A more accurate linear conceptual model for flow in an un- +confined aquifer with a horizontal confining layer can be obtained based on +the Dupuit–Forchheimer approximation in the form [32] +∇ · [K(x)∇v(x)] = 0, x ∈ D, +(20) +where v(x) = u2(x) and K(x) is the depth-averaged conductivity. Mathe- +matically, Eqs. (1) and (20) are identical, although the field u(x) computed +using these two equations will be different. Therefore, solving the inverse +14 + +Umtanum Ridge +Cold Creek Valley +Dry Creek Valley +Rattlesnake Hills +Rattlesnake Springs +Recharge Area +Gable Butte +Gable Mountain +Columbia River +Yakima River +well locations +Dirichlet boundary conditions +Neumann boundary conditions +no-flow boundary condition +Figure 4: The coarse-resolution mesh of (NF V = 1475) cells with well locations marked, +and the parts of boundaries colored for different types of prescribed boundary conditions. +15 + +problem for Eq. (1) is equivalent in complexity to solving the inverse prob- +lem for Eq. (20). We also note that applying the Dupuit–Forchheimer ap- +proximation to the Hanford Site aquifer will produce additional linear terms +in Eq. (20) due to the variations in the elevation of the bottom confining +layer of the aquifer. +The implementation of CKLEMAP and MAP are written in Python using +the NumPy and SciPy packages. All CKLEMAP and MAP simulations are +performed using a 3.2 GHz 8-core Intel Xeon W CPU and 32 GB of 2666 MHz +DDR4 RAM. +The weight γ in the CKLEMAP and MAP minimization problems is +empirically found to minimize the error with respect to the reference y fields +as γ = 10−6. When a reference field is not known, these weights can be found +using cross-validation [33]. +5.2. Performance of CKLEMAP as a function of the number of KL terms +Table 1: Performance of CKLEMAP in estimating the coarse-resolution (NF V = 1475) +mesh with Nys = 100 as functions of number of KL terms Ny. +Ny +200 +400 +600 +800 +1000 +least square +iterations +99–218 +44–335 +25–69 +28–177 +20–65 +execution +time (s) +17.55– +42.14 +12.37– +86.31 +9.76– +24.73 +14.60– +94.86 +14.25– +36.29 +relative +ℓ2 error +0.265– +0.568 +0.137– +0.239 +0.081– +0.098 +0.072– +0.082 +0.072– +0.083 +absolute +ℓ∞ error +13.08– +42.69 +6.56– +16.32 +3.71–5.63 +3.68–5.22 +3.46–5.31 +First, we study the relative ℓ2 and absolute ℓ∞ errors in the CKLEMAP +solution for y as well as the time-to-solution and the number of iterations +of the minimization algorithm as functions of Ny, the number of terms in +the CKLE of y for Nys = 100. The relative ℓ2 and absolute ℓ∞ errors are +16 + +200 +400 +600 +800 +1000 +10−1 +10−0.5 +Number of KL terms +ℓ2 errors +Figure 5: Relative ℓ2 errors versus the number of KL terms. +computed on the FV mesh, respectively, as +ε2(y) ≡ ∥ˆy − ˜y∥2 +∥˜y∥2 +. +(21) +and +ε∞(y) ≡ ∥ˆy − ˜y∥∞. +(22) +We find that for the considered inverse problem, all these quantities +strongly depend on the locations of y measurements. Therefore, we compute +these quantities for 10 different distributions of the measurement locations. +The ranges of the ℓ2 and ℓ∞ errors, execution times, and the numbers of +iterations are reported in Table 1. The ℓ2 error and its bounds as functions +of Ny are also plotted in Figure 5. We find that the ℓ2 errors decrease with +increasing Ny and converge to asymptotic values for Ny ≈ 800. The lower +bound of ℓ∞ continues to decrease even for Ny greater than 800, while the +upper bound increases from 5.22 to 5.31 as Ny increases from 800 to 1000. +However, the relative changes of ℓ∞ are insignificant for Ny > 800. What +is surprising is that the execution time does not significantly change with +increasing Ny. While the time per iteration increases with Ny, the number of +iterations tends to decrease. Therefore, in the rest of the numerical examples, +we set Ny = 1000, which corresponds to rtoly on the order of 10−8. +5.3. CKLEMAP and MAP errors versus the number of y measurements +Next, we study the accuracy of the CKLEMAP and MAP methods in +estimating y as the function of the number of y measurements. We assume +17 + +that u measurements are available at all wells. +We start with the low-resolution model. Figure 6 shows the locations of y +measurements, the y fields estimated by the MAP and CKLEMAP methods +for Nys = 25, 50, 100, and 200, and the distributions of point errors in the +MAP and CKLEMAP estimates of y relative to the reference field ˜y. For the +considered measurement locations, we observe that the MAP and CKLEMAP +methods have comparable accuracy for all Nys. +Table 2 shows the ranges of relative ℓ2 and absolute ℓ∞ errors in the +MAP and CKLEMAP y estimates as well as the number of iterations in the +minimization algorithm and the execution times (in seconds) for Nys ranging +from 25 to 200. Also included in this table are the execution times of the +accelerated CKLEMAP method. We note that the accuracy (including the +ℓ2 and absolute ℓ∞ errors) and the number of iterations in the accelerated +CKLEMAP and CKLEMAP methods are the same. +As expected, the accuracy of the MAP and CKLEMAP methods increases +with Nys. The MAP and CKLEMAP methods are almost equally accurate, +with ℓ2 and ℓ∞ errors in the CKLEMAP method being slightly smaller. How- +ever, we observe that CKLEMAP is faster than MAP for all considered values +of Nys except for Nys = 25, where the MAP’s lower bound of the execution +time is less than that of the CKLEMAP. Accelerated CKLEMAP is about +20% faster than CKLEMAP and for all considered values of Nys. Accelerated +CKLEMAP is also faster than MAP for all considered cases; however, the +speedup depends on Nys. +In all examples reported in Table 2, the number of unknowns in the +CKLEMAP method is 1000 (the number of terms in the CKLE expansion), +while in the MAP method, this number is 1475 (the number of cells in the FV +model). The reason for CKLEMAP being slower than MAP for Nys = 25 and +certain y measurement locations is that for such locations MAP converges +much faster. For example, the lower execution time bands in MAP and CK- +LEMAP correspond to 29 and 50 iterations, respectively. However, because +there are fewer unknowns in the CKLEMAP method, the CKLEMAP com- +putational time per iteration is smaller than that in MAP. As a result, the +computational time in the CKLEMAP is only 20% larger than that of MAP +for these limiting cases. The time per iteration is further reduced in the +accelerated CKLEMAP method, resulting in the execution time of acceler- +ated CKLEMAP being less than that of MAP by 20%. We also note that +for Nys > 25, MAP requires more iterations than CKLEMAP, making the +computational advantages of CKLEMAP even more significant. +18 + +reference +0 +2 +4 +6 +8 +10 +12 +Nys +25 +50 +100 +200 +observation +locations +CKLEMAP +estimates +0 +2 +4 +6 +8 +10 +12 +CKLEMAP +point errors +0 +1 +2 +3 +4 +5 +6 +MAP +estimates +0 +2 +4 +6 +8 +10 +12 +MAP +point errors +0 +1 +2 +3 +4 +5 +6 +Figure 6: The fine-resolution (NF V = 5900) reference y fields, the CKLEMAP and MAP +estimates of the y field and their point errors as functions of Nys. +19 + +Next, we perform a similar study for the medium-resolution model with +N = 5900 cells. Table 3 provides a comparative summary of the models con- +sidered for this case. Here, we find that CKLEMAP is slightly more accurate +than MAP for all considered values of Nys and one to two orders of magni- +tude faster than MAP. Accelerated CKLEMAP is approximately 10% faster +than CKLEMAP. The computational advantage of CKLEMAP significantly +increases with the problem size as the number of unknown parameters in the +MAP linearly increases with the problem size while the number of parameters +in the CKLEMAP is independent of the problem size. +5.4. Scaling of the execution time with the problem size +Table 2: +Performance of MAP and CKLEMAP in estimating the coarse-resolution +(NF V = 1475) mesh as functions of Nys. +Nys +solver +25 +50 +100 +200 +least square +iterations +MAP +29–95 +29–106 +41–60 +28–80 +CKLEMAP +50–96 +26–70 +20–65 +33–62 +execution +time (s) +MAP +31.36– +91.50 +57.98– +199.61 +76.39– +123.08 +32.88– +80.32 +CKLEMAP +37.01– +71.04 +21.71– +51.86 +14.25– +36.29 +17.73– +40.57 +accelerated +CKELMAP +25.45– +47.97 +17.00– +40.00 +12.07– +30.41 +12.78– +29.09 +relative +ℓ2 error +MAP +0.092– +0.111 +0.084– +0.101 +0.073– +0.084 +0.068– +0.073 +CKLEMAP +0.091– +0.109 +0.082– +0.101 +0.072– +0.083 +0.064– +0.071 +absolute +ℓ∞ error +MAP +5.38–6.61 +4.95–6.55 +4.06–6.35 +3.88–6.74 +CKLEMAP +4.96–6.25 +4.73–6.11 +3.46–5.31 +5.63–5.71 +The comparison of Tables 2 and 3 shows that the execution times of +the MAP, CKLEMAP, and accelerated CKLEMAP increase with the mesh +resolution; however, the execution times of CKLEMAP and accelerated CK- +LEMAP increase slower than that of MAP. To study the scalability of these +20 + +Table 3: +Performance of MAP and CKLEMAP in estimating the fine-resolution +(NF V = 5900) mesh as functions of Nys. +Nys +solver +25 +50 +100 +200 +least square +iterations +MAP +78–99 +71–97 +69–83 +23–76 +CKLEMAP +53–114 +20–142 +36–60 +15–83 +execution +time (s) +MAP +3907.00– +4868.21 +3528.90– +4580.40 +3533.06– +4190.20 +1247.37– +3733.05 +CKLEMAP +88.76– +181.67 +48.45– +200.08 +62.59– +100.04 +42.86– +148.03 +accelerated +CKELMAP +77.05– +141.90 +39.50– +156.63 +52.14– +81.19 +38.28– +120.18 +relative +ℓ2 error +MAP +0.0954– +0.112 +0.081– +0.105 +0.074– +0.088 +0.065– +0.073 +CKLEMAP +0.0906– +0.111 +0.081– +0.105 +0.068– +0.079 +0.061– +0.069 +absolute +ℓ∞ error +MAP +4.96–7.21 +5.45–7.28 +4.00–6.48 +4.37–5.20 +CKLEMAP +4.21–6.66 +4.94–6.74 +3.79–5.71 +3.82–5.28 +21 + +1475 +5900 +23600 +101 +102 +103 +104 +105 +3.7 · 10−8x2.91 +1.11 · 10−3x1.33 +8.11 · 10−4x1.35 +Number of FV cells +Execution time (s) +MAP +CKLEMAP +accelerated CKLEMAP +Figure 7: Execution times of MAP, CKLEMAP, and accelerated CKLEMAP methods +versus the number of FV cells. The execution times of MAP for the mesh with 23600 FV +cells are estimated by extrapolation. +methods with the problem size, we use these methods to estimate y in the +high-resolution FV model with N = 23600 and, in Figure 7, we plot the +execution times of these methods as functions of N. The number of y mea- +surements in all simulations reported in this figure is set to Nys = 100. +We also show the power-law models fitted to the scalability curves com- +puted using MAP, CKLEMAP, and accelerated CKLEMAP. We note that +for N = 23600, the MAP method did not converge after running for two +days. Therefore, the power law relationship for the MAP method is obtained +based on the execution times for N = 1475 and 5900 and used to estimate +the MAP’s execution time for the highest resolution by extrapolation. We +find that the MAP, CKLEMAP, and accelerated CKLEMAP execution times +scale as N 2.91, N 1.33, and N 1.35, respectively. +Therefore, the CKLEMAP +methods have a computational advantage over the MAP method for large +problems. The CKLEMAP and accelerated CKLEMAP methods have ap- +proximately the same scalability, but for the same problem size, the acceler- +ated CKLEMAP method is 10–20% faster than the CKLEMAP method. +22 + +6. Discussion and Conclusions +We proposed the CKLEMAP method as an alternative to the MAP meth- +ods for solving inverse PDE problems and used it for estimating the trans- +missivity and hydraulic head in a two-dimensional steady-state groundwater +model of the Hanford Site. The CKLEMAP method is based on the ap- +proximation of unknown parameters (log-transmissivity in this case) with +CKLEs. The advantage of using a CKLE over other representations (like +DNNs in [13]) is that it enforces (i.e., exactly matches) the field measure- +ments and the covariance structure, that is, it models the field as a realization +of the conditional Gaussian field with a prescribed covariance function. As a +general conclusion, we found that the accuracy of the MAP and CKLEMAP +methods is essentially the same (with CKLEMAP being a few percents more +accurate under most tested conditions), but CKLEMAP is faster than MAP. +Specifically, we demonstrated that the CKLEMAP and MAP execution +times scale with the problem size as N 1.33 and N 2.91, respectively, where N is +the number of FV cells. The close-to-linear scaling of CKLEMAP’s execution +time with problem size gives CKLEMAP a computational advantage over +the MAP method for large-scale problems. We consider this to be the main +advantage of the CKLEMAP method. +For the same number of measurements, the accuracy of MAP and CK- +LEMAP can depend on the measurement locations. +Both the MAP and +the CKLEMAP methods are, on average, equally accurate in terms of abso- +lute ℓ∞ errors. The CKLEMAP method is slightly more accurate than the +MAP method in terms of relative ℓ2 errors. The execution times of MAP +and CKLEMAP increase, and their accuracy decreases, as the number of y +measurements decreases. +In the CKLEMAP method, execution time and accuracy increase with +the increasing number of CKL terms. In this work, as a baseline, we used +Ny = 1000, which corresponds to rtol < 10−8. We stipulate that this criterion +is sufficient to obtain a convergent estimate of y with respect to the number +of CKL terms. +To further reduce the computational time, we proposed the accelerated +CKLEMAP method, which takes advantage of the sparse structure of the +stiffness matrix in the FV discretization of the residual term. We demon- +strated that the scalability of the accelerated CKLEMAP and CKLEMAP +methods is approximately the same; however, for the same problem size, +accelerated CKLEMAP is 10–20% faster than the CKLEMAP method. +23 + +7. Acknowledgments +This research was partially supported by the U.S. Department of Energy +(DOE) Advanced Scientific Computing program and the United States Geo- +logical Survey. Pacific Northwest National Laboratory is operated by Battelle +for the DOE under Contract DE-AC05-76RL01830. The data and codes used +in this paper are available at https://github.com/yeungyh/cklemap.git. +References +[1] P. K. Kitanidis, On the geostatistical approach to the inverse problem, +Advances in Water Resources 19 (6) (1996) 333–342. +doi:10.1016/ +0309-1708(96)00005-X. +[2] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica +19 (2010) 451–559. doi:10.1017/S0962492910000061. +[3] D. A. Barajas-Solano, B. E. Wohlberg, V. V. Vesselinov, D. M. Tar- +takovsky, Linear functional minimization for inverse modeling, Water +Resources Research 51 (2014) 4516–4531. doi:10.1002/2014WR016179. +[4] S. M. Wild, POUNDERS in TAO: Solving Derivative-Free Nonlinear +Least-Squares Problems with POUNDERS, Society for Industrial and +Applied Mathematics, 2017, Ch. 40, pp. 529–539. +doi:10.1137/1. +9781611974683.ch40. +[5] S. Mo, N. Zabaras, X. Shi, J. Wu, Integration of adversarial autoen- +coders with residual dense convolutional networks for estimation of +non-gaussian hydraulic conductivities, Water Resources Research 56 (2) +(2020) e2019WR026082. doi:10.1029/2019WR026082. +[6] S. Mo, Y. Zhu, N. Zabaras, X. Shi, J. Wu, Deep convolutional encoder- +decoder networks for uncertainty quantification of dynamic multiphase +flow in heterogeneous media, Water Resources Research 55 (1) (2019) +703–728. doi:10.1029/2018WR023528. +[7] S. Mo, N. Zabaras, X. Shi, J. Wu, Deep autoregressive neural net- +works for high-dimensional inverse problems in groundwater contami- +nant source identification, Water Resources Research 55 (5) (2019) 3856– +3881. doi:10.1029/2018WR024638. +24 + +[8] N. K. Chada, Y. Chen, D. Sanz-Alonso, Iterative ensemble kalman meth- +ods: A unified perspective with some new variants, Foundations of Data +Science 3 (3) (2021) 331–369. doi:10.3934/fods.2021011. +[9] J. Zhang, G. Lin, W. Li, L. Wu, L. Zeng, An iterative local updating +ensemble smoother for estimation and uncertainty assessment of hydro- +logic model parameters with multimodal distributions, Water Resources +Research 54 (3) (2018) 1716–1733. doi:10.1002/2017WR020906. +[10] C. Certes, G. de Marsily, Application of the pilot point method to the +identification of aquifer transmissivities, Advances in Water Resources +14 (5) (1991) 284–300. doi:10.1016/0309-1708(91)90040-U. +[11] J. E. Doherty, M. N. Fienen, R. J. Hunt, Approaches to highly parame- +terized inversion: Pilot-point theory, guidelines, and research directions, +US Geological Survey scientific investigations report 5168 (2010) 36. +doi:10.3133/sir20105168. +[12] M. J. Tonkin, J. Doherty, A hybrid regularized inversion methodology +for highly parameterized environmental models, Water Resources Re- +search 41 (10) (2005). doi:10.1029/2005WR003995. +[13] K. Xu, A. M. Tartakovsky, J. Burghardt, E. Darve, Learning viscoelas- +ticity models from indirect data using deep neural networks, Computer +Methods in Applied Mechanics and Engineering 387 (2021) 114124. +doi:10.1016/j.cma.2021.114124. +[14] D. O’Malley, J. K. Golden, V. V. Vesselinov, Learning to regularize +with a variational autoencoder for hydrologic inverse analysis (2019). +doi:10.48550/ARXIV.1906.02401. +[15] A. M. Tartakovsky, C. O. Marrero, P. Perdikaris, G. D. Tartakovsky, +D. Barajas-Solano, Physics-informed deep neural networks for learn- +ing parameters and constitutive relationships in subsurface flow prob- +lems, Water Resources Research 56 (5) (2020) e2019WR026731. doi: +10.1029/2019WR026731. +[16] Q. He, D. Barajas-Solano, G. Tartakovsky, A. M. Tartakovsky, Physics- +informed neural networks for multiphysics data assimilation with ap- +plication to subsurface transport, Advances in Water Resources (2020) +103610doi:10.1016/j.advwatres.2020.103610. +25 + +[17] L. Yang, S. Treichler, T. Kurth, K. Fischer, D. Barajas-Solano, +J. Romero, V. Churavy, A. Tartakovsky, M. Houston, M. Prabhat, +G. Karniadakis, Highly-scalable, physics-informed GANs for learning +solutions of stochastic PDEs, in: 2019 IEEE/ACM Third Workshop +on Deep Learning on Supercomputers (DLS), 2019, pp. 1–11. +doi: +10.1109/DLS49591.2019.00006. +[18] T. van Leeuwen, F. J. Herrmann, A penalty method for PDE- +constrained optimization in inverse problems, Inverse Problems 32 (1) +(2015) 015007. doi:10.1088/0266-5611/32/1/015007. +[19] T. Kadeethum, D. O’Malley, J. N. Fuhg, Y. Choi, J. Lee, H. S. +Viswanathan, N. Bouklas, A framework for data-driven solution and +parameter estimation of pdes using conditional generative adversar- +ial networks, Nature Computational Science 1 (12) (2021) 819–829. +doi:10.1038/s43588-021-00171-3. +[20] L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear +operators via DeepONet based on the universal approximation theorem +of operators, Nature Machine Intelligence 3 (3) (2021) 218–229. doi: +10.1038/s42256-021-00302-5. +[21] R. Tipireddy, D. A. Barajas-Solano, A. M. Tartakovsky, Conditional +Karhunen-Lo`eve expansion for uncertainty quantification and active +learning in partial differential equation models, Journal of Computa- +tional Physics 418 (2020) 109604. doi:10.1016/j.jcp.2020.109604. +[22] A. Tartakovsky, D. Barajas-Solano, Q. He, Physics-informed machine +learning with conditional Karhunen-Lo`eve expansions, Journal of Com- +putational Physics (2020) 109904doi:10.1016/j.jcp.2020.109904. +[23] Y.-H. Yeung, D. A. Barajas-Solano, A. M. Tartakovsky, Physics- +informed machine learning method for large-scale data assimilation +problems, Water Resources Research 58 (5) (2022) e2021WR031023. +doi:10.1029/2021WR031023. +[24] P. K. Kitanidis, J. Lee, Principal component geostatistical approach for +large-dimensional inverse problems, Water Resources Research 50 (7) +(2014) 5428–5443. doi:10.1002/2013WR014630. +26 + +[25] J. Lee, P. K. Kitanidis, Large-scale hydraulic tomography and joint +inversion of head and tracer data using the principal component geo- +statistical approach (PCGA), Water Resources Research 50 (7) (2014) +5410–5427. doi:10.1002/2014WR015483. +[26] Y. M. Marzouk, H. N. Najm, Dimensionality reduction and polynomial +chaos acceleration of bayesian inference in inverse problems, Journal of +Computational Physics 228 (6) (2009) 1862–1902. doi:10.1016/j.jcp. +2008.11.024. +[27] C. R. Cole, M. P. Bergeron, S. K. Wurstner, P. D. Thorne, S. Orr, M. I. +Mckinley, Transient inverse calibration of Hanford site-wide groundwa- +ter model to Hanford operational impacts - 1943 to 1996, Tech. rep., +Pacific Northwest National Laboratory (PNNL), Richland, Washington, +United States (2001). doi:10.2172/965656. +[28] C. +E. +Rasmussen, +Gaussian +processes +in +machine +learning, +Springer +Berlin +Heidelberg, +2004, +pp. +63–71. +doi:10.1007/ +978-3-540-28650-9_4. +[29] M. A. Branch, T. F. Coleman, Y. Li, A subspace, interior, and con- +jugate gradient method for large-scale bound-constrained minimization +problems, SIAM Journal on Scientific Computing 21 (1) (1999) 1–23. +doi:10.1137/S1064827595289108. +[30] J. R. Gilbert, Predicting structure in sparse matrix computations, SIAM +Journal on Matrix Analysis and Applications 15 (1) (1994) 62–79. doi: +10.1137/S0895479887139455. +[31] Y.-H. Yeung, J. Crouch, A. Pothen, Interactively cutting and constrain- +ing vertices in meshes using augmented matrices, ACM Transactions on +Graphics 35 (2) (February 2016). doi:10.1145/2856317. +[32] Y. Zhang, Nonlinear inversion of an unconfined aquifer: simultaneous +estimation of heterogeneous hydraulic conductivities, recharge rates, and +boundary conditions, Transport in porous media 102 (2) (2014) 275–299. +doi:10.1007/s11242-014-0275-x. +[33] R. R. Picard, R. D. Cook, Cross-validation of regression models, Journal +of the American Statistical Association 79 (387) (1984) 575–583. doi: +10.2307/2288403. +27 + diff --git a/T9FIT4oBgHgl3EQffis3/content/tmp_files/load_file.txt b/T9FIT4oBgHgl3EQffis3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a2acc6eb98311dd029938bba738dbe4a020280f --- /dev/null +++ b/T9FIT4oBgHgl3EQffis3/content/tmp_files/load_file.txt @@ -0,0 +1,776 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf,len=775 +page_content='Highlights Gaussian process regression and conditional Karhunen-Lo´eve mod- els for data assimilation in inverse problems⋆ Yu-Hong Yeung, David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, Alexandre M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky We propose CKLEMAP as an efficient alternative to the maximum a posteriori probability (MAP) method of parameter estimation for partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The efficiency is due to the use of a conditional Karhunen-Lo´eve repre- sentation of the parameter field and an acceleration scheme for Jacobian computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' CKLEMAP and MAP scale as N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='3 and N 3, where N is the number of nodes of degrees of freedom in the discretization of the governing partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' CKLEMAP is as accurate as MAP but significantly faster for large- scale parameter estimation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='11279v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='LG] 26 Jan 2023 Gaussian process regression and conditional Karhunen-Lo´eve models for data assimilation in inverse problems Yu-Hong Yeunga, David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solanoa, Alexandre M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovskya,b,∗ aPhysical and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, 99354, WA, USA bDepartment of Civil and Environmental Engineering, University of Illinois Urbana-Champaign, Urbana, 61801, IL, USA Abstract We present a model inversion algorithm, CKLEMAP, for data assimilation and parameter estimation in partial differential equation models of physi- cal systems with spatially heterogeneous parameter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' These fields are approximated using low-dimensional conditional Karhunen-Lo´eve expansions (CKLEs), which are constructed using Gaussian process regression (GPR) models of these fields trained on the parameters’ measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We then assimilate measurements of the state of the system and compute the max- imum a posteriori (MAP) estimate of the CKLE coefficients by solving a nonlinear least-squares problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' When solving this optimization problem, we efficiently compute the Jacobian of the vector objective by exploiting the sparsity structure of the linear system of equations associated with the forward solution of the physics problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The CKLEMAP method provides better scalability compared to the stan- dard MAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In the MAP method, the number of unknowns to be estimated is equal to the number of elements in the numerical forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' ⋆This research was partially supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Department of Energy (DOE) Ad- vanced Scientific Computing program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' ∗Corresponding author Email addresses: Yu-Hong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='Yeung@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='gov (Yu-Hong Yeung), David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='Barajas-Solano@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='gov (David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano), amt1998@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='edu (Alexandre M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky) Preprint submitted to Journal of Computational Physics January 27, 2023 On the other hand, in CKLEMAP, the number of unknowns (CKLE coeffi- cients) is controlled by the smoothness of the parameter field and the num- ber of measurements, and is in general much smaller than the number of discretization nodes, which leads to a significant reduction of computational cost with respect to the standard MAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' To show this advantage in scalability, we apply CKLEMAP to estimate the transmissivity field in a two-dimensional steady-state subsurface flow model of the Hanford Site by assimilating synthetic measurements of transmissivity and hydraulic head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We find that the execution time of CKLEMAP scales nearly linearly as N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='33, where N is the number of discretization nodes, while the execution time of standard MAP scales as N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The CKLEMAP method improved execu- tion time without sacrificing accuracy when compared to the standard MAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Keywords: Model inversion, Gaussian process regression, conditional Karhunen-Lo´eve expansion, maximum a posteriori (MAP) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Introduction Parameter estimation is a critical part of developing partial differential equation (PDE) models of natural or engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In heterogeneous systems, parameters vary in space (and, possibly, time), and the destructive nature and high cost of collecting measurements limit the number of direct parameter measurements that can be gathered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' As a consequence, modelers are tasked with solving the inverse problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', estimating parameters from a limited number of direct measurements and, usually, a larger number of indirect measurements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', measurements of the states in the PDE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In the context of subsurface flow and transport, such observables include hydraulic head and tracer breakthrough measurements at observation wells, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The heterogeneity of parameters gives rise to two challenges: (1) spa- tial heterogeneity must be parameterized, either naively, using the grid dis- cretization of the PDE’s domain, or through some other scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' and (2) sparse-in-space measurements are often not enough to fully characterize spa- tial heterogeneity, thus it is necessary to introduce assumptions about spatial heterogeneity that regularize the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Once parameterization and regularization schemes have been selected, one can compute the maximum a posteriori (MAP) estimate of the model 2 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The MAP estimate is computed by solving a PDE-constrained optimization problem consisting of minimizing a certain norm of the differ- ence between predicted and measured observables (data misfit term) plus a regularizing penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Assuming that the solution is obtained at a global min- imum, the MAP estimate is equivalent to the largest mode of the Bayesian posterior with the data misfit term corresponding to the (negative) Bayesian log-likelihood and the regularizing penalty corresponding to the (negative) Bayesian log-prior [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' One can drop the PDE constraint by modeling the predicted observables via a “surrogate” model, at the cost of constructing said model either on the fly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', [4]) or ahead of tackling the inverse prob- lem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', [5, 6, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Alternatives to MAP estimation for nonlinear problems include iterative linear filtering and smoothing [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In this work, by “MAP method” we will refer to MAP estimation via nonlinear least-squares using the parameterization in terms of the degrees of freedom of the spatial grid discretization of the forward solver scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The pilot point method (PPM) [10, 11, 12] provides parameterization and regularization by modeling parameter fields as a regressor computed from a set of spatially discrete values (“pilot points”) of the parameter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' These pilot points then become the parameters to be estimated via PDE- constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The choice of the number and locations of pilot points is not trivial and significantly affects the quality and time-to-solution of the inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' To address these challenges, [12] proposed to use the singular value decomposition of the sensitivities of observables with re- spect to the pilot points to reduce the effective dimension of the pilot point parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Beyond PPM, other parameterizations and regularization schemes have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For example, [13] represented the parameter field with a deep neural network and [14, 5] used the latent space represen- tation of the parameter fields defined by a variational autoencoder and a convolutional adversarial autoencoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Scientific machine learning (SciML) algorithms provide both an alter- native and a complement to the PDE-constrained optimization and linear filtering-based approaches to inverse problems described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' SciML ap- proaches for inverse problems can be roughly classified into two families: physics-informed deep learning (DL) and DL for constructing surrogate mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In physics-informed DL methods [15, 16, 17], the parameters and states of PDE models are represented by DL models such as feed-forward or con- volutional neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' then, the parameters of these DL models are estimated by minimizing an objective consisting of the data misfit term plus 3 a weighted penalty on the PDE model residuals evaluated at certain points in the simulation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' This objective corresponds to the so-called “penalty” approximation of the corresponding constrained minimization problem with a fixed penalty weight [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The physics-informed DL approaches rely on the expressive capacity of DL models to accurately represent parameters and states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' On the other hand, DL surrogate modeling approaches use DL models to approximate the map from parameters to observables [5, 6, 7, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' These approaches rely on the capacity of DL models to approximate functions of high-dimensional inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Other recent developments include “neural opera- tor” methods, which aim to learn the PDE solution as an explicit function of the model parameters [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Karhunen-Lo`eve expansions (KLEs) are extensively employed to param- eterize spatially heterogeneous fields for both uncertainty quantification and model inversion tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In [21], the conditional KLE of the parameter field was conditioned on the direct field’s measurements, leading to conditional KL expansions (CKLEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' It was demonstrated that using CKLEs instead of KLEs reduces the variance of the stochastic model of the parameter field and reduces uncertainty in the forward models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In [22, 23], CKLEs were used to represent both parameter and state fields for solving inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The CKLE parameters were estimated by minimizing the residuals of the govern- ing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The resulting “physics-informed CKLE” algorithm (PICKLE) was shown to provide approximate solutions to the inverse problem of accu- racy comparable to PDE-constrained optimization-based methods but at a significantly lower computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Here, we propose solving inverse problems in PDE models by representing the parameter fields using CKLEs conditioned on available direct measure- ments of these fields and then estimating the CKLE coefficients via nonlinear least-squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We refer to this combination of MAP estimation and CKLEs as “CKLEMAP.” Compared to PICKLE, CKLEMAP is free of the errors introduced by the approximation of the state with the CKLE expansions and the penalty approximation of the PDE constraint, which leads to more accurate solutions to the inverse problem at the cost of having to solve the forward problem during the nonlinear least-squares minimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Nevertheless, we significantly reduce the execution time of model inversion with respect to the MAP method by drastically reducing the number of pa- rameters to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We note that while KLEs, and more generally the spectrum of Gaussian process covariance models, have been extensively used to parameterize heterogeneous fields in Bayesian parameter estimation 4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', [24, 25, 26, 6]), the application of KLE in deterministic inverse meth- ods has not been explored and is the subject of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Furthermore, we demonstrate the advantage of using the CKLE representation as opposed to the one based on KLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We apply CKLEMAP to a high-dimensional (approximately 1000 param- eters in the CKLE are needed to accurately represent the transmissivity field) stationary groundwater flow model of the Hanford Site, a former nuclear pro- duction complex on the west shore of the Columbia River in the Columbia Basin in the southeast part of the state of Washington in the United States and currently operated by the United States Department of Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We use CKLEMAP to estimate the transmissivity field from synthetic measure- ments of the transmissivity and hydraulic head fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' These measurements are generated using the hydraulic conductivity measurements and boundary conditions obtained in the Hanford Site calibration study [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We compare the CKLEMAP and MAP methods and find that both meth- ods are very close in accuracy with respect to the reference field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' On the other hand, we find that the computational cost of MAP increases with the prob- lem size (the number N of finite volume cells) as N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='91, while the cost of CKLEMAP increases as N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We also observe that for N = 5900, the ex- ecution time of CKLEMAP is one order of magnitude smaller than that of MAP, and for NFV = 23600, we estimate that CKLEMAP would be more than two orders of magnitude faster than MAP (the execution time of CK- LEMAP is found to be ≈ 8 × 102 s, and the execution time of MAP of approximately 2 × 105 s is estimated from the scaling relationship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The choice of synthetic (as opposed to the field) measurements of the hydraulic head allows us to have a reference transmissivity field for comparing the accu- racy of the MAP and CKLEMAP methods while preserving the complexity of boundary conditions and the transmissivity field of the Hanford Site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Groundwater flow model We consider two-dimensional flow in a heterogeneous porous medium in the domain D ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Given some sparse measurements of the transmissivity T(x): D → R+ and the hydraulic head u(x): D → R, our goal is to estimate the spatial distribution of transmissivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Flow in porous media is described 5 by the boundary value problem (BVP) ∇ · [T(x)∇u(x)] = 0, x ∈ D, (1) T(x)∇u(x) · ⃗n(x) = −qN(x), x ∈ ΓN, (2) u(x) = uD(x), x ∈ ΓD, (3) where ΓN and ΓD are the disjoint subsets of the boundary of the domain D, where the Neumann and Dirichlet boundary conditions (BCs) are prescribed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The flux qN ∈ R at the Neumann boundary ΓN is in the direction of the outward-pointing unit vector ⃗n ∈ R2 normal to ΓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The prescribed hydraulic head at ΓD is denoted as uD ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In groundwater models, Dirichlet BCs describe water levels in the lakes and rivers connected to the aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Since it is possible to measure the water levels relatively accurately, we treat the Dirichlet boundary conditions as deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Furthermore, we assume that the homogeneous Neumann boundary condition (qN = 0) is imposed over the subset of ΓN formed by the impermeable boundaries of the aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The rest of ΓN is assumed to be formed by recharge areas where the values of qN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The boundary fluxes from recharge areas are difficult to measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' therefore, we treat the non-zero fluxes as random variables and estimate them along with the transmissivity field T as part of the inverse solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The MAP method (described in detail in Section 3) requires solving the governing equation for different BCs and realizations of T, which in general must be done numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In this study, we solve the governing equation using a cell-centered finite volume (FV) scheme with N quadrilateral cells, and the fluxes across cell faces are approximated using the two-point flux ap- proximation (TPFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For simplicity, we assume that ΓN and ΓD are entirely composed of cell faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Let ˆxi denote the ith cell center, with i ∈ [1, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We denote by ui ≡ u(ˆxi) and yi ≡ y(ˆxi) the discrete values of the hydraulic head field u and log-transmissivity y ≡ ln T field evaluated at the ith FV cell centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' These discrete values are organized into the column vectors u ≡ [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' , uN]⊤ ∈ RN and y ≡ [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' , yN]⊤ ∈ RN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Then, the FV-TPFA discretization of the BVP (1)–(3) yields the system of equations linear in u, l(u, y) ≡ A(y)u − b(y) = 0, (4) with stiffness matrix A: RN → RN×N and right-hand vector side b: RN → RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Here, l: RN ×RN → RN denotes the vector of discretized BVP residuals 6 whose entries correspond to the mass balance for each FV cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The set of FV cells C can be partitioned into three subsets: N, the set NN of cells adjacent to ΓN, D, the set ND of cells adjacent to ΓD, and the set of “interior” cells I = C \\ (D ∪ N) (that is, the cells to which boundary conditions do not contribute directly to their mass balance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The set I has cardinality NI = N − NN − ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' MAP formulation We assume that Nus and Nys measurements of u and y, denoted by us and ys, respectively, are collected at the cell centers indicated by the vectors of observation indices Iu and Iy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' That is, [us]i ≡ u(ˆx[Iu]i), [ys]i ≡ y(ˆx[Iu]j), i ∈ [1, Nus], j ∈ [1, Nys].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Using these measurements, we aim to estimate y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The MAP estimator [1] of y is computed by minimizing the sum of the ℓ2-norm of the discrepancy between measurements and model predictions, plus a regularization penalty on y, that is, by solving the PDE-constrained minimization problem min u,y 1 2∥us − Huu∥2 2 + 1 2∥ys − Hyy∥2 2 + γR(y), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' l(u, y) = 0, (5) where R(y) is the regularization penalty, γ > 0 is a regularization weight, and Hu : RNus×N and Hy : RNys×N are observation matrices, which downsample u and y using the observation indices Iu and Iy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Specifically, Hu ≡ IN[Iu], and Hy ≡ IN[Iy] are submatrices of the N ×N identity matrix IN corresponding to the rows of indices Iu and Iy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For y, we employ the so-called “H1 regularization,” which penalizes the H1 seminorm of y (the ℓ2-norm of the gradient of y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In the discrete case, the H1 seminorm penalty is of the form ∥Dy∥2 2, where D is the TPFA dis- cretization of the gradient operator such that Dy is equal to the gradients of y across the interior faces of the FV discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The resulting PDE- constrained minimization reads min u,y 1 2∥us − Huu∥2 2 + 1 2∥ys − Hyy∥2 2 + γ 2∥Dy∥2 2, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' l(u, y) = 0, (6) 7 The MAP estimates ˆu and ˆy obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (6) are equivalent to the largest mode (ˆu, ˆy) of the joint posterior distribution of (u, y) in a Bayesian interpretation of the inverse problem, in which the data misfit terms corre- spond to a Gaussian negative log-likelihood and the regularization penalty to a Gaussian negative log-prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' CKLEMAP method for inverse problems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Parameterizing y(x) via conditional Karhunen-Lo´eve expansions As in the PICKLE method [22, 23], we represent the unknown parameter field y(x) using the truncated CKLE yc(x, ξ) ≡ ¯yc(x) + Ny � i=1 φy i (x) � λy i ξi, (7) where ξ ≡ (ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' , ξNy)⊤ is the vector of CKLE coefficients and the eigen- pairs {φy i (x), λy i }Ny i=1 are the solutions of the eigenvalue problem � D Cc y(x, x′)φy(x′) dx′ = λyφy(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (8) Here, ¯yc(x) and Cc y(x, x′) denote the mean and covariance of y(x) conditioned on the measurements yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The CKLE is truncated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', Ny is selected) such as to achieve a desired relative tolerance rtoly ≡ N � i=Ny+1 λy i / N � i=1 λy i , (9) where N is the number of FV cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The GPR (or Kriging) equations are used to compute yc(x) and Cc y(x, y): ¯yc(x) = C(x)C−1 s ys, (10) Cc y(x, x′) = Cy(x, x′) − C(x)C−1 s C(x′), (11) where Cs is the Nys × Nys observation covariance matrix with elements [Cs]ij = Cy(ˆx[Iy]i, ˆx[Iy]j) and C(x) is the Nys-dimensional vector function with components [C(x)]i = Cy(x, ˆx[Iy]i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 8 The prior covariance kernel Cy(x, y) is estimated as in the GPR method by choosing a parameterized covariance model and computing its hyperpa- rameters by minimizing the marginal log-likelihood of the data ys [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In this work, we employ the 5/2-Mat´ern kernel as the prior covariance model, Cy(x, y) = σ2 � 1 + √ 5|x − y| l + 5 3 |x − y|2 l2 � exp � − √ 5|x − y| l � , with hyperparameters σ and λ, which correspond to the standard deviation and the correlation length, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' By representing y(x) via the CKLE (7), we replace the discrete vector y as the unknown of the inverse problem with the CKLE coefficients ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Specifically, we propose parameterizing y in the MAP problem (6) via the discrete CKLE yc(ξ) ≡ ¯yc + Ψyξ, (12) where [¯yc]i ≡ ¯yc(ˆxi), [Ψy]ij ≡ � λy jφy j(ˆxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We refer to this approach as the “CKLEMAP” method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Given that, for sufficiently smooth log-transmissivity fields, the number of CKLE coefficients required to accurately represent yc is much smaller than the number of FV cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', Ny ≪ N, the CKLEMAP method is less computationally expensive than the MAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' CKLEMAP minimization problem formulation By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (4) with y = yc(ξ), it can be seen that u can be expressed as a function of ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' specifically, u(ξ) = [A(ξ)]−1 b(ξ), (13) where A (ξ) = A (yc(ξ)) and b (ξ) = b (yc(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' By expressing u as a func- tion of ξ, we can remove the PDE constraint from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (6), leading to the CKLEMAP unconstrained minimization problem min ξ 1 2∥us − Huu(ξ)∥2 2 + 1 2∥ys − Hyyc(ξ)∥2 2 + γ 2∥Dyc(ξ)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (14) To solve the CKLEMAP problem Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (14), we recast it as the nonlinear least-squares minimization problem min ξ 1 2 ∥f(ξ)∥2 2 , f(ξ) = � � us − Huu(ξ) ys − Hyyc(ξ) √γ Dyc(ξ) � � , 9 which we solve using the Trust Region Reflective algorithm [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The least- square minimization algorithm requires the evaluation of the Jacobian Jξ of the objective vector of the least-squares problem, f, which is also the most computationally demanding part of the least-square minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' This Jacobian evaluation is done in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' First, we evaluate the Jacobian of the objective vector with respect to yc, which reads Jξ = Jyc � ∂yc ∂ξ I � = � � −Hu ∂u(yc) ∂yc −Hy √γ D � � �Ψy I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (15) The partial derivative ∂u/∂yc is evaluated via the chain rule [3, 23] as de- scribed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We note that most elements of Jyc are constant over iterations except the partial derivatives in the first block row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' These con- stant values are computed once before the least-square minimization and reused in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' With Jyc computed, Jξ can then be evaluated by postmultiplying the first block column by Ψy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Computations of partial derivatives in the evaluation of Jacobian In this section we describe how the partial derivative ∂u/∂yc, required to evaluate the Jacobian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (15), are evalauted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Let p denote yc i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Differen- tiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (4) with respect to p yields dl dp = ∂l ∂u ∂u ∂p + ∂l ∂p = A∂u ∂p + �∂A ∂p u − ∂b ∂p � = 0, (16) which can be readily solved for ∂u/∂p, leading to the expression ∂u ∂p = −A−1 �∂A ∂p u − ∂b ∂p � = −A−1 ∂l ∂p ���� u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (17) It can be seen that evaluating ∂u/∂yc requires evaluating the sensitivities of the TPFA stiffness matrix A and right-hand side vector b with respect to yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (17) into the first row block of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (15) and taking the transpose yields � ∂l ∂yc ���� u �⊤ A−1H⊤ u, (18) by the fact that A is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 10 Note that in the MAP method, the Jacobian is given as Jy = � � −Hu ∂u(y) ∂y −Hy √γ D � � , (19) and the partial derivatives are computed as in the CKLEMAP method, with y being treated the same way as yc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Accelerated CKLEMAP method In the “accelerated” CKLEMAP method, we compute A−1H⊤ u efficiently by exploiting the sparsity structure of the Cholesky factor of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Recall that each column of H⊤ u = (IN[Iu])⊤ has only one non-zero entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Therefore, if the sparsity structure of the Cholesky factor L of A is known, the sparsity structure of each column of Z = L−1H⊤ u is {closureL(i) | i ∈ Iu}, that is, the subset of vertices in the graph G(L) that have a path from each vertex i ∈ Iu [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Figure 1 shows an example of a closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Furthermore, the graph of a Cholesky factor L is a directed tree, and any closure induced by a vertex i is all the vertices along the path from i to the root of the tree [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' This enables a simple algorithm to find the sparsity structure of the solution of LZ = H⊤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Figure 2 illustrates this algorithm together with a graphical example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Once we have the sparsity structure Zi of zi, the column i of Z, we only need the submatrix L[Zi, Zi] instead of the whole matrix L to solve for zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Such submatrix is highlighted in blue dots in the lower triangular matrix L in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' This eliminates the unnecessary computations involving the part of L that does not contribute to the final solutions, thus accelerating the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Furthermore, since the topology of the FV discretization is static, the sparsity structure of the Cholesky factor L is fixed throughout the entire least-square minimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Given this, together with the fact that Hu is constant, it follows that Zi is also fixed and only needs to be computed once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Figure 3 shows the closures of two observation locations in Iu on the Hanford Site experiment to be discussed in detail in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The gray lines indicate the cells that do not contribute to the columns of the Jacobian corresponding to either of these two locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We note that, although the computations of the Jacobian can be acceler- ated by 3–4 times using the procedure described above, the overall execution time reduction in solving the minimization problems exhibited by the nu- merical experiments of Section 5 is 10–20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' This is because the nonlinear 11 1 2 3 4 5 6 7 8 Figure 1: Closure of a unit column vector e3 ≡ [0, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' ]⊤ in a graph G(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The nonzero entries of A−1e3 are those nodes in the closure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', {3, 4, 6, 7, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 1: procedure FindSparsity(L, x) 2: j ← x 3: S ← {j} 4: while j ̸= N do 5: j ← argmini>jL[i, j] ̸= 0 6: S ← S ∪ {j} 7: end while 8: return S 9: end procedure (a) Algorithm L × z = ex (b) Graphical Example Figure 2: Algorithm for finding the sparsity structure S of z = L−1ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' least-squares minimization algorithm, the Trust Region algorithm, dominates most of the execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The execution times can be further reduced by optimizing the implementation of the Trust Region algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Numerical experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Case study We evaluate the performance of the proposed CKLEMAP formulation against MAP with a case study of parameter estimation in a steady-state two-dimensional groundwater model of the Hanford Site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The reference log- transmissivity field ˜y and boundary conditions uD and qN are based on the data obtained from a three-dimensional Hanford Site calibration study [27] and are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The details of the reference transmissivity field generation are given in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' To study the scalability of the CKLEMAP and MAP methods with the problem size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', the number of cells in the FV model), we generate the reference field at two additional resolutions with four times and 16 times the number of cells in the base FV model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 12 directed tree G(L) and its root closure 1 closure 2 common closure Figure 3: The directed tree G(L) structure on with two closures from different cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 13 The numbers of cells in the low, medium, and high-resolution models are 1475, 5900, and 23600, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For a higher resolution mesh, we divide each cell in a lower resolution model into four equiareal subcells and interpolate ˜y at the centers of each subcell, as well as uD and qN at the midpoints of each boundary edge of the boundary subcells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' There are 558 wells at the Hanford Site where u can be potentially mea- sured [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Some of these wells are located in the same coarse or fine cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Figure 4 shows the locations of the cells in the low-resolution FV model that contain at least one well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Since our model uses exclusively cells but not points to specify spatial locations, multiple wells are treated as a single well if they are located in the same cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' As a result, there are 323 wells in the low-resolution FV model, while the medium-resolution model has 408 wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The aforementioned Hanford Site calibration study defined the Dirichlet and Neumann boundaries ΓD and ΓN as shown in Figure 4, and provides the estimates of the heads uD and the fluxes qN at these boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In setting boundary conditions for our comparison study, we assume that uD and qN are both known and are given by the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For each reference log-transmissivity field ˜y, we generate the hydraulic head field ˜u by solving the Darcy flow equation on the corresponding FV mesh with the Dirichlet and (deterministic) Neumann boundary conditions that are set as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The values of the reference y and u fields at all cell locations ˆxi are organized into the vectors ˜y and ˜u, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Then, we randomly pick Nys well locations and treat the values of ˜y at these locations as y measurements to form ys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Similarly, we draw Nus measurements of the hydraulic head u from ˜u to form us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' These measurements are treated as synthetic data sets and used in the CKLEMAP and MAP methods to estimate the entire y and u fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We note that the aquifer at the Hanford Site is unconfined, and the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (1) to describe flow at the Hanford Site relies on a conceptual simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' A more accurate linear conceptual model for flow in an un- confined aquifer with a horizontal confining layer can be obtained based on the Dupuit–Forchheimer approximation in the form [32] ∇ · [K(x)∇v(x)] = 0, x ∈ D, (20) where v(x) = u2(x) and K(x) is the depth-averaged conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Mathe- matically, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (1) and (20) are identical, although the field u(x) computed using these two equations will be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Therefore, solving the inverse 14 Umtanum Ridge Cold Creek Valley Dry Creek Valley Rattlesnake Hills Rattlesnake Springs Recharge Area Gable Butte Gable Mountain Columbia River Yakima River well locations Dirichlet boundary conditions Neumann boundary conditions no-flow boundary condition Figure 4: The coarse-resolution mesh of (NF V = 1475) cells with well locations marked, and the parts of boundaries colored for different types of prescribed boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 15 problem for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (1) is equivalent in complexity to solving the inverse prob- lem for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We also note that applying the Dupuit–Forchheimer ap- proximation to the Hanford Site aquifer will produce additional linear terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (20) due to the variations in the elevation of the bottom confining layer of the aquifer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The implementation of CKLEMAP and MAP are written in Python using the NumPy and SciPy packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' All CKLEMAP and MAP simulations are performed using a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2 GHz 8-core Intel Xeon W CPU and 32 GB of 2666 MHz DDR4 RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The weight γ in the CKLEMAP and MAP minimization problems is empirically found to minimize the error with respect to the reference y fields as γ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' When a reference field is not known, these weights can be found using cross-validation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Performance of CKLEMAP as a function of the number of KL terms Table 1: Performance of CKLEMAP in estimating the coarse-resolution (NF V = 1475) mesh with Nys = 100 as functions of number of KL terms Ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Ny 200 400 600 800 1000 least square iterations 99–218 44–335 25–69 28–177 20–65 execution time (s) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='55– 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='14 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='37– 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='31 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='76– 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='73 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='60– 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='86 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='25– 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='29 relative ℓ2 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='265– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='137– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='081– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='072– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='072– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='083 absolute ℓ∞ error 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='08– 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='69 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='56– 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='71–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='68–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='46–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='31 First, we study the relative ℓ2 and absolute ℓ∞ errors in the CKLEMAP solution for y as well as the time-to-solution and the number of iterations of the minimization algorithm as functions of Ny, the number of terms in the CKLE of y for Nys = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The relative ℓ2 and absolute ℓ∞ errors are 16 200 400 600 800 1000 10−1 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='5 Number of KL terms ℓ2 errors Figure 5: Relative ℓ2 errors versus the number of KL terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' computed on the FV mesh, respectively, as ε2(y) ≡ ∥ˆy − ˜y∥2 ∥˜y∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (21) and ε∞(y) ≡ ∥ˆy − ˜y∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' (22) We find that for the considered inverse problem, all these quantities strongly depend on the locations of y measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Therefore, we compute these quantities for 10 different distributions of the measurement locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The ranges of the ℓ2 and ℓ∞ errors, execution times, and the numbers of iterations are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The ℓ2 error and its bounds as functions of Ny are also plotted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We find that the ℓ2 errors decrease with increasing Ny and converge to asymptotic values for Ny ≈ 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The lower bound of ℓ∞ continues to decrease even for Ny greater than 800, while the upper bound increases from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='22 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='31 as Ny increases from 800 to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' However, the relative changes of ℓ∞ are insignificant for Ny > 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' What is surprising is that the execution time does not significantly change with increasing Ny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' While the time per iteration increases with Ny, the number of iterations tends to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Therefore, in the rest of the numerical examples, we set Ny = 1000, which corresponds to rtoly on the order of 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' CKLEMAP and MAP errors versus the number of y measurements Next, we study the accuracy of the CKLEMAP and MAP methods in estimating y as the function of the number of y measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We assume 17 that u measurements are available at all wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We start with the low-resolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Figure 6 shows the locations of y measurements, the y fields estimated by the MAP and CKLEMAP methods for Nys = 25, 50, 100, and 200, and the distributions of point errors in the MAP and CKLEMAP estimates of y relative to the reference field ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For the considered measurement locations, we observe that the MAP and CKLEMAP methods have comparable accuracy for all Nys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Table 2 shows the ranges of relative ℓ2 and absolute ℓ∞ errors in the MAP and CKLEMAP y estimates as well as the number of iterations in the minimization algorithm and the execution times (in seconds) for Nys ranging from 25 to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Also included in this table are the execution times of the accelerated CKLEMAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We note that the accuracy (including the ℓ2 and absolute ℓ∞ errors) and the number of iterations in the accelerated CKLEMAP and CKLEMAP methods are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' As expected, the accuracy of the MAP and CKLEMAP methods increases with Nys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The MAP and CKLEMAP methods are almost equally accurate, with ℓ2 and ℓ∞ errors in the CKLEMAP method being slightly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' How- ever, we observe that CKLEMAP is faster than MAP for all considered values of Nys except for Nys = 25, where the MAP’s lower bound of the execution time is less than that of the CKLEMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Accelerated CKLEMAP is about 20% faster than CKLEMAP and for all considered values of Nys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Accelerated CKLEMAP is also faster than MAP for all considered cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' however, the speedup depends on Nys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In all examples reported in Table 2, the number of unknowns in the CKLEMAP method is 1000 (the number of terms in the CKLE expansion), while in the MAP method, this number is 1475 (the number of cells in the FV model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The reason for CKLEMAP being slower than MAP for Nys = 25 and certain y measurement locations is that for such locations MAP converges much faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For example, the lower execution time bands in MAP and CK- LEMAP correspond to 29 and 50 iterations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' However, because there are fewer unknowns in the CKLEMAP method, the CKLEMAP com- putational time per iteration is smaller than that in MAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' As a result, the computational time in the CKLEMAP is only 20% larger than that of MAP for these limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The time per iteration is further reduced in the accelerated CKLEMAP method, resulting in the execution time of acceler- ated CKLEMAP being less than that of MAP by 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We also note that for Nys > 25, MAP requires more iterations than CKLEMAP, making the computational advantages of CKLEMAP even more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 18 reference 0 2 4 6 8 10 12 Nys 25 50 100 200 observation locations CKLEMAP estimates 0 2 4 6 8 10 12 CKLEMAP point errors 0 1 2 3 4 5 6 MAP estimates 0 2 4 6 8 10 12 MAP point errors 0 1 2 3 4 5 6 Figure 6: The fine-resolution (NF V = 5900) reference y fields, the CKLEMAP and MAP estimates of the y field and their point errors as functions of Nys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 19 Next, we perform a similar study for the medium-resolution model with N = 5900 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Table 3 provides a comparative summary of the models con- sidered for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Here, we find that CKLEMAP is slightly more accurate than MAP for all considered values of Nys and one to two orders of magni- tude faster than MAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Accelerated CKLEMAP is approximately 10% faster than CKLEMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The computational advantage of CKLEMAP significantly increases with the problem size as the number of unknown parameters in the MAP linearly increases with the problem size while the number of parameters in the CKLEMAP is independent of the problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Scaling of the execution time with the problem size Table 2: Performance of MAP and CKLEMAP in estimating the coarse-resolution (NF V = 1475) mesh as functions of Nys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Nys solver 25 50 100 200 least square iterations MAP 29–95 29–106 41–60 28–80 CKLEMAP 50–96 26–70 20–65 33–62 execution time (s) MAP 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='36– 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='50 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='98– 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='61 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='39– 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='08 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='88– 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='32 CKLEMAP 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='01– 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='04 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='71– 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='86 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='25– 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='29 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='73– 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='57 accelerated CKELMAP 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='45– 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='97 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='00– 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='07– 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='41 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='78– 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='09 relative ℓ2 error MAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='092– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='084– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='073– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='068– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='073 CKLEMAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='091– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='082– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='072– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='064– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='071 absolute ℓ∞ error MAP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='38–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='95–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='06–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='88–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='74 CKLEMAP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='96–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='73–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='46–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='63–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='71 The comparison of Tables 2 and 3 shows that the execution times of the MAP, CKLEMAP, and accelerated CKLEMAP increase with the mesh resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' however, the execution times of CKLEMAP and accelerated CK- LEMAP increase slower than that of MAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' To study the scalability of these 20 Table 3: Performance of MAP and CKLEMAP in estimating the fine-resolution (NF V = 5900) mesh as functions of Nys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Nys solver 25 50 100 200 least square iterations MAP 78–99 71–97 69–83 23–76 CKLEMAP 53–114 20–142 36–60 15–83 execution time (s) MAP 3907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='00– 4868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='21 3528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='90– 4580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='40 3533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='06– 4190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='20 1247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='37– 3733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='05 CKLEMAP 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='76– 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='67 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='45– 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='08 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='59– 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='04 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='86– 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='03 accelerated CKELMAP 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='05– 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='90 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='50– 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='63 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='14– 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='19 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='28– 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='18 relative ℓ2 error MAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='0954– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='081– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='074– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='065– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='073 CKLEMAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='0906– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='081– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='068– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='061– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='069 absolute ℓ∞ error MAP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='96–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='45–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='00–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='37–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='20 CKLEMAP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='21–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='94–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='79–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='82–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='28 21 1475 5900 23600 101 102 103 104 105 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='7 · 10−8x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='11 · 10−3x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='11 · 10−4x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='35 Number of FV cells Execution time (s) MAP CKLEMAP accelerated CKLEMAP Figure 7: Execution times of MAP, CKLEMAP, and accelerated CKLEMAP methods versus the number of FV cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The execution times of MAP for the mesh with 23600 FV cells are estimated by extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' methods with the problem size, we use these methods to estimate y in the high-resolution FV model with N = 23600 and, in Figure 7, we plot the execution times of these methods as functions of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The number of y mea- surements in all simulations reported in this figure is set to Nys = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We also show the power-law models fitted to the scalability curves com- puted using MAP, CKLEMAP, and accelerated CKLEMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We note that for N = 23600, the MAP method did not converge after running for two days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Therefore, the power law relationship for the MAP method is obtained based on the execution times for N = 1475 and 5900 and used to estimate the MAP’s execution time for the highest resolution by extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We find that the MAP, CKLEMAP, and accelerated CKLEMAP execution times scale as N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='91, N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='33, and N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='35, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Therefore, the CKLEMAP methods have a computational advantage over the MAP method for large problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The CKLEMAP and accelerated CKLEMAP methods have ap- proximately the same scalability, but for the same problem size, the acceler- ated CKLEMAP method is 10–20% faster than the CKLEMAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 22 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Discussion and Conclusions We proposed the CKLEMAP method as an alternative to the MAP meth- ods for solving inverse PDE problems and used it for estimating the trans- missivity and hydraulic head in a two-dimensional steady-state groundwater model of the Hanford Site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The CKLEMAP method is based on the ap- proximation of unknown parameters (log-transmissivity in this case) with CKLEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The advantage of using a CKLE over other representations (like DNNs in [13]) is that it enforces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', exactly matches) the field measure- ments and the covariance structure, that is, it models the field as a realization of the conditional Gaussian field with a prescribed covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' As a general conclusion, we found that the accuracy of the MAP and CKLEMAP methods is essentially the same (with CKLEMAP being a few percents more accurate under most tested conditions), but CKLEMAP is faster than MAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Specifically, we demonstrated that the CKLEMAP and MAP execution times scale with the problem size as N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='33 and N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='91, respectively, where N is the number of FV cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The close-to-linear scaling of CKLEMAP’s execution time with problem size gives CKLEMAP a computational advantage over the MAP method for large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We consider this to be the main advantage of the CKLEMAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' For the same number of measurements, the accuracy of MAP and CK- LEMAP can depend on the measurement locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Both the MAP and the CKLEMAP methods are, on average, equally accurate in terms of abso- lute ℓ∞ errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The CKLEMAP method is slightly more accurate than the MAP method in terms of relative ℓ2 errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The execution times of MAP and CKLEMAP increase, and their accuracy decreases, as the number of y measurements decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In the CKLEMAP method, execution time and accuracy increase with the increasing number of CKL terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' In this work, as a baseline, we used Ny = 1000, which corresponds to rtol < 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We stipulate that this criterion is sufficient to obtain a convergent estimate of y with respect to the number of CKL terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' To further reduce the computational time, we proposed the accelerated CKLEMAP method, which takes advantage of the sparse structure of the stiffness matrix in the FV discretization of the residual term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' We demon- strated that the scalability of the accelerated CKLEMAP and CKLEMAP methods is approximately the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' however, for the same problem size, accelerated CKLEMAP is 10–20% faster than the CKLEMAP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Acknowledgments This research was partially supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Department of Energy (DOE) Advanced Scientific Computing program and the United States Geo- logical Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' The data and codes used in this paper are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='com/yeungyh/cklemap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Kitanidis, On the geostatistical approach to the inverse problem, Advances in Water Resources 19 (6) (1996) 333–342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1016/ 0309-1708(96)00005-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Stuart, Inverse problems: A Bayesian perspective, Acta Numerica 19 (2010) 451–559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1017/S0962492910000061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Wohlberg, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Vesselinov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tar- takovsky, Linear functional minimization for inverse modeling, Water Resources Research 51 (2014) 4516–4531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1002/2014WR016179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Wild, POUNDERS in TAO: Solving Derivative-Free Nonlinear Least-Squares Problems with POUNDERS, Society for Industrial and Applied Mathematics, 2017, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 40, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 529–539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1137/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 9781611974683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='ch40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Mo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zabaras, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Wu, Integration of adversarial autoen- coders with residual dense convolutional networks for estimation of non-gaussian hydraulic conductivities, Water Resources Research 56 (2) (2020) e2019WR026082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1029/2019WR026082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Mo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zhu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zabaras, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Wu, Deep convolutional encoder- decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media, Water Resources Research 55 (1) (2019) 703–728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1029/2018WR023528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Mo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zabaras, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Wu, Deep autoregressive neural net- works for high-dimensional inverse problems in groundwater contami- nant source identification, Water Resources Research 55 (5) (2019) 3856– 3881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1029/2018WR024638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 24 [8] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Chada, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Sanz-Alonso, Iterative ensemble kalman meth- ods: A unified perspective with some new variants, Foundations of Data Science 3 (3) (2021) 331–369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='3934/fods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2021011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Lin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zeng, An iterative local updating ensemble smoother for estimation and uncertainty assessment of hydro- logic model parameters with multimodal distributions, Water Resources Research 54 (3) (2018) 1716–1733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1002/2017WR020906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Certes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' de Marsily, Application of the pilot point method to the identification of aquifer transmissivities, Advances in Water Resources 14 (5) (1991) 284–300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1016/0309-1708(91)90040-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Doherty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Fienen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Hunt, Approaches to highly parame- terized inversion: Pilot-point theory, guidelines, and research directions, US Geological Survey scientific investigations report 5168 (2010) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='3133/sir20105168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tonkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Doherty, A hybrid regularized inversion methodology for highly parameterized environmental models, Water Resources Re- search 41 (10) (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1029/2005WR003995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Xu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Burghardt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Darve, Learning viscoelas- ticity models from indirect data using deep neural networks, Computer Methods in Applied Mechanics and Engineering 387 (2021) 114124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='cma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='114124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' O’Malley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Golden, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Vesselinov, Learning to regularize with a variational autoencoder for hydrologic inverse analysis (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='02401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Marrero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Perdikaris, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, Physics-informed deep neural networks for learn- ing parameters and constitutive relationships in subsurface flow prob- lems, Water Resources Research 56 (5) (2020) e2019WR026731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1029/2019WR026731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [16] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' He, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, Physics- informed neural networks for multiphysics data assimilation with ap- plication to subsurface transport, Advances in Water Resources (2020) 103610doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='103610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 25 [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Treichler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Kurth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Fischer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Romero, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Churavy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Houston, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Prabhat, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Karniadakis, Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs, in: 2019 IEEE/ACM Third Workshop on Deep Learning on Supercomputers (DLS), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1109/DLS49591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='00006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' van Leeuwen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Herrmann, A penalty method for PDE- constrained optimization in inverse problems, Inverse Problems 32 (1) (2015) 015007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1088/0266-5611/32/1/015007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Kadeethum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' O’Malley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Fuhg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Choi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Viswanathan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Bouklas, A framework for data-driven solution and parameter estimation of pdes using conditional generative adversar- ial networks, Nature Computational Science 1 (12) (2021) 819–829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1038/s43588-021-00171-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Jin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Pang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nature Machine Intelligence 3 (3) (2021) 218–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1038/s42256-021-00302-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tipireddy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, Conditional Karhunen-Lo`eve expansion for uncertainty quantification and active learning in partial differential equation models, Journal of Computa- tional Physics 418 (2020) 109604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='109604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' He, Physics-informed machine learning with conditional Karhunen-Lo`eve expansions, Journal of Com- putational Physics (2020) 109904doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='109904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Yeung, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Barajas-Solano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Tartakovsky, Physics- informed machine learning method for large-scale data assimilation problems, Water Resources Research 58 (5) (2022) e2021WR031023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1029/2021WR031023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [24] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Kitanidis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Lee, Principal component geostatistical approach for large-dimensional inverse problems, Water Resources Research 50 (7) (2014) 5428–5443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1002/2013WR014630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 26 [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Lee, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Kitanidis, Large-scale hydraulic tomography and joint inversion of head and tracer data using the principal component geo- statistical approach (PCGA), Water Resources Research 50 (7) (2014) 5410–5427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1002/2014WR015483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Marzouk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Najm, Dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems, Journal of Computational Physics 228 (6) (2009) 1862–1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Cole, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Bergeron, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Wurstner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Thorne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Orr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Mckinley, Transient inverse calibration of Hanford site-wide groundwa- ter model to Hanford operational impacts - 1943 to 1996, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=', Pacific Northwest National Laboratory (PNNL), Richland, Washington, United States (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2172/965656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Rasmussen, Gaussian processes in machine learning, Springer Berlin Heidelberg, 2004, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 63–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1007/ 978-3-540-28650-9_4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Branch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Coleman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Li, A subspace, interior, and con- jugate gradient method for large-scale bound-constrained minimization problems, SIAM Journal on Scientific Computing 21 (1) (1999) 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1137/S1064827595289108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Gilbert, Predicting structure in sparse matrix computations, SIAM Journal on Matrix Analysis and Applications 15 (1) (1994) 62–79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1137/S0895479887139455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Yeung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Crouch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Pothen, Interactively cutting and constrain- ing vertices in meshes using augmented matrices, ACM Transactions on Graphics 35 (2) (February 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1145/2856317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Zhang, Nonlinear inversion of an unconfined aquifer: simultaneous estimation of heterogeneous hydraulic conductivities, recharge rates, and boundary conditions, Transport in porous media 102 (2) (2014) 275–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='1007/s11242-014-0275-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Picard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' Cook, Cross-validation of regression models, Journal of the American Statistical Association 79 (387) (1984) 575–583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content='2307/2288403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf'} diff --git a/TNFAT4oBgHgl3EQf2R4K/vector_store/index.pkl b/TNFAT4oBgHgl3EQf2R4K/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2bb021a502c0be641b07c0354182a2786ff94cff --- /dev/null +++ b/TNFAT4oBgHgl3EQf2R4K/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fd23cf844293e5eccb221e0b54bea2cb9993c6c3fcf6802983acfe530e4f3221 +size 135569 diff --git a/WdFRT4oBgHgl3EQfMjc9/content/tmp_files/2301.13506v1.pdf.txt b/WdFRT4oBgHgl3EQfMjc9/content/tmp_files/2301.13506v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b78cb9a73c77d323c687255b60c82164967cd7d --- /dev/null +++ b/WdFRT4oBgHgl3EQfMjc9/content/tmp_files/2301.13506v1.pdf.txt @@ -0,0 +1,5735 @@ +DNN Explanation for Safety Analysis: an Empirical +Evaluation of Clustering-based Approaches +MOHAMMED OUALID ATTAOUI, SnT Centre, University of Luxembourg, Luxembourg +HAZEM FAHMY, SnT Centre, University of Luxembourg, Luxembourg +FABRIZIO PASTORE, SnT Centre, University of Luxembourg, Luxembourg +LIONEL BRIAND, SnT Centre, University of Luxembourg, Luxembourg and School of EECS, University of +Ottawa, Canada +The adoption of deep neural networks (DNNs) in safety-critical contexts is often prevented by the lack of +effective means to explain their results, especially when they are erroneous. In our previous work, we proposed +a white-box approach (HUDD) and a black-box approach (SAFE) to automatically characterize DNN failures. +They both identify clusters of similar images from a potentially large set of images leading to DNN failures. +However, the analysis pipelines for HUDD and SAFE were instantiated in specific ways according to common +practices, deferring the analysis of other pipelines to future work. +In this paper, we report on an empirical evaluation of 99 different pipelines for root cause analysis of +DNN failures. They combine transfer learning, autoencoders, heatmaps of neuron relevance, dimensionality +reduction techniques, and different clustering algorithms. Our results show that the best pipeline combines +transfer learning, DBSCAN, and UMAP. It leads to clusters almost exclusively capturing images of the same +failure scenario, thus facilitating root cause analysis. Further, it generates distinct clusters for each root cause +of failure, thus enabling engineers to detect all the unsafe scenarios. Interestingly, these results hold even for +failure scenarios that are only observed in a small percentage of the failing images. +CCS Concepts: • Software and its engineering → Software defect analysis; • Computing methodolo- +gies → Machine learning. +Additional Key Words and Phrases: DNN Explanation, DNN Functional Safety Analysis, DNN Debugging, +Clustering, Transfer Learning +ACM Reference Format: +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand. 2023. DNN Explanation for +Safety Analysis: an Empirical Evaluation of Clustering-based Approaches. 1, 1 (February 2023), 44 pages. +https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +Deep neural networks (DNNs) have achieved extremely high predictive accuracy in various domains, +such as computer vision [3, 64], autonomous driving [42, 80], and natural language processing [18, +54]. Despite their superior performance, the lack of explainability of DNN models remains an issue +Authors’ addresses: Mohammed Oualid Attaoui, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, +mohammed.attaoui@uni.lu; Hazem Fahmy, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, +hazem.fahmy@uni.lu; Fabrizio Pastore, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, fabrizio. +pastore@uni.lu; Lionel Briand, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, School of EECS, +University of Ottawa, Ottawa, Canada, lionel.briand@uni.lu. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee +provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and +the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. +Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires +prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2023 Association for Computing Machinery. +XXXX-XXXX/2023/2-ART $15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +, Vol. 1, No. 1, Article . Publication date: February 2023. +arXiv:2301.13506v1 [cs.SE] 31 Jan 2023 + +2 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +in many contexts. While they can approximate complex and arbitrary functions, studying their +structure often provides little or no insight into the underlying prediction mechanisms. There +seems to be an intrinsic tension between Machine Learning (ML) performance and explainability. +Often the highest-performing methods (for example, Deep Learning) are the least explainable, and +the most explainable (for example, decision trees) are the least accurate [30]. +For DNNs to be trustworthy, in many critical contexts where they are used, we must understand +why they behave the way they do [7]. Explanation methods aim at making neural network decisions +trustworthy [27]. Several explanation methods are proposed in the literature (see Section 5). In our +work, because of our focus on safety analysis, we focus on explanation methods for root cause +analysis, which concerns identifying the underlying reason of a DNN failure (root cause), which is, +in our context, an incorrect DNN prediction or classification. +Root cause analysis techniques based on unsupervised learning have proven their effective- +ness [78, 87]. These methods group failure samples (e.g., data collected during hardware testing) +without requiring diagnostic labels, such that the samples in each cluster share similar root causes. +Our previous work is the first application of unsupervised learning to perform root cause analysis +targeting DNN failures. Precisely, we proposed two DNN explanation methods: SAFE (Safety +Analysis based on Feature Extraction) [4] and HUDD (Heatmap-based Unsupervised Debugging of +DNNs) [20]. They both process a set of failure-inducing images and generate clusters of similar +images. Commonalities across images in each cluster provide information about the root cause +of the failure. For example, applying our approaches to failure-inducing images for a DNN that +classifies car seat occupancy may include a cluster of images with child seats containing a bag; +such cluster may help engineers determine that bags inside child seat are likely to be misclassified +and, therefore, the training set should be improved accordingly (e.g., more child seats with objects +should be considered). Both SAFE and HUDD also support the identification of additional images +to be used to retrain the DNN. +HUDD and SAFE differ with respect to the kind of data used to perform clustering and the pipeline +of steps they rely on. HUDD applies clustering based on internal DNN information; precisely, for +all failure-inducing images, it generates heatmaps capturing the relevance of DNN neurons on the +DNN output. Finally, it applies a hierarchical clustering algorithm relying on a distance metric based +on the generated heatmaps. SAFE is black-box as it does not rely on internal DNN information. +It generates clusters based on the visual similarity across failure inducing images. To this end it +relies on feature extraction based on transfer learning, dimensionality reduction, and the DBSCAN +clustering algorithm. +SAFE and HUDD rely on a pipeline that has been configured in specific ways according to best +practices. However, several variants exist for each component of both approaches (e.g., different +transfer learning models, different clustering algorithms). +In this paper, we aim to evaluate these pipeline variants for both SAFE and HUDD. Therefore, we +propose an empirical evaluation of 99 alternative configurations for SAFE and HUDD (pipelines). +These pipelines were obtained using different combinations of feature extraction methods, clustering +algorithms, dimensionality reduction techniques, in addition, we assessed the effect of fine tuning +the transfer learning models used by feature extraction methods. +For our empirical evaluation we considered six case study subjects, two of which were provided +by our industry partner in the automotive domain, IEE Sensing [36]. Our subjects’ applications +include head pose classification, eye gaze detection, drowsiness detection, steering angle prediction, +unattended child detection, and car position detection. +We present a systematic and extensive evaluation scheme for these pipelines, which entails +generating failure causes that resemble realistic scenarios (e.g., poor lighting conditions or camera +misconfiguration). Since the reason of failure in these scenarios are known a priori, such an +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +3 +evaluation scheme enables us to objectively analyze and evaluate the performance and robustness +of these pipelines. +Our empirical results conclude that the best pipelines support and facilitate the process of +functional safety analysis such that they 1) can generate RCCs that group together a very high +proportion of images capturing a same root cause (94.3%, on average), 2) can capture most of the +root causes of failures for all case study subjects (96.7%, on average), and 3) are robust to the rarity +of failure instances in a data set (i.e., when some causes of failures affect less than 10% of the +failure-inducing images). +The remainder of this paper is organized as follows. In Section 2, we briefly present the main +features and limitations of SAFE and HUDD, along with other feature extraction models (Autoen- +coders and Backpropagation-based Heatmaps). In Section 3, we describe the different models and +algorithms we use in our evaluated pipelines. In Section 4, we present the research questions, the +experiment design and results, including a comparison between pipelines. In Section 5, we discuss +and compare related work. Finally, we conclude this paper in Section 6. +2 +BACKGROUND +This section provides an overview of our previous work that inspired this research. We focus on +clustering methods, heatmap-based DNN Explanations, the HUDD and SAFE DNN explanation +methods, and Autoencoders. +2.1 +Clustering +Clustering is a data analysis method that mines essential information from a dataset by grouping +data into several groups called clusters. In clustering, similar data points are grouped into the same +cluster, while non-similar data points are put into different clusters. There are two main objectives +in data clustering; the first objective is to minimize the dissimilarity within the cluster, and the +second objective is to maximize the inter-cluster dissimilarity. HUDD and SAFE rely on hierarchical +agglomerative clustering (HAC [63]) and density-based clustering (DBSCAN [19]), respectively. +In HAC, each observation starts in its own cluster and pairs of clusters are iteratively merged to +minimize an objective function (e.g., error sum of squares [85]). DBSCAN works by considering +dense regions as clusters; it is detailed in Section 3. +2.2 +Heatmap-based DNN Explanations +Approaches that aim to explain DNN results have been developed in recent years [26]. Most +of these concern the generation of heatmaps that capture the importance of pixels in image +predictions. They include black-box [13, 60] and white-box approaches [51, 68, 72, 90, 91]. Black- +box approaches generate heatmaps for the input layer and do not provide insights regarding internal +DNN layers. White-box approaches rely on the backpropagation of the relevance score computed +by the DNN [51, 68, 72, 90, 91]. +In this Section, we focus on a white-box technique called Layer-Wise Relevance Propagation +(LRP) [51] because it has been integrated into HUDD. LRP was selected because it does not present +the shortcomings of other heatmap generation approaches [20]. +LRP redistributes the relevance scores of neurons in a higher layer to those of the lower layer. +Figure 1 illustrates how LRP operates on a fully connected network used to classify inputs. In the +forward pass, the DNN receives an input and generates an output (e.g., classifies the gaze direction +as TopLeft) while recording the activations of each neuron. In the backward pass, LRP generates +internal heatmaps for a DNN layer 𝑘, which consists of a matrix with the relevance scores computed +for all the neurons of layer 𝑘. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +4 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +TopRight +Input +Layer +Output +Layer +DNN with LRP +Layer k +DNN +output: +Top +Right +TopCenter +TopLeft +Layer j +wj1k1 +wj1k2 +wj1k3 +I +O +I +Input of DNN +Output of LRP +O +Legend: +Neuron connections +Image to Classify +Heatmap of Input Layer +Heatmaps +of Internal Layers +7,5 +5,3 +2,3 +O +Fig. 1. Layer-Wise Relevance Propagation. +Fig. 2. An example image of HPD subject (on the left) and applied LRP (on the right) showing that the mouth +had a large influence on the DNN behavior. +The heatmap in Figure 1 shows that the pupil and part of the eyelid, which are the non-white +parts in the heatmap, had a significant effect on the DNN output. Furthermore, the heatmap in +Figure 2 shows that the mouth and part of the nose are the input pixels that mostly impacted on +the DNN output. +A heatmap is a matrix with entries in R, i.e., it is a triple (𝑁, 𝑀, 𝑓 ) where 𝑁, 𝑀 ∈ N and 𝑓 is +a map [𝑁] × [𝑀] → R. We use the syntax 𝐻 [𝑖, 𝑗]𝐿 +𝑥 to refer to an entry in row 𝑖 (i.e., 𝑖 < 𝑁) and +column j (i.e., 𝑗 < 𝑀) of a heatmap 𝐻 computed on layer 𝐿 from an image 𝑥. The size of the heatmap +matrix (i.e., the number of entries) is 𝑁 · 𝑀, with 𝑁 and 𝑀 are determined by the dimensions of +the DNN layer 𝐿. For convolution layers, 𝑁 represents the number of neurons in the feature map, +whereas 𝑀 represents the number of feature maps. For example, the heatmap for the eighth layer +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +5 +of AlexNet has size 169 × 256 (convolution layer), while the the heatmap for the tenth layer has +size 4096 × 1 (linear layer). +2.3 +Heatmap-based Unsupervised Debugging of DNNs (HUDD) +Step1. +Heatmap +based +clustering +Root cause clusters +C1 +Step 5. +Label images +Step 4. Identify +Unsafe Images +Failure-inducing +test set images +Unsafe Set: +improvement set +images +belonging +to the root cause +clusters +C2 +C3 +Simulator execution +Step 3. Generate +new images +Collection of field data +Improvement +set: new images +(unlabeled) +C1 +C2 +C3 +Labeled +Unsafe Set +C1 C2 C3 +Step 7. DNN +Retraining +Step 2. Functional +Safety Analysis +Training set +images +Balanced Labeled +Unsafe Set +C1 C2 C3 +Improved +DNN +model +Step 6. +Bootstrap +DNN +model +Fig. 3. Overview of HUDD. +Although heatmaps may provide useful information to determine the characteristics of an image +that led to an erroneous result from the DNN, they are of limited applicability because, to determine +the cause of all DNN errors observed in the test set, engineers may need to visually inspect all the +error-inducing images, which is practically infeasible. To overcome such limitations, we recently +developed HUDD [20], a technique that facilitates the explanation and removal of the DNN errors +observed in a test set. HUDD generates clusters of images that lead to a DNN error because of +the same root cause. The root cause is determined by the engineer who visualizes a subset of the +images belonging to each cluster and identifies the commonality across each image (e.g., for a Gaze +detection DNN, all the images present a closed eye). To further support DNN debugging, HUDD +automatically retrains the DNN by selecting a subset from a pool of unlabeled images that will +likely lead to DNN errors because of the same root causes observed in the test set. +Figure 3 provides an overview of HUDD, which consists of six steps. In Step 1, root cause clusters +are identified by relying on a hierarchical clustering algorithm applied to heatmaps generated +for each failure inducing image. Step 2 involves a visual inspection of clustered images. In this +step, engineers visualize a few representative images for each RCC; the inspection enables the +engineers to determine which are the commonalities across the images in each cluster and, therefore, +determine the failure root cause. Example root causes include the presence of an object inside a child +seat (as reported in the Introduction) or a face turned left thus making an eye not visible and causing +misclassification in a gaze detection system. HUDD’s Step 2 supports functional safety analysis +because each failure root cause represents a usage scenario in which the DNN is likely to fail, and, +based on domain knowledge, engineers can determine the likelihood of each failure scenario, its +safety impact, and possible countermeasures, as required by functional safety analysis standards +[37, 38]. For example, objects inside child seats might be very common but they lead to false alarms +not hazards; misclassified gaze may instead instead prevent the system from determining that the +driver is not pay attention to the road. Countermeasures include the retraining of the DNN, which +is supported by HUDD’s Step 3. In Step 3, a new set of images, referred to as the improvement set, is +provided by the engineers to retrain the model. In Step 4, HUDD automatically selects a subset of +, Vol. 1, No. 1, Article . Publication date: February 2023. + +6 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +images from the improvement set called the unsafe set. The engineers label the images in the unsafe +set in Step 5. Finally, in Step 6, HUDD automatically retrains the model to enhance its prediction +accuracy. +Heatmap-based Clustering in HUDD. Clustering based on heatmaps is a key component of +HUDD, an its functioning is useful to understand some of the pipelines considered in this paper. +HUDD relies on LRP to generate an heatmap for every internal layer of the DNN, for each failure- +inducing image. However, since distinct DNN layers lead to entries defined on different value +ranges [52], to enable the comparison of clustering results across different layers, we generate +normalized heatmaps by relying on min-max normalization [31]. +For each DNN layer 𝐿, a distance matrix is constructed using the generated heatmaps; it captures +the distance between every pair of failure-inducing image in the test set. The distance between a +pair of images ⟨𝑎,𝑏⟩, at layer 𝐿, is computed as follows: +heatmapDistance𝐿(𝑎,𝑏) = EuclideanDistance( ˜𝐻𝐿 +𝑎 , ˜𝐻𝐿 +𝑏 ) +(1) +where ˜𝐻𝐿 +𝑥 is the heatmap computed for image 𝑥 at layer 𝐿. EuclideanDistance is a function that +computes the euclidean distance between two 𝑁 × 𝑀 matrices according to the formula +EuclideanDistance(𝐴, 𝐵) = +� +� +� +� 𝑁 +∑︁ +𝑖=1 +𝑀 +∑︁ +𝑗=1 +(𝐴𝑖,𝑗 − 𝐵𝑖,𝑗)2 +(2) +where 𝐴𝑖,𝑗 and 𝐵𝑖,𝑗 are the values in the cell at row 𝑖 and column 𝑗 of the matrix. +HUDD applies the HAC clustering algorithm multiple times, once for every DNN layer. For each +DNN layer, HUDD selects the optimal number of clusters using the knee-point method applied to +the weighted average intra-cluster distance (WICD). WICD is defined according to the following +formula: +WICD(𝐿𝑙) = +�|𝐿𝑙 | +𝑗=1 +� +𝐼𝐶𝐷(𝐿𝑙,𝐶𝑗) ∗ |𝐶𝑗 | +|𝐶 | +� +|𝐿𝑙 | +(3) +where 𝐿𝑙 is a specific layer of the DNN, |𝐿𝑙 | is the number of clusters in the layer 𝐿𝑙, 𝐼𝐶𝐷 is the +intra-cluster distance for cluster 𝐶𝑖 belonging to layer 𝐿𝑙, |𝐶𝑗 | represents the number of elements in +cluster 𝐶𝑗, whereas |𝐶| represents the number of images in all the clusters. +In Formula 3, ICD(𝐿𝑙,𝐶𝑗) is computed as follows: +ICD(𝐿𝑙,𝐶𝑗) = +�𝑁𝑗 +𝑖=0 heatmapDistance𝐿𝑙 (𝑝𝑎 +𝑖 , 𝑝𝑏 +𝑖 ) +𝑁𝑗 +(4) +where 𝑝𝑖 is a unique pair of images in cluster 𝐶𝑗, and 𝑁𝑗 is the total number of pairs it contains. The +superscripts 𝑎 and 𝑏 refer to the two images of the pair to which the distance formula is applied. +HUDD then select the layer 𝐿𝑚 with the minimal WICD. By definition, the clusters generated +for layer 𝐿𝑚 are the ones that maximize cohesion and we therefore anticipate that they will group +together images that exhibit similar characteristics. +In our study, we rely on HUDD as a feature extraction method; precisely, we use the heatmaps +generated by the layer selected by HUDD as features. +2.4 +Safety Analysis based on Feature Extraction (SAFE) +SAFE is based on a combination of a transfer learning-based feature extraction method, a clustering +algorithm, and a dimensionality reduction technique. The workflow of SAFE matches HUDD’s, +except for Step 1 and Step 4. In SAFE’s Step 1 RCCs are identified by relying on non-convex +clustering (DBSCAN) applied to features extracted from failure-inducing images; HUDD, instead, +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +7 +applies hierarchical clustering to heatmaps. In Step 4, SAFE selects the improvement step using a +procedure that relies on DBSCAN’s outputs. +The pipelines evaluated in this paper had been inspired by the pipeline implemented in SAFE’s +Step 1, which consists of three stages (see Figure 4): Feature Extraction, Dimensionality Reduction, and +Clustering. In this paper we investigate variants of the SAFE pipeline using different combinations +of these components. Additionally, we introduce a fine-tuning stage where we fine-tune the pre- +trained transfer learning models to generate more domain-specific models. Excluding clustering, +which was introduced in Section 2.1, the components of SAFE’s pipeline are briefly described below. +2.4.1 +Transfer Learning and Feature Extraction. To maximize the accuracy of image-processing +DNNs in a cost-effective way, engineers often rely on the transfer learning approach, which consists +of transferring knowledge from a generic domain, usually ImageNet [73], to another specific domain, +(e.g., safety analysis, in our case). In other terms, we try to exploit what has been learned in one +task and generalize it to another task. Researchers have demonstrated the efficiency of transfer +learning from ImageNet to other domains [77]. +Transfer learning-based Feature Extraction is an efficient method to transform unstructured data +into structured raw data to be exploited by any machine learning algorithm. In this method, the +features are extracted from images using a pre-trained CNN model [15]. +The standard CNN architecture comprises three types of layers: convolutional layers, pooling +layers, and fully connected layers. The convolutional layer is considered the primary building block +of a CNN. This layer extracts relevant features from input images during training. Convolutional +and pooling layers are stacked to form a hierarchical feature extraction module. The CNN model +receives an input image of size (224, 224, 3). This image is then passed through the network’s layers +to generate a vector of features. The feature extraction process, for each image, generates raw data +represented by a 2𝐷 matrix (denoted as 𝑋) formalized below: +𝑋 = + +𝑥11 +𝑥12 +... +𝑥1𝑚 +𝑙1 +𝑥21 +𝑥22 +... +𝑥2𝑚 +𝑙2 +... +... +... +... +... +𝑥𝑘1 +𝑥𝑘2 +... +𝑥𝑘𝑚 +𝑙𝑘 + +,𝑙𝑖 ∈ {𝐶1,𝐶2, ...,𝐶𝑐} +(5) +where 𝐶𝑖 represent the class categories, 𝑐 is the number of categories, 𝑚 = 𝑁 × 𝑁 is the number of +features, and 𝑘 is the size of the dataset. +SAFE uses the VGG16 model pre-trained on the ImageNet dataset as a feature extraction method. +2.4.2 +Dimensionality Reduction. Dimensionality reduction aims at approximating data in high- +dimensional vector spaces [29]. It is important in our context since we extract a high number of +Features Extraction +Detection of root causes +Failure +inducing +images +Step1.1. Data +Preprocessing +Step1.2. +Features +Extraction +Step1.4. +Clustering +Step1.5. +Root cause +clusters +Step1.3. +Dimensionality +Reduction +Fig. 4. Generation of root cause clusters with SAFE +, Vol. 1, No. 1, Article . Publication date: February 2023. + +211 +C12 +1m +12 +21 +C22 +C2m +li e{l, l2, .., le], i e [1, c +Ck1 +k2 +... +CkmPC 1 +PC 2 +X +PCA +PC2 +王 +中 +国 +PC18 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +features from the images (512 to 2048). In SAFE, we used the Principal Component Analysis (PCA) +dimensionality reduction method to reduce the number of features from 2048 to 100. +2.5 +Autoencoders +Autoencoders (AE) are unsupervised artificial neural networks that learn how to compress and +encode the data before reconstructing it from the compressed encoded version to a representation +that resembles the original input as much as possible. AEs can extract features that can be used to +improve downstream tasks, such as clustering or supervised learning, that benefit from dimension- +ality reduction and higher-level features. In other words, AEs try to learn an approximation to the +identity function and, by placing various constraints on the network’s architecture and activation +functions, they extract useful representations [23]. +Figure 5 illustrates the neural network architecture of a simple AE. It consist of four main +components: +• Encoder: learns how to compress the input data and reduce its dimensions into an encoded +representation. +• Bottleneck: contains the encoded representation of the input data (i.e., the extracted features +vector). +• Decoder: reconstructs the input data from the encoded version (retrieved from the Bottleneck) +such that it resembles the original input data as much as possible. +• Reconstruction Loss: the difference between the Encoder’s input and the reconstructed +version (the Decoder’s output). The objective is to minimize such loss during training. +The objective of an AE’s training process is to minimize its reconstruction loss, measured as either +the mean-squared error or the cross-entropy loss between original inputs and its constructed inputs. +x +Encoder +Decoder +x’ +Bottleneck +Fig. 5. Autoencoder Architecture +3 +THE PROPOSED PIPELINES +This section presents the different pipelines that can be used to implement variants of SAFE and +HUDD. The evaluated pipelines differ from the original SAFE and HUDD variants with respect to +four components: Feature Extraction, Dimensionality Reduction, Clustering, and Fine-Tuning. Each +pipeline is a combination of a feature extraction method (FE), a dimensionality reduction technique +(DR), and a clustering algorithm (CA). When feature extraction is based on transfer learning, we +distinguish between models that are fine-tuned and not fine-tuned (FT/NoFT); feature extraction +approaches not based on transfer learning cannot be fine-tuned. We refer to each pipeline with +the pattern FE/{FT,NoFT}/DR/CA, with each keyword being replaced with the name of the selected +method. We depict in Figure 6 all the pipelines evaluated in our study; the different components +are described in the following sections. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +9 +PCA +UMAP +NONE +4x Transfer +Learning Models +LRP +HUDD +Fine-tuning +No Fine-tuning +DBSCAN +HDBSCAN +K-Means +99 +Pipelines +Feature Extraction +Method +Fine-tuning +Dimensionality Reduction +Technique +Clustering +Algorithm +AE +Fig. 6. Pipelines evaluated in our experiments. +3.1 +Feature Extraction +3.1.1 +Feature Extraction based on Transfer Learning. Several DNN architectures to extract features +based on transfer learning have been proposed: Inception-V3 [75], VGGNet [70], ResNet-50 [33], +and Xception [9]. These DNNs were trained on ImageNet [14], which is a dataset with more than +14 millions annotated images. The number of extracted features depends on the selected DNN +architecture; Inception-V3, VGGNet-16, ResNet-50, and Xception generate 2048, 512, 2048, and 2048 +features, respectively. They are described in the following paragraphs. +• VGG-16: VGG-16 is a Convolution Neural Network (CNN) architecture and the winner of the +ILSVR (Imagenet) competition in 2014. VGG-16 focuses on convolution layers of 3 × 3 filters +with a stride of 1 and always uses the same padding and maxpooling layer of 2 × 2 with a +stride of 2 instead of having a large number of hyper-parameters. It follows this arrangement +of convolution and max pool layers consistently throughout the whole architecture. VGG-16 +has two fully connected layers followed by a softmax layer as an output. The network has an +image input size of 224 × 224. +• ResNet-50: ResNet [33] is a CNN based on residual blocks. This architecture aims to solve +vanishing gradient problems in deep neural networks. During the backpropagation process, +the gradient diminishes dramatically in deep networks. Small values of gradients prevent the +weights from changing their values, which slows the training process. To solve this issue, +ResNet introduces residual blocks. These building blocks present skip connections between +the previous convolutional layer’s input and the current convolutional layer’s output. Similar +to VGG-16, the network has an image input size of 224 × 224. +• Inception-V3: Inception-V3 is a refined version of Inception [76]. This network proposes +additional variants of Inception blocks to reduce the number of multiplications in the convolu- +tion and minimize computational complexity. These variants are based on two factorizations: +factorization into smaller convolutions and factorization into asymmetric convolutions. The +network has an image input size of 224 × 224. +• Xception: Xception is a pre-trained CNN that is 71 layers deep and can classify images +into 1, 000 different classes such as animals, objects, and humans. This allowed the model to +learn various feature representations for a wide range of images. The Xception’s input size is +299 × 299. +3.1.2 +Fine-tuning. Fine-tuning is a typical strategy for extracting features using transfer learning. +Specifically, fine-tuning relies on taking a model that has already been trained for a particular task +𝐴 as a starting point and then removing, adding, freezing, or unfreezing some layers to improve +, Vol. 1, No. 1, Article . Publication date: February 2023. + +10 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +the performance for a similar task 𝐵. It aims to benefit from the knowledge gained from a source +task and generalize it to a target task. +Fine-tuning consists of freezing the shallow layers (close to the input), which learn more generic +features (edges, shapes, and textures), and retraining the deeper layers (i.e., we let the DNN algorithm +update the weights of the layers close to the output), which learn more specific features from the +input data[16]. To fine-tune a pretrained DNN model, we follow four steps: +(1) Create a new model whose layers (along with their weights) are cloned from the pre-trained +model, except for the output layer. +(2) Add a new fully connected output layer with a number of outputs equal to the number of +classes in the target dataset, and initialize its weights with random values. +(3) Freeze shallow layers in the network, which are responsible for the feature extraction process +(to guarantee that all the important features, previously learned by the pre-trained model, +are not eliminated). +(4) Start training the new model on the target dataset, where the weights of all the non-frozen +layers will keep updating using the backpropagation process. For the termination criterion, +we use 100 epochs or until the loss stops improving (whichever criterion is met first). +3.1.3 +Feature Extraction based on Autoencoders. As explained in Section 2.5, an Encoder plus a +Decoder make up an autoencoder (AE). The input is compressed by the Encoder, and the Decoder +reconstructs the input using the Encoder’s compressed version (at the Bottleneck). +Since AEs extracts only the few input features necessary to aid the reconstruction of the output, +the encoder might ignore other features which are not prioritized. For example in case of face +images, the AE can discard the color of the skin because it is a non-prioritized feature to the AE. +However, the encoder often learns useful properties of the data [28]. The model can then receive +input data from any domain, and a fixed-length feature vector obtained at the Bottleneck can be +used for clustering. Such a vector offers a compressed version of the input data representation +containing sufficient information about this data. +3.1.4 +Feature Extraction based on Heatmaps. In our work, we rely on heatmaps as an additional +method for feature extraction. Since heatmaps represent the relevance of each neuron on DNN +outputs, failure-inducing inputs sharing the same underlying cause should show similar heatmaps. +Precisely, we rely on two different methodologies for extracting features using heatmaps, we refer +to them as LRP and HUDD, according to the name of the technique driving feature extraction. LRP +and HUDD have been introduced in Section 2. Feature extraction based on LRP, which generates +heatmaps for internal layers but does not integrate a mechanism to select the most informative +layer, considers the heatmap computed by the LRP technique for the input layer. Feature extraction +based on HUDD, instead, considers the heatmap generated for the DNN internal layer selected by +HUDD as the best for clustering. +3.2 +Dimensionality Reduction +Several dimensionality reduction techniques exist in the literature. In this paper, we rely on two +state-of-the-art techniques: Principal Component Analysis (PCA) [57] and Uniform Manifold +Approximation and Projection (UMAP) [49]. PCA is used for its simplicity of implementation and +because it doesn’t require much time and memory resources. UMAP is used for its effectiveness +when applied before clustering. UMAP groups data points based on relative proximity, which +optimizes the clustering results. PCA and UMAP are described below. +3.2.1 +Principal Component Analysis (PCA). To reduce dimensionality, PCA creates a 2-dimensional +matrix of variables and observations. Then, for this matrix, it constructs a variable space with a +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +11 +dimension corresponding to the number of variables available. Finally, it projects each data point +onto the first few maximum variance directions in the variable space. This procedure allows PCA to +obtain a lower-dimensional data representation while maximizing data variation. The first principal +component can equivalently be defined as the direction that maximizes the variance of the projected +data [69]. In our context, we reduce the features for all our evaluated pipelines to 10 components. +We empirically obtained this number in a preliminary investigation conducted with one of our case +study subjects (i.e., HPD). Precisely, we executed a clustering algorithm (K-means) multiple times; +each execution was performed with a set of features obtained by applying PCA with a different +number of components. We evaluated all the clustering solutions using the Silhouette Index [66] +(see Section 3.3) and chose the number of components yielding the highest index value. +3.2.2 +Uniform Manifold Approximation and Projection (UMAP). Uniform Manifold Approximation +and Projection (UMAP) is a dimension reduction technique that can be used for visualization but +also for general non-linear dimension reduction. UMAP is fast, and scaling well in terms of both +dataset size and dimensionality. The main limitation of UMAP is that it doesn’t preserve the density +of the data, which is, instead, better preserved by PCA. +First, UMAP forms a weighted graph representation between each pair of data points, where the +edge weights are the probability of two data points being connected to each other. This graph is +obtained by extending a radius outward each data point such that two data points are connected if +their radii overlap. However, since an underestimation of such a radius can lead to the generation +of small, isolated clusters, and its overestimation can lead to connecting all data points together, +UMAP selects such a radius locally. The radius selection is performed based on the distance from +each data point to its ’𝑛 − 𝑡ℎ’ neighbor. Finally, UMAP decreases the likelihood of two data points +getting connected past the first neighbor (as the radius grows larger), which preserves the balance +between the high-dimensional and low-dimensional representations. Once the high-dimensional +graph is constructed, UMAP optimizes the layout of a low-dimensional representation to be as +similar as possible. The general idea is to initialize the low-dimensional data points and then move +them around until they form clusters that have the same structure as the high-dimensional data, +preserving the connectedness of the data points. UMAP calculates Similarity Scores (distances) in +the high dimensional graph to help identify the clustered points and tries to preserve that clustering +in the low dimensional graph. +Since UMAP can keep the structure of the data, even in a 2-dimensional space, we reduce the +number of features to 2 components. +3.3 +Clustering algorithms +In this study, we rely on three well-known clustering algorithms, K-means [46], DBSCAN [19], and +HDBSCAN [48] described below. These three clustering algorithms were chosen after preliminary +experiments including also the Hierarchical Agglomerative Clustering (HAC) [53] and the Mean +Shift algorithm [25]. When generating clusters for one of our subjects (i.e., HPD, see Section 4.1), +HAC and Mean Shift yielded much lower values of the Silhouette Index than the DBSCAN, HDB- +SCAN, and K-means algorithms; therefore, we discarded HAC and Mean Shift from our selection. +Since a clustering algorithm may require the manual selection of parameters’ values, such as the +number of clusters (K-means) or the minimum distance between data points (DBSCAN), we rely on +an internal evaluation metric (the Silhouette Index [66]) and the knee-point method [67] to automate +the selection of such values. +The Silhouette Index is a standard practice in cluster analysis that maximizes cohesion (i.e., how +closely related objects are in a cluster) and separation (i.e., how well-separated a cluster is from +other clusters). +, Vol. 1, No. 1, Article . Publication date: February 2023. + +12 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Fig. 7. Approximating the optimal number of clusters 𝐾 using the Knee-point method. In this case the optimal +𝐾 is equal to 6. +The knee-point method automates the elbow method heuristics [79] by fitting a spline to the raw +data using univariate interpolation, normalizing min/max values of the fitted data, and selecting +the knee-points at which the curve most significantly deviates from the straight line segment that +connects the first and last data point. We rely on the knee-point method to automatically select the +optimal number of clusters for the K-means algorithm. +3.3.1 +K-means: K-means is a well-known clustering algorithm. It takes a number 𝐾 as input and +divides the data into 𝐾 clusters based on the distance calculated from the data points to the center +of the cluster. This algorithm’s main function is to minimize the distance between the data points +and their cluster center as much as possible. In the original K-means algorithm, the number of +clusters (𝐾) is set manually, which can affect the quality of the clusters since we don’t have any +prior knowledge of the data (i.e., in our context, engineers cannot know in advance how many root +causes of failures should be identified). +To select an optimal value of 𝐾, we rely on the knee-point method. Precisely, we cluster the data +with different values of 𝐾 (in our evaluation, we consider the range [5 − 35]). For each clustering +result, we compute the within-cluster sum of squared errors (SSD), which is the sum of the distances +of each point to its cluster center. We then apply the knee-point approach to these SSDs and their +respective 𝐾. Figure 7 shows an example of 𝐾-approximation using the knee-point method. +3.3.2 +DBSCAN:. DBSCAN (Density-Based Spatial Clustering of Applications with Noise) [19], is +an algorithm that defines clusters using local density estimation. It can be divided into four steps: +(1) The 𝜖-neighborhood of a data point is determined as the set of data points that are at most 𝜖 +distant from it. +(2) If a data point has a number of neighbors, above a configurable threshold (called MinPts), it +is then considered a core point, and a high-density area has been detected. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +14000 +12000 +10000 +8000 +SSE +6000 +4000 +2000 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +kDNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +13 +(3) Since core points can be in each other’s neighborhoods, a cluster consists of the set of core +points that can be reached through their 𝜖-neighborhoods and all the data points in these +𝜖-neighborhoods. +(4) Any data point that is not a core point and does not have a core point in its neighborhood is +considered noise. +To obtain clusters using DBSCAN, we need to select two configuration parameters: (1) the distance +threshold, 𝜖, to determine the 𝜖-neighborhood of each data point, and (2) the minimum number of +neighbors, MinPts, needed for a data point to be considered a core point. For the identification of +the values for 𝜖 and MinPts, we rely on the same strategy integrated in SAFE, described below. +We determine the optimal value for 𝜖 by first computing the Euclidean distance from each data +point to its closest neighbor. Then, we identify the optimal 𝜖 value as the knee-point of the curve +obtained by considering those distances in ascending order. +To select an optimal MinPts value, we execute DBSCAN multiple times with varying 𝑀𝑖𝑛𝑃𝑡𝑠 +values and with an 𝜖 equal to the optimal value determined above. We then select the clustering +configuration that corresponds to the highest Silhouette Index value. +3.3.3 +HDBSCAN:. HDBSCAN (Hierarchical Density-Based Spatial Clustering of Applications with +Noise) is an extension of DBSCAN to solve its main limitation: selecting a global 𝜖. DBSCAN uses a +single global 𝜖 value to determine the clusters. When the clusters have varying densities, using +one global value can lead to a suboptimal partitioning of the data. Instead, HDBSCAN overcomes +such a limitation by relying on different 𝜖 values for each cluster, thus finding clusters of varying +densities. +HDBSCAN first builds a hierarchy using varying 𝜖 to figure out which clusters end up merging +together and in which order. Based on the hierarchy of the clusters, HDBSCAN selects the most +persisting clusters as final clusters. Cluster persistence represents how long a cluster stays without +splitting when decreasing the value of 𝜖. After selecting a cluster, all its descendants are ignored. +Figure 8 shows an example of the clusters’ hierarchy found by HDBSCAN. The 𝑦-axis represents +the values of 𝜖. Vertical bars represent clusters; the color and width of each vertical bar depict +the size of the cluster. We can notice that certain clusters split after the value of 𝜖 is increased, +while others persist. HDBSCAN decides which subclusters to select based on their persistence. +The persistence of a subcluster is captured by the length of the colored vertical bars in the plot. +HDBSCAN selects the clusters having the highest persistence. The unselected data points are +considered noise. In our example, only 6 clusters are selected (circled bars); they are the longest +vertical bars in the hierarchy. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +14 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Fig. 8. Example clusters selected by HDBSCAN. +4 +EMPIRICAL EVALUATION +In this Section, we aim to evaluate the pipelines presented in Section 3. A pipeline leads to the +generation of clusters of images that are visually inspected by safety engineers to determine the root +cause captured by each. We assume that a root cause can be described in terms of the commonalities +across the images in a cluster; each root cause is thus a distinct scenario in which the DNN may fail +(hereafter, failure scenario). The pipeline that best support such process should be the one requiring +minimal effort towards accurate identification of root causes. Therefore, the best pipeline is the one +that generates clusters having a high proportion of similar images (to facilitate the identification of +the root cause, based on analyzing similarities across images in a cluster), enable the detection of all +the root causes of failures, and be robust to the rarity of a particular root cause (to avoid ignoring +infrequent but unsafe failure causes). Based on the above, we defined three research questions to +drive our empirical evaluation: +RQ1 Which pipeline generates root cause clusters with the highest purity? We define a pure cluster +as one that contains only images representing the same failure scenario. Such clusters are expected +to be easier to interpret; indeed, the engineer should more easily determine the root cause of failures +if all the images share the same characteristics. Therefore, the best pipeline is the one that leads to +clusters with the highest degree of purity. The purity of a cluster is computed as the maximum +proportion of images belonging to a same failure scenario in this cluster. +RQ2 Which pipelines generate root cause clusters covering more failure scenarios? This research +question investigates to which extent the different pipelines miss failure failure scenarios. Ideally, +all failure scenarios should be captured by one or more clusters. We say that a failure scenario +is covered by a cluster if a majority of the images in the cluster belong to the scenario; indeed, +commonalities shared by most of the images in a cluster should be noticed by engineers during +visual inspection. We aim to determine which pipeline maximizes such coverage. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +0 +2000 +20 +1500 +40 +of points +3 +imber +60 +80 +500 +100DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +15 +RQ3 How is the quality of root cause clusters generated affected by infrequent failure scenarios? +Some failure scenarios may be infrequent but are nevertheless important to identify. Ideally, a +pipeline should be able to produce high-quality clusters even when a small number of images +belong to one or more failure scenarios. In this research question, we vary the number of images +belonging to failure scenariosand study how the effectiveness of pipelines (purity and coverage of +the generated clusters) is affected. +To perform our empirical evaluation, we have implemented the transfer learning models using +Tensorflow [1] and Keras [10]. The clustering algorithms and the dimensionality reduction methods +were implemented using the Scikit-Learn library [58]. All the experiments were carried out on +an Intel Core i9 processor running macOS with 32GB RAM. Additionally, in our experiments, +we relied on the LRP implementation provided by LRP authors [50] for well-known types of +layers (i.e., MaxPooling, AvgPooling, Linear, and Convolutional layers). Further, we extended the +implementation of LRP to include DNN models implemented in PyTorch [62], Tensorflow, and +Keras libraries. +4.1 +Subjects of the study +To evaluate our pipelines, we consider four different DNNs that process synthetic images in the +automotive domain. These DNNs support gaze detection, drowsiness detection, headpose detection, +and unattended child detection, which are subjects of ongoing innovation projects at IEE Sensing, +our industry partner developing sensing components for automotive. Additionally, we consider two +DNNs that process real-world images to support autonomous driving: steering angle prediction +and car position detection. +The gaze detection DNN (GD) performs gaze tracking; it can be used to determine a driver’s +focus and attention. It divides gaze directions into eight categories: TopLeft, TopCenter, TopRight, +MiddleLeft, MiddleRight, BottomLeft, BottomCenter, and BottomRight. The drowsiness detection +DNN (OC) has the same architecture as the gaze detection DNN and relies on the same dataset, +except that it predicts whether the driver’s eyes are open or closed. +The head-pose detection DNN (HPD) is an important cue for scene interpretation and computer +remote control, such as in driver assistance systems. It determines the pose of a driver’s head in an +image based on nine categories: straight, rotated left, rotated left, rotated top left, rotated bottom +right, rotated right, rotated top right, tilted, and headed up. +The unattended child detection DNN is trained with the Synthetic dataset for Vehicle Interior +Rear seat Occupancy detection (SVIRO) [12]. SVIRO is a dataset generated by IEE that represents +scenes in the passenger compartment of ten different vehicles. The dataset has been used by IEE to +train DNNs performing rear seat occupancy detection using a camera system. We use it to train a +DNN for unattended child detection. We consider a seat empty when there is an object, an empty +infant/child seat, or nothing. We consider the presence of a child/infant as a class and the presence +of an adult as another class. As a result, we have labelled the dataset with three classes (i.e., empty +seats, children/infants not accompanied by adults, and presence of an adult). +For Steering Angle Prediction (SAP), we rely on the pre-trained Autumn DNN model [61], +which follows the DAVE-2 architecture [6] provided by NVIDIA. It is a DNN to automate steering +commands of self-driving vehicles [81]; it predicts the angle at which the wheel should be turned. +It has been trained on a dataset of road images captured by a dashboard camera mounted in the car. +Car Position Detection (CPD) DNNs are used by most Advanced-Driver Assistance Systems +(ADAS) to predict the positions of nearby cars. We rely on the CenterNet DNN [17], which is +an accurate DNN used by most competition-winning approaches for object detection [82]. It has +been trained on images from the ApolloScape dataset [35] collected using a dashboard camera to +estimate the absolute position of vehicles with respect to the ego-vehicle. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +16 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Table 1. Case Study Systems +DNN Data +Training Test +Failure +# +# +# +# +# +# +# +# +# +# +Source +Set Size +Set Size (Ac- +curacy) +inducing +images +M1 +N2 +H3 +B5 +SG6 +EG7 +EO8 +S9 +D10 +NF11 +GD +UnityEyes +61,063 +132,630 (96%) +5,371 +- +80 +- +80 +- +- +- +80 +80 +80 +OC +UnityEyes +1,704 +4,232 (88%) +506 +- +20 +- +20 +- +- +- +20 +20 +20 +HPD +Blender +16,013 +2,825 (44%) +1,580 +90 +90 +90 +90 +90 +90 +- +90 +90 +90 +SVIRO Blender +15,489 +3,427 (74%) +884 +- +30 +- +30 +- +- +30 +30 +30 +30 +SAP +Autopilot [8] +33,808 +45,406 (84%) +7,169 +- +90 +- +90 +- +- +- +90 +90 +90 +CPD +Apollo [35] +5,208 +4,996 (91%) +444 +- +90 +- +90 +- +- +- +90 +90 +90 +1 Mask 2 Noise 3 Hand 5 Blurriness 6 SunGlasses 7 EyeGlasses 8 Everyday Object 9 Scaling 10 Darkness 11 No Injected Fault +For each subject DNN, we apply our pipelines to a set of failure-inducing images. Such sets +consist of (1) images belonging to a provided test set and leading to a DNN failure and (2) test set +images that were not leading to a DNN failure but had been modified to cause a DNN failure; the +latter are images with injected root causes of failures and are described in Section 4.2. In classifier +DNNs (i.e., OC, GD, HPD, and SVIRO) a failure occurs in the presence of an image being incorrectly +classified. For SAP and CPD, which are regression DNNs, we set a threshold to determine DNN +failures. For SAP, we observe a DNN failure when the squared error between the predicted and +the true angle is above 0.18 radian (10.3◦), which is deemed to be an acceptable error in related +work [80]. For CPD, since it tackles a multi-object detection problem, we report a DNN failure +when the result contains at least one false positive (i.e., the distance between the predicted position +of the car and the ground truth is above 10 pixels [71]). +In Table 1, we provide details about the case study subjects used to evaluate our pipelines. For +each subject, we report the source of the data set (e.g., the simulator used to generate the data), +the training and test set sizes, the accuracy of the DNN on the original test set, the number of +failure-inducing images and the number of images for each injected root cause (they are detailed in +Section 4.2). +We fine-tune the pipelines relying on transfer learning using the test sets of the respective case +studies. We use the resulting fine-tuned model to extract the features from the failure-inducing +sets. We train on the test sets because the number of images in each set is sufficient for the model +to learn the features. Also, we train the autoencoders on the training set, and use the test set of the +respective case study to validate the results. The termination criterion is 50 epochs unless we reach +an early stopping point (the model stops improving). After training, we use only the encoder part +to extract the features from the images in the failure inducing set. +4.2 +Injected Failure Scenarios +To assess the ability of different pipelines to generate clusters that are pure and cover all the root +causes of failures, we need to know the root causes of failures in the test set. Since such root causes +may vary (e.g., lack of sufficient illumination, presence of a shadow in a specific part of the image) +and it is not possible to objectively demonstrate that a failure cause has been correctly captured by +a cluster (e.g., some readers may not agree that certain images show lack of sufficient illumination), +to avoid introducing bias and subjectivity in our results, we modify a subset of the provided test +set images so that they will fail because of known root causes of failures. In total, we considered +nine different root causes to be injected in our test set images and refer to them as injected failure +scenarios (i.e., failure scenarios with injected root causes). +We derive an image belonging to an injected failure scenario by modifying a test set image +according to the specific root cause we aim to inject; for example, by covering the mouth of a +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +17 +person with a mask. To ensure that a modified image leads to a DNN failure because of the injected +root cause, we modify only test set images that, before modification, lead to a correct DNN output. +Figure 9 illustrates the different injected failure scenarios. Below, we describe the nine root +causes considered in our study: +• Hand: The presence of a hand blocking the full view of the driver’s face could affect the +DNN result, leading it to mispredict the driver’s head direction. We simulate a hand that is +partially covering the face appearing in the image. +• Mask: Similar to Hand, the presence of a mask covering the nose or the mouth may affect a +DNN that recognizes the driver’s head pose. Using image key points, we drew the shape of a +white mask to simulate a mask covering the nose and the mouth. +• Sunglasses: As for the Mask, we use the eyes’ key points to draw sunglasses covering the +driver’s eyes. +• Eyeglasses: Different from the Sunglasses, we draw glasses with the eyes being still visible. +• Noise: A noisy image is one that contains random perturbations of colors or brightness. This +failure scenario has been considered in related work to evaluate DNN robustness against +adversarial attacks [43]. In real-world automotive systems, such a failure scenario resembles a +defective camera or a high signal-to-noise ratio (SNR) in the communication channel between +different electronic control units (ECUs), resulting in a noisy input. We use the Scikit-Image +library [84] to add Gaussian Noise, a statistical noise with a probability density function +equal to a normal distribution, also known as Gaussian Distribution. +• Blurriness: As for Noise, this failure scenario was used to evaluate DNN robustness [80]. +We use the Pillow library [11] to add blurriness to images using a radius of 30 pixels. +• Darkness: Once again, this failure scenario was used in related work to evaluate DNN +robustness [59] . In practice, poor lighting conditions could make the DNN fail because it +cannot clearly recognize what is depicted in an image. We use the Pillow library [11] to +decrease the brightness of images by a factor of 0.3; we selected 0.3 because it is the lowest +value introducing failures in our subject results. +• Scaling: Such a failure scenario mimics the situation where a camera is misconfigured, +leading to rescaled images being fed to the DNN. We reduce the size of an image by a value +based on the image size (i.e., large 1200px × 1200px images are scaled by 400px, small 320px +× 320px images by 70px). and insert a black background using the Pillow library [11]. +• Everyday Object: For the SVIRO dataset, we introduce, in the car’s rear seat, an object (e.g., +a washing machine or a handbag) never observed in the training set, thus simulating the +effect of an incomplete training dataset. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +18 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Mask +Noise +Blurriness +Darkness +Hand +Sunglasses +Eyeglasses +Scaling +No injected fault +Everyday object +Fig. 9. Injected failure scenarios in our study. +For regression DNNs (SAP and CPD), we randomly selected 90 images to be copied and modified +for each failure scenario. For classifier DNNs, for each failure scenario, we randomly selected 10 +images for each class label. +Please note that in addition to the injected failure scenarios explained above, our DNNs are +affected by other natural failure causes (e.g., borderline images that are misclassified because they +are very similar to the ones belonging to another class). Such cases are observed with any machine +learning model since it is usually not possible to achieve perfect accuracy through training. We +refer to these images as belonging to natural failure scenarios. In our analysis, we include a number +of images belonging to natural failure scenarios equal to those with injected failure scenarios. This +is because natural failure scenarios are usually observed with any DNN and, therefore, should be +considered when generating RCCs. +4.3 +RQ1: Which pipeline generates root cause clusters with the highest purity? +4.3.1 +Design and measurements. A pure cluster includes only images presenting the same root +cause (i.e., common cause leading to a DNN failure); for example, a hand covering a person’s mouth. +Pure clusters simplify root cause analysis because they should make it easier for an engineer to +determine the commonality across images and therefore the cause of failures. +Since the likely root cause of the failure in our injected failure scenarios is known, we focus on +these scenarios to respond to RQ1. For each RCC, we compute the proportion of images belonging +to each injected failure scenario. Therefore, we measure the purity 𝑃 of a cluster 𝐶 (hereafter, 𝑃𝐶) +as the highest proportion of images belonging to one injected failure scenario 𝑓 ∈ 𝐹 assigned to +cluster 𝐶, where 𝐹 is the set of all failure scenarios. 𝑃𝐶 is computed as follows: +P𝐶 = max +𝑓 ∈𝐹 +�𝐶𝑓 +|𝐶| +� +(6) +The proportion of a failure scenario 𝑓 in a cluster 𝐶 is computed as the number of images +belonging to 𝑓 assigned to cluster 𝐶 (𝐶𝑓 ), divided by the size of cluster 𝐶. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +19 +Clusters that do not include any image belonging to an injected failure scenario are assumed +to capture root causes due to natural failure scenarios and, consequently, are excluded from our +analysis. +We study the purity distribution across RCCs generated for the different case study subjects. +Since, ideally, we would like to obtain pure clusters, the best pipeline is the one that maximizes the +average purity across the generated RCCs. +4.3.2 +Results. Figure 10 depicts a regression tree illustrating how the different components of +a pipeline (feature extraction methods, fine-tuning, dimensionality reduction techniques and +clustering algorithms) determine the purity of the clusters generated by a pipeline. We use the +Conditional Inference Tree (CTree) algorithm [34] to generate this decision tree with a maximum +depth set to 4 (we have four components in a pipeline) and a minimum split set to 10 (i.e., the +weight of a node to be considered for splitting). The dataset used to build the tree consists of the +components of each pipeline as attributes, and the purity of the generated clusters as the predicted +outcome. The dataset size is equal to 99, the number of pipelines. +Each node of the tree represents a feature of the pipeline. Leaves (terminal nodes) depict box plots +representing distributions of the average purity across RCCs generated by the pipelines belonging +to each leaf. Each point in the box plot is the average purity of one pipeline (i.e., the average of the +purity of all the RCCs generated across all case study subjects). To split a node, the CTree algorithm +first identifies the feature with the highest association (covariance) with the response variable +(purity, in our case). Precisely, it relies on a permutation test of independence (null hypothesis) +between any of the features and the response [74]; the feature with the lowest significant p-value +is then selected (alpha = 0.05, in our case). Once a feature has been selected, a binary split is then +performed by identifying the value that maximizes the test statistics across the two subsets. Since +we are in the presence of multiple hypothesis (assume 𝑚, for each node), to prevent a Type I error, +for each feature 𝑗, CTree computes its Bonferroni-adjusted [89] 𝑝-value𝑗 as +adjusted 𝑝-value𝑗 = 1 − (1 − 𝑝-value𝑗)𝑚 +In Figure 10, we notice that the pipelines with fine-tuned models (Node 3 and 4) generate lower- +purity clusters than those without any fine-tuning (Node 6 and 7). This is because these models +were fine-tuned on a test set that did not include any injected root cause (i.e., only natural failure +scenarios); recall that fine tuning is performed with labeled images (e.g., training set) and since +our injected root causes capture scenarios not foreseen at training time, it would be unrealistic +to consider such scenarios for fine tuning. Fine-tuning a model on a set of images means that it +will learn all the features of those images. Therefore, clustering based on fine-tuned models will +generate clusters based on the features observed during training, excluding injected features (i.e, +the injected root causes). As a result, in our experiments, images are clustered based only on their +natural fault (e.g., borderline class) instead of the injected faults. +The pipelines using non-fine-tuned transfer learning models as a feature extraction method +(Node 7) generate purer clusters (min = 50%, median = 80%, max = 96%) than the pipelines using an +autoencoder model, HUDD, or LRP (Node 6) (min = 50%, median = 65%, max = 70%). The purpose +of the Autoencoder model is to provide a condensed representation of the image to be used for +reconstruction. This is done by ignoring the features that the model considers insignificant and only +keeping the features that help the encoder reconstruct the image accurately. Therefore, since the +autoencoder is trained on the training set, the injected faults are ignored. Given that clustering is +based on the condensed representation, the generated clusters are less pure than the ones generated +by the pipelines with transfer learning models. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +20 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +As for HUDD and LRP, it seems that their main limitation is that heatmaps cannot capture the +presence of root causes affecting all the pixels in an image (i.e., the result of noise, blurriness, +darkness, scaling). Heatmaps mainly capture which pixels of the image drive the DNN output, thus +leading clustering to group images where the same pixels affected the output. For instance, the +DNN’s response to a blurred image with a shadow on the mouth could be different from that of +another blurred image with a shadow on the eyes, thus leading to different clusters for these images +although they represent the same injected failure scenario (blurriness). +Finally, we notice that the pipelines using HDBSCAN and DBSCAN (Node 3) as a clustering +algorithm yield purer clusters (min = 25%, median = 40%, max = 80%) than those using K-means +(Node 4, min = 22%, median = 27%, max = 29%). This is because K-means faces difficulty dealing +with non-convex clusters. A cluster is convex if, for every pair of points belonging to it, it also +includes every point on the straight line segment between them [41], which gives the cluster a +spherical form. Nevertheless, in many practical cases, the data leads to clusters with arbitrary, +non-convex shapes. Such clusters, however, cannot be appropriately detected by a centroid-based +algorithm (e.g., K-means), as they are not designed for arbitrary-shaped clusters. +DBSCAN and HDBSCAN are density-based clustering algorithms. They consider high-density +regions as clusters (see Section 2). The root cause clusters generated by DBSCAN and HDBSCAN +are arbitrary-shaped and more homogeneous (i.e., clusters with higher within-cluster similarity) +with very similar images. In contrast, a convex cluster generated by K-means tends to be less dense +and can group rather dissimilar images. As a result, a convex cluster is less pure than a non-convex +one. +Fig. 10. Decision Tree illustrating how the different features of a pipeline determine the average purity of +root-cause clusters. +We report the significance of these results in Table 2, including the values of the Vargha and +Delaney’s ˆ𝐴12 effect size and the 𝑝-values resulting from performing a Mann-Whitney U-test to +compare the average purity of the pipelines using transfer learning models (Node 7 in the decision +tree) and the pipelines represented by the other nodes. Typically, an ˆ𝐴12 effect size above 0.56 +is considered practically significant with higher thresholds for medium (0.64) and large (0.71) +effects [39], thus suggesting the effect sizes between the pipelines using transfer learning models +and other pipelines are large. Further, 𝑝-values suggest these differences are statistically significant. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +finetuning +p< 0.001 +NoFT +5 +clusteringalgs + transfermodel +p = 0.002 +p = 0.023 +Dbscan,HDBSCAN +K-means +AE, HUDD, LRP +InceptionV3, ResNet50, VGG16, Xception +Node 3 (n = 24) +Node 4 (n = 12) +Node 6 (n = 27) +Node 7 (n = 36) +100 - +100 - +100 - +100 - +0 +oo +80 +80 +80 - +80 +60 - +60 - + 09 +- 09 +40 - +40 - +40 - +40 -DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +21 +Table 3. RQ1: Pipelines with a purity greater than 90%. The last column represents the average of averages. +Pipelines +Avg. purity across RCCs +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +19 +VGG-16 +NO +None +K-Means +91.7% +92.1% +95.5% +82.5% +97.3% +99.7% +93.2% +25 +VGG-16 +NO +UMAP +K-Means +97.6% +84.4% +93.7% +82.4% +90.3% +97.8% +91.0% +26 +VGG-16 +NO +UMAP +DBSCAN +99.0% +93.0% +99.6% +79.7% +98.1% +96.6% +94.3% +39 +ResNet-50 +NO +None +HDBSCAN +96.4% +100.0% +100.0% +78.8% +87.5% +100.0% +93.8% +43 +ResNet-50 +NO +UMAP +K-Means +99.4% +93.0% +82.3% +79.6% +99.6% +97.7% +91.9% +44 +ResNet-50 +NO +UMAP +DBSCAN +100.0% +95.8% +95.8% +79.0% +99.7% +99.3% +94.9% +62 +Inception-V3 +NO +UMAP +DBSCAN +93.4% +95.2% +98.1% +76.6% +97.4% +83.1% +90.7% +𝐹𝐸 Feature Extraction 𝐹𝑇 Fine-tuning 𝐷𝑅 Dimensionality Reduction 𝐶𝐴 Clustering Algorithm +Table 2. RQ1: p-values and and effect size values when comparing the results of the pipelines with the best +purity of clusters (according to the decision tree) to the other pipelines. +Node 3 +Node 4 +Node 6 +p-value +7e-11 +2e-7 +5e-6 +ˆ𝐴12 +1.00 +1.00 +0.80 +Finally, in Table 3, we report the pipelines that generated clusters with an average purity above +90% across all case study subjects, along with the purity obtained for each subject; the complete +results obtained for all pipelines appear in Appendix A, Table 11. An average purity of 100% means +that all the clusters generated by the pipeline are pure. Interestingly, all the pipelines in Table 3 +belong to Node 7 in Figure 10, thus confirming our main finding. Five of these seven best pipelines, +rely on UMAP, without fine-tuning but with a transfer learning model, which is therefore our +suggestion to perform root cause analysis. The best result is obtained with ResNet-50 combined +with UMAP and DBSCAN. +4.4 +RQ2: Which pipelines generate root cause clusters covering more failure +scenarios? +4.4.1 +Design and measurements. This research question investigates the extent to which our +pipelines identify all failure scenarios. We compare pipelines in terms of the percentage of injected +failure scenarios being covered by at least one RCC. A failure scenario is covered by an RCC if it +enables the engineer to determine the root cause of the failure. Precisely, when images belonging +to a failure scenario 𝑓 represent a sufficiently large share of images in a cluster 𝐶, it is easier for +an engineer to notice that 𝑓 is a likely root cause. Therefore, we assume that an injected failure +scenario 𝑓 is covered by a cluster 𝐶 if it contains at least 90% of images with 𝑓 . Since this threshold +is relatively high, our results can be considered conservative. +Given that our injected failure scenarios are represented by the same number of images in the +failure-inducing test set, every failure scenario has the same likelihood of being observed. Therefore, +we expect to obtain RCCs corresponding to each failure scenario. +4.4.2 +Results. Figure 11 shows a decision tree illustrating how the different components of a +pipeline determine the coverage of failure scenarios. As for RQ1, we used the Conditional Inference +Tree CTree algorithm to generate this decision tree (with the same parameter settings). +Each leaf node depicts a box plot with the distribution of the percentages of failure scenarios +covered by the set of pipelines that include the components listed in the decision nodes. For +instance, Node 9 provides the distribution of the percentage of failure scenarios covered by the +, Vol. 1, No. 1, Article . Publication date: February 2023. + +22 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +RCCs generated by pipelines using UMAP as a dimensionality reduction technique and non-fine- +tuned transfer learning models as feature extraction methods (12 pipelines). Ideally, the root-cause +clusters generated by a pipeline should cover 100% of the failure scenarios. +The decision tree in Figure 11 confirms RQ1 results. The pipelines without fine-tuning (Nodes 6, +8 and 9) outperform the pipelines with fine-tuning (Nodes 3 and 4). The pipelines with transfer +learning models (Nodes 8 and 9) generate clusters that cover more failure scenarios than those +generated by the pipelines using HUDD, LRP, and AE (Node 6). Also, the pipelines using the +DBSCAN and HDBSCAN clustering algorithms (Node 3) yield better results than the ones using +K-means (Node 4). +Further, the decision tree in Figure 11 gives us more insights into which dimensionality reduction +method is more effective. We notice that the root-cause clusters generated by the pipelines using +UMAP (Node 9) lead to a better distribution (min = 45%, median = 85%, max = 100%) than the +pipelines using PCA or not using any dimensionality reduction (Node 8, min = 25%, median = +55%, max = 90%). This is because UMAP yields a better separation of the clusters (i.e., less clusters +overlap) compared to PCA. When using UMAP, all the data points converge towards their closest +neighbor (the most similar data point). Therefore, neighboring data points in higher dimensions +end up in the same neighborhood in lower dimensions, resulting in a compact and well-separated +clusters where it is easier for the clustering algorithms to distinguish them. +Fig. 11. Decision Tree illustrating how the different features of a pipeline determine the coverage of the +failure scenarios.) +We report the significance of these results in Table 4, including the values of the Vargha and +Delaney’s ˆ𝐴12 effect size and the 𝑝-values resulting from performing a Mann-Whitney U-test to +compare the percentages of covered failure scenarios resulting from the pipelines using UMAP +(Node 9 in the decision tree in Figure 11), and the other pipelines. Table 4 shows that the 𝑝-values, +when comparing the pipelines using UMAP to the other pipelines, are always below 0.05. This +implies that in all the cases, differences are statistically significant with large effect sizes (above +0.77). +, Vol. 1, No. 1, Article . Publication date: February 2023. + +finetuning +p < 0.001 +NoFT +2 +5 + clusteringalgs + transfermodel +p = 0.009 +p = 0.013 +AE, HUDD, LRP +InceptionV3, ResNet50, VGG16, Xception +7 + dimreduction +Dbscan, HDBSCAN +K-means +p = 0.031 +None, PCA +UMAP +Node 3 (n = 24) +Node 4 (n = 12) +Node 6 (n = 27) +Node 8 (n = 24) +Node 9 (n = 12) +100 +100 +100 +100 +100 +80 - +80 +80 - +80 - +80 - +60 +60 +60 +60 +60 +40 - +40 +40 - +40 - +40 - +20 +20 +20 +20 +20 +- 0 +0 +.0DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +23 +Table 5. RQ2: Pipelines with a coverage greater than 90%. The last column represents the average of averages. +Pipelines +Percentage of covered faulty scenarios +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +26 +VGG-16 +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +80.0% +100.0% +100.0% +96.7% +44 +ResNet-50 +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +60.0% +100.0% +100.0% +93.3% +62 +Inception-V3 +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +100.0% +100.0% +100.0% +100.0% +80 +Xception +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +40.0% +100.0% +100.0% +90.0% +𝐹𝐸 Feature Extraction 𝐹𝑇 Fine-tuning 𝐷𝑅 Dimensionality Reduction 𝐶𝐴 Clustering Algorithm +Table 4. RQ2: p-values and and effect size values when comparing the results of the pipelines with the best +coverage of the faulty scenarios (according to the decision tree) to the other pipelines. +Node 3 +Node 4 +Node 6 +Node 8 +p-value +1e-5 +1e-5 +4e-5 +8e-3 +ˆ𝐴12 +0.95 +1.00 +0.91 +0.77 +In Table 5, we report the pipelines that generated clusters covering at least 90% of the failure +scenarios across all case study subjects, along with the coverage obtained for each case study +subject (complete results for all the pipelines are reported in Appendix B, Table 12). If the coverage +is equal to 100%, all the failure scenarios are covered by the RCCs. Unsurprisingly, the pipelines +in Table 5 belong to Node 7 in Figure 11: they rely on a non-fine-tuned transfer learning model +for feature extraction, and UMAP for dimensionality reduction. Further, they all use DBSCAN +for clustering. These pipelines consistently yielded the best results for all individual case studies +(confirming the results obtained in RQ1). +Such findings are further supported by the results in Table 11 and Table 12, where we notice that +the combination of UMAP with DBSCAN always achieves higher purity and coverage (in bold) +than its alternatives, regardless of the used feature extraction method. +4.5 +RQ3: How is the quality of root cause clusters generated affected by infrequent +failure scenarios? +4.5.1 +Design and measurements. We study the effect of infrequent failure scenarios on the quality +of the RCCs generated by the pipelines. We consider a failure scenario infrequent when it is +observed in a low proportion of the images in the failure-inducing set. To be practically useful, a +good pipeline should be able to generate root-cause clusters even for infrequent failure scenarios; +indeed, in safety-critical contexts, infrequent failure scenarios may lead to hazards and thus should +be detected when testing the system. For instance, if only five out of hundred failure-inducing +images belong to a failure scenario and we have three failure scenarios in total, a robust pipeline +should still generate an RCC containing only the images of the infrequent failure scenario. +We generate 10 different failure-inducing sets for each case study subject (a total of 60 failure- +inducing sets). To construct a failure-inducing set, for each root cause that might affect the case +study (see Table 1, Page 16), we generate a number 𝑛 of images affected by the injected root cause. +We randomly select a number 𝑛 that is lower than the number of images selected for the same root +cause in RQ1. Further, for classifier DNNs, we select a value higher than the number of classes +of the corresponding case study (we enforce one root cause of failures for one image per class, +at least); for regression DNNs, we select a value above 2. Since 𝑛 is randomly selected (uniform +distribution), we obtained failure-inducing sets containing failure scenarios whose number vary. +In addition, we also include a randomly selected number of images belonging to natural failure +, Vol. 1, No. 1, Article . Publication date: February 2023. + +24 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +scenarios, to mimic what happens in practice (see RQ1). The number of images belonging to natural +failure scenarios varies between two and the total number of injected failure scenario images. +In total, we generated 60 failure-inducing sets (10 × 6 subject DNNs). For each failure-inducing +set, we randomly selected the number of images representing a failure scenario (injected or natural +scenario). Such number should be higher than the number of classes (to ensure that there is at +least one scenario for each class) and lower than the total number of images representing a failure +scenario generated in RQ1 (see Table 1). In the case of regression DNNs, the minimum number +of images representing a failure scenario is set to 2 since a cluster is formed by grouping at least +two images. For instance, the number of images representing a failure scenario for each failure- +inducing set of the HPD case study (9 classes) is randomly selected between 9 and 90 (see Table 13, +Appendix C). +Since we aim to study the effect of infrequent failure scenarios on the quality of the generated +RCCs, we categorize our 290 failure scenarios into infrequent and frequent. Infrequent failure +scenarios are the ones that include a proportion of injected images that is lower than the median +proportion in all the generated failure-inducing sets (equals to 18% in our study). For example, +noise is frequent in the dataset GD_1 (64 > 18) but infrequent in the dataset OC_2 (4 < 18). +We consider only the best pipelines resulting from the experiments in RQ1 and RQ2 (i.e., having +purity or coverage above 90% as shown in Tables 3 and 5); they are pipeline 26 (VGG16/DB- +SCAN/UMAP/NoFT), 44 (ResNet50/DBSCAN/UMAP/NoFT), 62 (InceptionV3/DBSCAN/UMAP/NoFT), +19 (VGG16/K-means/None/NoFT), 25 (VGG16/K-means/UMAP/NoFT), 39 (ResNet50/HDBSCAN/None/NoFT), +43 (ResNet50/K-means/UMAP/NoFT), and 80 (Xception/DBSCAN/UMAP/NoFT). The first three pipelines +(i.e., 26, 44, 62) were the best for both RQ1 and RQ2, the next four (i.e., 19, 25, 39, 43) were selected +based on RQ1 results while the latter (i.e., 80) based those of RQ2. We compute the purity and +coverage of the RCCs generated by each of these pipelines, following the same procedures adopted +for RQ1 and RQ2. We then compare the distribution of purity and coverage for infrequent and +frequent failure scenarios. The most robust pipelines are the ones being affected the least, in terms +of purity and coverage, by infrequent failure scenarios. +4.5.2 +Results. In Figure 12, for each selected pipeline, we report the average purity across all +the RCCs1 with the injected failure scenarios having a certain frequency. The 𝑥-axis reports the +proportion of images for failure scenarios whereas the 𝑦-axis reports the average purity of the +RCCs associated to each failure scenario. +Figure 12 shows that when the frequency of the failure scenarios is below the median (infrequent +scenario), the cluster purity obtained by pipelines tends to significantly lower and decrease rapidly +as the frequency decreases. This is expected because when a failure scenario is infrequent, the +clustering algorithm tends to either cluster its images as noise or distribute them over the other +clusters. For density-based clustering algorithms, images belonging to infrequent scenarios may +not become core points when the identification of a core point requires more data points in their +neighborhood. In such case, images belonging to infrequent scenarios will be either labeled as noise +points or border points (belonging to other clusters). The same is true for K-means, where these +points are usually spread across other clusters because they cannot form a cluster. +To strengthen our findings, in Table 6, we report the results when comparing the purity of the +selected pipelines for frequent and infrequent failure scenarios; further, we report the Vargha and +Delaney’s ˆ𝐴12 effect size and the 𝑝-values resulting from performing a Mann-Whitney U-test. We +notice that for all pipelines, the difference between frequent and infrequent scenarios are significant +1As discussed in Section 4.3.1. The red vertical line represents the median frequency of failure scenarios. We say that an +RCC is associated with (or captures) an injected failure scenario 𝑓 when the majority of the images in the cluster belong to +scenario 𝑓 . +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +25 +Table 6. RQ3: p-values and effect size values when comparing the purity of the best pipelines with the +frequent and infrequent failure scenarios. +Pipelines +26 +44 +62 +39 +80 +19 +25 +43 +Average Purity for infrequent +failure scenarios +94% +87% +91% +79% +87% +76% +70% +65% +Average Purity for frequent fail- +ure scenarios +100% +100% +100% +92% +99% +96% +96% +93% +p-value +4e-6 +2e-10 +1e-6 +2e-9 +8e-9 +8e-5 +2e-10 +3e-14 +ˆ𝐴12 +0.58 +0.64 +0.59 +0.60 +0.63 +0.68 +0.70 +0.75 +Table 7. RQ3: p-values and effect size values when comparing the best pipeline in Table 6 (i.e., Pipeline 26, +VGG16/Dbscan/UMAP/NoFT) to the other pipelines based on the average purity of the clusters associated to +infrequent failure scenarios. +Pipelines +44 +62 +39 +80 +19 +25 +43 +p-value +0.002 +0.51 +4e-5 +0.006 +3e-12 +2e-14 +4e-21 +ˆ𝐴12 +0.57 +0.51 +0.60 +0.56 +0.69 +0.71 +0.77 +(p-value < 0.05). However, the effect sizes for Pipelines 26, 62, 45, and 80 are small, while they are +medium for Pipelines 19 and 44, which indicates that pipelines including DBSCAN (i.e., Pipelines 26, +62, 45, and 80) are much more robust to infrequent scenarios than others (i.e., the difference between +frequent and infrequent scenarios is less pronounced). Actually, the pipelines using DBSCAN fare +better than the rest also in the general case. Indeed, almost all the injected failure scenarios with +frequency above 18% have 100% purity (see Figure 12); further for infrequent failure scenarios they +include less data points below 100% than the other pipelines. This is because DBSCAN tends to find +clusters with different sizes if these clusters are dense enough; K-means, instead, derives clusters +that are of similar size. +Further, we notice that the purity of the clusters generated by Pipeline 26 (VGG16/Dbscan/UM- +AP/NoFT), for infrequent failure scenarios, is higher (average is 94%) than the purity of the clusters +generated by the other pipelines; differences are significant (see Table 7), thus suggesting Pipeline +26 might be the best choice. +Concerning coverage, Figure 13 shows, for each pipeline, histograms with the average coverage +obtained for failure scenarios having proportions of failure inducing images within specific ranges. +In general, we observe that coverage is higher for frequent scenarios. This is due to the correlation +between pure clusters and coverage; the less pure the generated clusters, the fewer failure scenarios +they cover. When the failure scenarios are infrequent, their images are distributed over the other +clusters, reducing their purity and, thus, reducing the probability of these scenarios being covered. +To demonstrate the significance of the difference between coverage results obtained with frequent +and infrequent scenarios, we apply the Fisher’s Exact test2 to compare the coverage of frequent +and infrequent scenarios for the clusters generated by the selected pipelines. We report the 𝑝- +values resulting from the Fisher’s Exact test in Table 8 and observe that differences are statistically +significant thus indicating that pipelines perform better with frequent failure scenarios. +Further, Figure 13 shows that pipeline 62 (InceptionV3/DBSCAN/UMAP/NoFT) is the one perform- +ing best with the least frequent scenarios (i.e., range 0-5%) but no pipeline fares well in that range. +2The Fisher’s Exact test [83] is a statistical test used to determine if there is a non-random association between two +categorical variables [86]. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +26 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Table 8. RQ3: Fisher exact test values when comparing the coverage of the lowly represented and highly +represented faulty scenarios by the clusters generated by the best pipelines. +Pipelines +26 +44 +62 +39 +80 +19 +25 +43 +Average +Coverage +for +infre- +quent failure scenarios +85% +71% +82% +66% +73% +51% +46% +34% +Average Coverage for frequent +failure scenarios +100% +98% +99% +86% +98% +86% +87% +77% +Fisher’s Exact test +1e-5 +1e-5 +1e-5 +1e-5 +1e-5 +2e-4 +1e-5 +1e-5 +Fig. 12. Purity of the clusters associated with frequent and infrequent failure scenarios. The x-axis captures +the frequency of a failure scenario (i.e., proportion of failure-inducing images for a failure scenario). Each data +point is the average of all the RCCs associated to one distinct failure scenario. The red vertical line represents +the median frequency of failure scenarios. +Pipeline 26 (VGG16/DBSCAN/UMAP/NoFT) is the one performing best with infrequent scenarios in +the range 5% to 20%; indeed, it is the only pipeline providing an average coverage above 90% for that +range. To further demonstrate the significance of the difference in performance between Pipeline +26 and the other pipelines, we apply Fisher’s exact test to the coverage obtained for infrequent +scenarios. We report the 𝑝-values resulting from this test in Table 9. We notice that all the 𝑝-values +are below 0.05 except when Pipeline 26 is compared to Pipeline 62; indeed, the results of these two +pipelines are similar as visible in Figure 13), even though Pipeline 26 performs slightly better on +average. +In conclusion, infrequent failure scenarios affect both purity and coverage; however, some +pipelines fare better than others. Our results suggest that the pipeline (26) relying on a non-fine- +tuned VGG16 model, with UMAP and DBSCAN (Pipeline 26) is the best choice because it yields +, Vol. 1, No. 1, Article . Publication date: February 2023. + +VGG16/Dbscan/UMAP/NoFT(26) +ResNet50/Dbscan/UMAP/NoFT (44) +InceptionV3/Dbscan/UMAP/NoFT(62) +ResNet50/HDBSCAN/None/NoFT (39) +100% +100% +100% +100% +80% +80% +80% +80% +60% +60% +%09 +%09 +40% +40% +40% +40% +20% +20% +20% +20% +0% +0% +0% +0% +0% +10% +20% +30% +40% +50% +60% +0% +10% +20% +30% +40% +50% +60% +0% +10% +20% +30% +40% +50% +60% +0% +10% +20% +30% +40% +50% +60% +Xception/Dbscan/UMAP/NoFT (80) +VGG16/K-means/None/NoFT(19) +VGG16/K-means/UMAP/NoFT (25) +ResNet50/K-means/UMAP/NoFT(43) +100% +100% +100% +100% +80% +80% +%08 +%08 +%09 +60% +%09 +%09 +40% +40% +40% +40% +20% +20% +20% +20% +0% +0% +0% +0% +0% +10% +20% +30% +40% +50% +60% +0% +10% +20% +30% +40% +50% +%09 +0% +10% +20% +30% +40% +50% +%09 +0% +10% +20% +30% +40% +50% +%09DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +27 +Fig. 13. Comparing the percentage of coverage across different ranges of proportions of failure scenarios in +each set. +Table 9. RQ3: Fisher exact test values when comparing the best pipeline "VGG16/Dbscan/UMAP/NoFT" to +the other pipelines based on the coverage of the infrequent failure scenarios. +Pipelines +44 +62 +39 +80 +19 +25 +43 +Fisher’s Exact test +4e-2 +0.55 +1e-5 +2e-3 +0.018 +1e-5 +1e-5 +significantly higher purity and coverage than the other pipelines. Pipeline 26 is also less negatively +affected by infrequent failure scenarios since coverage is above 90% when the frequency is above +5%, which is not the case for all the other pipelines. +4.6 +Discussion +The results of RQ1 and RQ2 show that there is a family of pipelines leading to higher purity (i.e., +they simplify the identification of root causes) and coverage (i.e., they enable the identification +of all root causes). Such pipelines rely on transfer learning, UMAP for dimensionality reduction, +DBSCAN for clustering, and are not using fine tuning. Among such pipelines, considering that it is +reasonable to expect unsafe scenarios to be infrequent, based on the results of RQ3, we suggest to +use the pipeline relying on VGG16 (Pipeline 26) as transfer learning model. +In our study, we focused on effectiveness, not cost; indeed, our main purpose is to identify the +pipeline that generates clusters that do not confuse the end-user (i.e., they are pure) and is likely +to help identify all the root causes of failures (i.e., they have high coverage). In contrast, cost is +related to the number of clusters being inspected. However, our root cause analysis toolset [21] +includes the generation of animated gifs, one for each cluster, thus enabling the quick visualization +of all the images in a cluster. With such toolset we conjecture that the number of clusters’s images +does not strongly impact cost as all the images are, anyway, quickly visualized. What is important, +instead, is the purity of clusters as with low purity the end-user will not find it easy to determine +commonalities among images. +Nevertheless, to further discuss cost, we measure the number of clusters to be inspected for +each pipeline considering the dataset used for RQ1 and RQ2. We count only clusters capturing the +injected failure scenarios since for the others we cannot precisely determine what is the expected +number of clusters. A lower number of clusters should indicate lower cost and, since a number +, Vol. 1, No. 1, Article . Publication date: February 2023. + +VGG16/Dbscan/UMAP/NoFT(26) +Inception_V3/Dbscan/UMAP/NoFT (62) +ResNet50/Dbscan/UMAP/NoFT (44) +ResNet50/HDBSCAN/None/NoFT (39) +120 +120 +120 +100 +90 +100 +100 +100 +80 +80 +erage +80 +Ranges of the proportion of faulty scenarios in a set +Ranges of the proportion of faulty scenarios in a set +Ranges of the proportion offaulty scenarios in a set +Ranges of the proportion of faulty scenarios in a set +Xception/Dbscan/UMAP/NoFT (80) +VGG16/K-means/None/NoFT(19) +VGG16/K-means/UMAP/NoFT(25) +ResNet50/K-means/UMAP/NoFT(43) +120 +120 +100 +120 +100 +100 +100 +age +80 +80 +Cover +40 +40 +60 +40 +20 +2 +10 +Ranges of the proportion of faulty scenarios in a set +Ranges of the proportion of faulty scenarios in a set +Ranges of the proportion of faulty scenarios in a set +Ranges of the proportion of faulty scenarios in a set28 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Fig. 14. Examples of clusters generated by pipeline 26 for the HPD case study subject. +of clusters higher than the number of failure scenarios to be discovered implies the presence of +redundant clusters, we compute the degree of redundancy as: +redundancy ratio = +number of clusters +covered failure scenarios +Finally, to discuss how well each pipeline improves current practice in industry, we estimate +the degree of savings with respect to the such practice, which entails the visual inspection of all +images. To do so, we assume that inspecting a single cluster using animated gifs is as inexpensive +as visualizing one single image. Indeed, though clusters involve several images, through animation, +they actually make it easier to quickly identify commonalities rather than guessing root causes from +a single image. Figure 14 shows four example clusters where all the images present a commonality +(i.e., the root cause of the DNN failure) that is easy to determine when visualizing all the images in +a sequence. Therefore, we estimate savings as: +savings = 1 − number of clusters +number of images +Table 10 shows our results; it reports the number of RCCs generated for each case study DNN +and across all of them. Further, it reports the percentage and number of failure scenarios covered by +each pipeline (used to compute redundancy and providing information about the effectiveness of a +pipeline), along with redundancy ratio and savings. We report only the results for the best pipelines +, Vol. 1, No. 1, Article . Publication date: February 2023. + +Cluster 1 (images with glasses) +Cluster 2 (images with noise) +Cluster 3 (images with mask) +Cluster 4 (images with scaling)DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +29 +Table 10. The number of redundant clusters generated by the best pipelines for each case study subject and +across all of them. The last columns represent the number and the percentage of failure scenarios covered by +the pipelines, the redundancy ratio, and the savings. +pipelines +Number of generated clusters +Covered failure scenarios (percentage %) +Redundancy ratio +Savings +GD +HPD +OC +SVIRO +CPD +SAP +TOTAL +VGG16/K-means/None/NoFT +3 +5 +3 +3 +3 +2 +19 +17 (59%) +1,12 +0,99 +VGG16/K-means/UMAP/NoFT +4 +8 +2 +4 +3 +3 +24 +20 (69%) +1,20 +0,99 +ResNet50/K-means/UMAP/NoFT +3 +4 +3 +2 +2 +4 +18 +15 (52%) +1,20 +0,99 +VGG16/Dbscan/UMAP/NoFT +26 +77 +13 +8 +13 +37 +174 +28 (97%) +6,21 +0,91 +ResNet50/Dbscan/UMAP/NoFT +42 +51 +5 +10 +27 +44 +179 +27 (93%) +6,63 +0,91 +Inception_V3/Dbscan/UMAP/NoFT +28 +60 +7 +9 +15 +42 +161 +29 (100%) +5,55 +0,92 +Xception/Dbscan/UMAP/NoFT +33 +30 +9 +2 +9 +14 +97 +25 (86%) +3,88 +0,95 +ResNet50/HDBSCAN/None/NoFT +14 +171 +15 +7 +74 +3 +284 +24 (83%) +11,83 +0,86 +identified when addressing RQ1 to RQ2 because there is no reason to select pipelines that do not +achieve high purity and coverage. +The number of clusters generated by the selected pipelines ranges between 18 and 284. The +pipelines leading to the lowest number of clusters are the ones including K-means: ResNet50/K- +means/UMAP/NoFT (18), VGG16/K-means/None/NoFT (19), and VGG16/K-means/UMAP/NoFT (24). +Pipelines with DBSCAN and HDBSCAN lead to a much higher number of clusters. To discuss the +practical impact of such a high number of clusters, we focus on the redundancy ratio, which ranges +between 1.12 and 11.8; the redundancy ratio indicates that the pipeline with the highest number of +clusters (i.e., ResNet50/HDBSCAN/None/NoFT), on average, presents 11 redundant clusters for each +identified failure scenario. Given that, in the presence of pure clusters, understanding the scenario +captured by one pipeline is quick with animated gifs, we consider that inspecting 11 redundant +clusters per fault has a limited impact on cost. Finally, if we focus on savings, we can observe that +respect to current practice, all the pipelines except (ResNet50/HDBSCAN/Only/NoFT) lead to savings +above 90%, thus showing that their adoption is highly beneficial. +Although the pipelines including K-means lead to the lowest cost, their coverage is particularly +low for infrequent scenarios (see Table 8, with coverage below 35% for the range [0-5], and below +60% for the range [5-10]), which is bound to be a common situation in practice. Since pipelines +leading to a small number of clusters can be highly ineffective in realistic safety-critical contexts +(i.e., when some failure causes are infrequent), assuming that redundant clusters are easy to +manage, we conclude that the best choice are the pipelines that maximize purity and coverage, as +discussed above (i.e., Pipeline 26, VGG16/DBSCAN/UMAP/NoFT). A possible tradeoff is Pipeline 80 +(Xception/DBSCAN/UMAP/NoFT), which is among the best performing for RQ3 (e.g., coverage above +40% for the range [0-5], and above 70% for the range [5-10]) and leads to 3.6 redundant clusters +only, on average. +4.7 +Threats to validity +We discuss internal, conclusion, construct, and external validity according to conventional prac- +tices [88]. +4.7.1 +Internal validity. Since 72 of our 99 pipelines use a Transfer Learning pre-trained model to +extract the features, a possible internal threat is that this model can negatively affect our results +if inadequate. Indeed, clustering relies on the similarity computed on the extracted features. To +mitigate this threat, we visually inspected the clusters to check their consistency. Having consistent +clusters means the features extracted by the models contain enough information to cluster the +images based on their similarity. +Another potential threat might be that the dataset (with the injected faults) was created with the +proposed approach in mind. Therefore, there might be a risk of bias. To mitigate this risk, all the +, Vol. 1, No. 1, Article . Publication date: February 2023. + +30 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +methods used in our pipelines (feature extraction methods, clustering algorithms, dimensionality +reduction techniques) are independent of the data. These methods do not require any a priori +knowledge on the data. We also publish our data to further mitigate this risk. All the experiments +can be reproduced with any injected faulty scenario. +4.7.2 +External validity. To alleviate the threats related to the choice of the case study DNNs, we +use six well-studied datasets with diverse complexity. Four out of six subject DNNs implement +tasks motivated by IEE business needs. These DNNs address problems that are quite common in +the automotive industry. The other two DNNs are also related to the automotive industry and were +used in many Kaggle challenges [56, 82]. +Although our pipelines were only tested on case study DNNs related to the automotive industry, +we believe they will perform well with other data sets. This is because the models used for the +feature extraction were pre-trained on ImageNet, which means that the model can capture features +related to 1, 000 classes, including humans, animals, and objects. As for AE, it can learn the aspects +of any data set during training and provide high-quality clusters. Finally, for HUDD and LRP, the +extraction of heatmap-based features is performed on well-known layer types that are part of any +DNN model, regardless of the task at hand (i.e., they can be extended to DNNs that were not studied +in this work). +4.7.3 +Construct validity. The construct considered in our work is effectiveness. We measure the +effectiveness through complementary indicators as follows: +For RQ1, we evaluate the effectiveness of our pipelines by computing the purity of the generated +clusters. The purity of a cluster is measured as the maximum proportion of images representing +one faulty scenario in this cluster. +For RQ2, we evaluate the effectiveness of our pipelines based on the coverage of the injected +faulty scenarios by the root cause clusters. A faulty scenario is covered by a cluster if at least 90% +of the images in this cluster represent such faulty scenario. +Finally, for RQ3, we consider both the purity and the coverage to measure the robustness of the +top-performing pipelines to rare faulty scenarios. +4.7.4 +Conclusion validity. Conclusion validity addresses threats that impact the ability to conclude +appropriately. To mitigate such threats and to avoid violating parametric assumptions in our +statistical analysis, we rely on a non-parametric test and effect size measure (i.e., Mann Whitney +U-test and the Vargha and Delaney’s ˆ𝐴12 statistics, respectively) to assess the statistical significance +of differences in our results. Additionally, we applied the Fisher’s exact test when comparing +coverage results related to different distributions of faulty scenarios (i.e., RQ3), which is commonly +used in similar contexts. All results were reported based on both purity and coverage parameters, +and six datasets were analyzed during our experiments. +4.8 +Data Availability +All our implementations, the failure-inducing sets, the generated root-cause clusters and the data +generated to address our research questions are available online [5]. +5 +RELATED WORK +Our paper is related to the literature on DNN debugging and applications of transfer learning to +perform root cause analysis [55, 78]. +Heatmap-based approaches [13, 51, 60, 68, 72, 90, 91] explain the DNN’s prediction of an image +by highlighting which region of that image influenced the most the DNN output. For example, +Grad-CAM generates a heatmap from the gradient flowing into the last layer. The heatmap is then +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +31 +superposed on the original image to highlight the regions of the image that activated the DNN +and influenced the decision [68]. The main limitation of these approaches is that they require +the inspection of all the heatmaps generated for the images processed by the DNN (e.g., error- +inducing images) and, different from our pipelines, do not provide engineers with guidance for their +inspection (i.e., one cluster for each failure root cause). SHAP (SHapley Additive exPlanation) [45] +generates explanations by calculating the contribution of each feature to predictions, thus explaining +what features are the most important for each prediction. In the case of an image CNN, SHAP +considers a group of pixels as a feature and calculates their contribution to the decision made by +the DNN. Like heatmap-based approaches, SHAP does not provide guidance for the investigation +of multiple failure-inducing images. +DeepJanus [65] helps identify misbehaviors in a Deep Learning system by finding a set of pairs +of inputs that are similar to each other and that trigger different behaviors of the Deep Learning +system. This set of pairs represents the border between the input regions where the DNN behaves +as expected or fails. Different from our work, DeepJanus characterizes the behaviour of a DNN +that can be tested with a simulator but cannot provide explanations for failures observed with +real-world images. +Some DNN testing approaches explain the input regions where DNN errors are observed by +relying on decision trees constructed using the simulator parameters used to generate test input +images [2, 32]. Although decision trees are an effective mean to provide explanations for failures +detected during simulator-based testing, they cannot be applied to provide explanations for failures +observed with real-world images. To overcome such a limitation, we have recently developed +SEDE [22], a technique that applies HUDD to failure-inducing real-world images to generate root +cause clusters and then relies on evolutionary algorithms to drive the generation, for each RCC, of +similar images using simulators. The simulator parameter values used to generate such images are +then fed into PART [24], a tree-based rule learning algorithm to characterize each RCCs in terms +of simulator parameters (i.e., it generates expressions that constrain simulator parameters). The +work in this paper is complementary to SEDE since the latter can be applied to clusters generated +with the best pipeline (i.e., Pipeline 26). +Pan et al. [55] combine Transfer Learning with clustering to find root causes of hardware +failures. The proposed method uses different clustering algorithms (K-means [47], decision tree +clustering [44], hierarchical clustering [40]) on hardware test data to cluster failures likely due to +the same causes. Different from their work, we aim to explain failures in DNNs that process images +(i.e., our feature space is much larger). Ter Burg et al. [78] explain DNNs based on a transfer learning +model that has been fine-tuned to detect geometric shapes connecting face landmarks. Such shapes +are treated as features and the contribution of each feature is computed by relying on SHAP. The +output should help end-users determine what influenced the DNN output. Unfortunately, similar +to heatmap-based approaches, this approach does not support the explanation of multiple failures +but require engineers to process them one by one. +To conclude, our previous works (i.e., HUDD [20] and SAFE [4]) have been the first to apply +clustering algorithms to white-box and black-box feature extraction approaches to explain failure +causes in DNN-based systems. This study is the first to systematically assess and compare the +performance of alternative white-box and black-box feature extraction approaches, dimensionality +reduction techniques, and clustering algorithms using a wide variety of practical, realistic failure +scenarios. +6 +CONCLUSION +In this paper, we presented an large-scale empirical evaluation of 99 different pipelines for root +cause analysis of DNN failures. Our pipelines receive as input a set of images leading to DNN +, Vol. 1, No. 1, Article . Publication date: February 2023. + +32 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +failures and generate as output cluster of images sharing similar characteristics. As demonstrated by +our previous work, by visualizing the images in each cluster, an engineer can notice commonalities +across the images in each cluster; such commonalities represent the root causes of failures, help +characterize failure scenarios and, thus, support engineers in improving the system (e.g., by selecting +additional similar images to retrain the DNN or by introducing countermeasures in the system). +We considered 99 pipelines resulting from the combination of five methods for feature extraction, +two techniques for dimensionality reduction and three clustering algorithms. Our methods for +feature extraction include white-box (i.e., heatmap generation techniques) and black-box approaches +(i.e., fine-tuned and non-finetuned transfer learning models). Additionally, we rely on PCA and +UMAP for dimensionality reduction and K-means, DBSCAN, and HDBSCAN for clustering. +We evaluated our pipelines in terms of clusters’ purity and coverage of failures based on failure +scenarios widely varying in terms of frequency, thus analyzing the impact of rare scenarios on our +best pipelines. Based on six case study subjects in the automotive domain, our results show that the +best results are obtained with pipelines relying on VGG-16 as transfer learning model, not using +fine tuning, leveraging UMAP as a dimensionality reduction technique, and using DBSCAN as +clustering algorithm. When the failure scenarios are equally distributed, the best pipeline achieved a +purity of 94.3% (i.e., almost all the images in RCCs present the same failure scenario) and a coverage +of 96.7%. The same pipeline also performs well with rare failure scenarios; indeed, when images +belonging to failure scenarios represent between 5 and 10% of the total number of images, it still +can cover 90% of the failure scenarios with a cluster purity above 70%. +ACKNOWLEDGMENTS +This project has received funding from IEE Luxembourg, Luxembourg’s National Research Fund +(FNR) under grant BRIDGES2020/IS/14711346/FUNTASY, and NSERC of Canada under the Dis- +covery and CRC programs. Authors would like to thank Thomas Stifter from IEE for his valuable +support. The experiments presented in this paper were carried out using the HPC facilities of the +University of Luxembourg (see http://hpc.uni.lu). +REFERENCES +[1] Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy +Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, +Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, +Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, +Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, +Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. 2015. TensorFlow: Large-Scale Machine Learning on Heterogeneous +Systems. https://www.tensorflow.org/ Software available from tensorflow.org. +[2] Raja Ben Abdessalem, Shiva Nejati, Lionel C Briand, and Thomas Stifter. 2018. Testing vision-based control systems +using learnable evolutionary algorithms. In 2018 IEEE/ACM 40th International Conference on Software Engineering +(ICSE). IEEE, 1016–1026. +[3] Mohammed Oualid Attaoui, Nassima Dif, Hanene Azzag, and Mustapha Lebbah. 2022. Regions of interest selection in +histopathological images using subspace and multi-objective stream clustering. The Visual Computer (2022), 1–19. +[4] Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand. 2022. Black-Box Safety Analysis +and Retraining of DNNs Based on Feature Extraction and Clustering. ACM Trans. Softw. Eng. Methodol. (jul 2022). +https://doi.org/10.1145/3550271 +[5] Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand. 2023. DNN Explanation for Safety Anal- +ysis: an Empirical Evaluation of Clustering-based Approaches - Replicability package. https://figshare.com/projects/ +DNN_Explanation_for_Safety_Analysis_an_Empirical_Evaluation_of_Clustering-based_Approaches/157973 +[6] Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D +Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, et al. 2016. End to end learning for self-driving cars. arXiv preprint +arXiv:1604.07316 (2016). +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +33 +[7] Diogo V Carvalho, Eduardo M Pereira, and Jaime S Cardoso. 2019. Machine learning interpretability: A survey on +methods and metrics. Electronics 8, 8 (2019), 832. +[8] Sully Chen. 2016. Autopilot-Tensorflow. https://github.com/SullyChen/Autopilot-TensorFlow +[9] François Chollet. 2017. Xception: Deep learning with depthwise separable convolutions. In Proceedings of the IEEE +conference on computer vision and pattern recognition. 1251–1258. +[10] Francois Chollet et al. 2015. Keras. https://github.com/fchollet/keras +[11] Alex Clark. 2015. Pillow (PIL Fork) Documentation. +https://buildmedia.readthedocs.org/media/pdf/pillow/latest/ +pillow.pdf +[12] Steve Dias Da Cruz, Oliver Wasenmüller, Hans-Peter Beise, Thomas Stifter, and Didier Stricker. 2020. SVIRO: Synthetic +Vehicle Interior Rear Seat Occupancy Dataset and Benchmark. In 2020 IEEE Winter Conference on Applications of +Computer Vision (WACV). IEEE, 962–971. https://doi.org/10.1109/WACV45572.2020.9093315 +[13] Piotr Dabkowski and Yarin Gal. 2017. Real Time Image Saliency for Black Box Classifiers. In Proceedings of the 31st +International Conference on Neural Information Processing Systems (NIPS?17). Curran Associates Inc., Red Hook, NY, +USA, 6970–6979. +[14] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009. Imagenet: A large-scale hierarchical image +database. In 2009 IEEE conference on computer vision and pattern recognition. Ieee, 248–255. +[15] Nassima Dif, Mohammed Oualid Attaoui, Zakaria Elberrichi, Mustapha Lebbah, and Hanene Azzag. 2021. Transfer +learning from synthetic labels for histopathological images classification. Applied Intelligence (2021), 1–20. +[16] Raghav Bali Dipanjan Sarkar. 2021. Transfer Learning in Action. Manning Publications. 69–76 pages. https://livebook. +manning.com/book/transfer-learning-in-action/chapter-1/v-1/69 (visited 2022-02-19). +[17] Kaiwen Duan, Song Bai, Lingxi Xie, Honggang Qi, Qingming Huang, and Qi Tian. 2019. CenterNet: Keypoint Triplets +for Object Detection. In 2019 IEEE/CVF International Conference on Computer Vision (ICCV). IEEE Computer Society, +6568–6577. +[18] Ekin EKİNCİ, Hidayet TAKCI, and Sultan ALAGÖZ. 2022. Poet Classification Using ANN and DNN. Electronic Letters +on Science and Engineering 18, 1 (2022), 10–20. +[19] Martin Ester, Hans-Peter Kriegel, Jörg Sander, Xiaowei Xu, et al. 1996. A density-based algorithm for discovering +clusters in large spatial databases with noise.. In Kdd, Vol. 96. 226–231. +[20] Hazem Fahmy, Fabrizio Pastore, Mojtaba Bagherzadeh, and Lionel Briand. 2021. Supporting Deep Neural Network +Safety Analysis and Retraining Through Heatmap-Based Unsupervised Learning. IEEE Transactions on Reliability +(2021), 1–17. https://doi.org/10.1109/TR.2021.3074750 +[21] Hazem Fahmy, Fabrizio Pastore, and Lionel Briand. 2022. HUDD: A tool to debug DNNs for safety analysis. In 2022 +IEEE/ACM 44th International Conference on Software Engineering: Companion Proceedings (ICSE-Companion). 100–104. +https://doi.org/10.1145/3510454.3516858 +[22] Hazem Fahmy, Fabrizio Pastore, Lionel Briand, and Thomas Stifter. 2022. Simulator-Based Explanation and Debugging +of Hazard-Triggering Events in DNN-Based Safety-Critical Systems. ACM Trans. Softw. Eng. Methodol. (oct 2022). +https://doi.org/10.1145/3569935 +[23] Florent Forest, Mustapha Lebbah, Hanene Azzag, and Jérôme Lacaille. 2021. Deep embedded self-organizing maps for +joint representation learning and topology-preserving clustering. Neural Computing and Applications 33, 24 (2021), +17439–17469. +[24] Eibe Frank and Ian H. Witten. 1998. Generating Accurate Rule Sets Without Global Optimization. In Fifteenth +International Conference on Machine Learning, J. Shavlik (Ed.). Morgan Kaufmann, 144–151. +[25] Keinosuke Fukunaga and Larry Hostetler. 1975. The estimation of the gradient of a density function, with applications +in pattern recognition. IEEE Transactions on information theory 21, 1 (1975), 32–40. +[26] Rafael Garcia, Alexandru C. Telea, Bruno Castro da Silva, Jim Torresen, and Joao Luiz Dihl Comba. 2018. A task-and- +technique centered survey on visual analytics for deep learning model engineering. Computers and Graphics 77 (2018), +30 – 49. https://doi.org/10.1016/j.cag.2018.09.018 +[27] Leilani H Gilpin, David Bau, Ben Z Yuan, Ayesha Bajwa, Michael Specter, and Lalana Kagal. 2018. Explaining +explanations: An overview of interpretability of machine learning. In 2018 IEEE 5th International Conference on data +science and advanced analytics (DSAA). IEEE, 80–89. +[28] Ian Goodfellow, Yoshua Bengio, and Aaron Courville. 2016. Deep Learning. MIT Press. http://www.deeplearningbook. +org. +[29] Alexander N Gorban and Andrei Y Zinovyev. 2010. Principal graphs and manifolds. In Handbook of research on machine +learning applications and trends: algorithms, methods, and techniques. IGI Global, 28–59. +[30] David Gunning and David Aha. 2019. DARPA’s explainable artificial intelligence (XAI) program. AI magazine 40, 2 +(2019), 44–58. +[31] Jiawei Han, Micheline Kamber, and Jian Pei. 2011. Data Mining: Concepts and Techniques (3rd ed.). Morgan Kaufmann +Publishers Inc., San Francisco, CA, USA. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +34 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +[32] Fitash Ul Haq, Donghwan Shin, Lionel C. Briand, Thomas Stifter, and Jun Wang. 2021. Automatic Test Suite Generation +for Key-Points Detection DNNs Using Many-Objective Search (Experience Paper). In Proceedings of the 30th ACM +SIGSOFT International Symposium on Software Testing and Analysis (ISSTA 2021). Association for Computing Machinery, +New York, NY, USA, 91–102. https://doi.org/10.1145/3460319.3464802 +[33] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. Deep residual learning for image recognition. In +Proceedings of the IEEE conference on computer vision and pattern recognition. 770–778. +[34] Torsten Hothorn, Kurt Hornik, and Achim Zeileis. 2015. ctree: Conditional inference trees. The comprehensive R +archive network 8 (2015). +[35] Xinyu Huang, Peng Wang, Xinjing Cheng, Dingfu Zhou, Qichuan Geng, and Ruigang Yang. 2020. The ApolloScape +Open Dataset for Autonomous Driving and Its Application. IEEE Transactions on Pattern Analysis and Machine +Intelligence 42, 10 (oct 2020), 2702–2719. https://doi.org/10.1109/tpami.2019.2926463 +[36] IEE. 2020. IEE Sensing solutions. www.iee.lu. +[37] International Organization for Standardization. 2020. ISO, ISO-24765-2017, Systems and software engineering - +Vocabulary. +[38] International Organization for Standardization. 2020. ISO, ISO26262-1:2018, Road vehicles: Functional safety. +[39] Barbara Kitchenham, Lech Madeyski, David Budgen, Jacky Keung, Pearl Brereton, Stuart Charters, Shirley Gibbs, +and Amnart Pohthong. 2017. Robust Statistical Methods for Empirical Software Engineering. Empirical Software +Engineering 22, 2 (2017), 579–630. https://doi.org/10.1007/s10664-016-9437-5 +[40] Hans-Friedrich Köhn and Lawrence J Hubert. 2014. Hierarchical cluster analysis. Wiley StatsRef: Statistics Reference +Online (2014), 1–13. +[41] Hans-Peter Kriegel, Peer Kröger, Jörg Sander, and Arthur Zimek. 2011. Density-based clustering. Wiley Interdisciplinary +Reviews: Data Mining and Knowledge Discovery 1, 3 (2011), 231–240. +[42] Zhong Li, Minxue Pan, Tian Zhang, and Xuandong Li. 2021. Testing DNN-based Autonomous Driving Systems under +Critical Environmental Conditions. In International Conference on Machine Learning. PMLR, 6471–6482. +[43] Bin Liang, Hongcheng Li, Miaoqiang Su, Xirong Li, Wenchang Shi, and Xiaofeng Wang. 2021. Detecting Adversarial +Image Examples in Deep Neural Networks with Adaptive Noise Reduction. IEEE Transactions on Dependable and Secure +Computing 18, 1 (jan 2021), 72–85. https://doi.org/10.1109/tdsc.2018.2874243 +[44] Bing Liu, Yiyuan Xia, and Philip S Yu. 2000. Clustering through decision tree construction. In Proceedings of the ninth +international conference on Information and knowledge management. 20–29. +[45] Scott M Lundberg and Su-In Lee. 2017. A unified approach to interpreting model predictions. Advances in neural +information processing systems 30 (2017). +[46] J. MacQueen. 1967. Some methods for classification and analysis of multivariate observations. Proceedings of BSMSP +(1967), 281–297. +[47] J. MacQueen. 1967. Some Methods for Classification and Analysis of Multivariate Observations. In Proceedings of +the 5th Berkeley Symposium on Mathematical Statistics and Probability - Vol. 1, L. M. Le Cam and J. Neyman (Eds.). +University of California Press, Berkeley, CA, USA, 281–297. +[48] Leland McInnes, John Healy, and Steve Astels. 2017. hdbscan: Hierarchical density based clustering. Journal of Open +Source Software 2, 11 (2017), 205. +[49] Leland McInnes, John Healy, Nathaniel Saul, and Lukas Grossberger. 2018. UMAP: Uniform Manifold Approximation +and Projection. The Journal of Open Source Software 3, 29 (2018), 861. +[50] Gregoire Montavon. 2019. WIFS 2017 Tutorial on Methods for Understanding DNNs and their Predictions. +http: +//heatmapping.org/wifs2017/ +[51] Grégoire Montavon, Alexander Binder, Sebastian Lapuschkin, Wojciech Samek, and Klaus Robert Müller. 2019. Layer- +Wise Relevance Propagation: An Overview. Springer International Publishing, Cham, 193–209. https://doi.org/10.1007/ +978-3-030-28954-6_10 +[52] Gregoire Montavon, Sebastian Lapuschkin, Alexander Binder, Wojciech Samek, and Klaus-Robert Muller. 2017. Ex- +plaining nonlinear classification decisions with deep Taylor decomposition. Pattern Recognition 65 (2017), 211 – 222. +https://doi.org/10.1016/j.patcog.2016.11.008 +[53] Fionn Murtagh and Pierre Legendre. 2014. Ward’s Hierarchical Agglomerative Clustering Method: Which Algorithms +Implement Ward’s Criterion? Journal of Classification 31, 3 (01 Oct 2014), 274–295. https://doi.org/10.1007/s00357- +014-9161-z +[54] A Kousar Nikhath, MD Abdul Rab, N Venkata Bharadwaja, L Goutham Reddy, K Saicharan, C Venkat Manas Reddy, +et al. 2022. An Intelligent College Enquiry Bot using NLP and Deep Learning based techniques. In 2022 International +Conference for Advancement in Technology (ICONAT). IEEE, 1–6. +[55] Renjian Pan, Xin Li, and Krishnendu Chakrabarty. 2021. Unsupervised Root-Cause Analysis with Transfer Learning +for Integrated Systems. In 2021 IEEE 39th VLSI Test Symposium (VTS). IEEE, 1–6. +[56] ADVAY PATIL. 2022. Car Object Detection. https://www.kaggle.com/code/advaypatil/car-object-detection +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +35 +[57] Karl Pearson. 1901. LIII. On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and +Dublin philosophical magazine and journal of science 2, 11 (1901), 559–572. +[58] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. +Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. 2011. Scikit-learn: Machine +Learning in Python. Journal of Machine Learning Research 12 (2011), 2825–2830. +[59] Kexin Pei, Yinzhi Cao, Junfeng Yang, and Suman Jana. 2017. Deepxplore: Automated whitebox testing of deep learning +systems. In proceedings of the 26th Symposium on Operating Systems Principles. 1–18. +[60] Vitali Petsiuk, Abir Das, and Kate Saenko. 2018. RISE: Randomized Input Sampling for Explanation of Black-box +Models. In Proceedings of the British Machine Vision Conference (BMVC). +[61] Autumn pre-trained model. 2017. Udacity Self-Driving Car Challenge 2. +https://github.com/udacity/self-driving- +car/tree/master/steering-models/community-models/autumn +[62] PyTorch. 2020. PyTorch DNN framework. https://pytorch.org +[63] Sizhe Rao, Minghui Wang, Cuixia Tian, Xiangqiao Ao, et al. 2021. A Hierarchical Tree-Based Syslog Clustering Scheme +for Network Diagnosis. In 2021 17th International Conference on Network and Service Management (CNSM). IEEE, 28–34. +[64] Satti RG Reddy, GP Varma, and Rajya Lakshmi Davuluri. 2022. Deep Neural Network (DNN) Mechanism for Identifica- +tion of Diseased and Healthy Plant Leaf Images Using Computer Vision. Annals of Data Science (2022), 1–30. +[65] Vincenzo Riccio and Paolo Tonella. 2020. Model-based exploration of the frontier of behaviours for deep learning system +testing. In Proceedings of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on +the Foundations of Software Engineering. 876–888. +[66] Peter J Rousseeuw. 1987. Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. Journal of +computational and applied mathematics 20 (1987), 53–65. +[67] V. Satopaa, J. Albrecht, D. Irwin, and B. Raghavan. 2011. Finding a "Kneedle" in a Haystack: Detecting Knee Points in +System Behavior. In 2011 31st International Conference on Distributed Computing Systems Workshops. 166–171. +[68] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra. 2017. Grad-CAM: Visual Explanations from +Deep Networks via Gradient-Based Localization. In 2017 IEEE International Conference on Computer Vision (ICCV). +618–626. https://doi.org/10.1109/ICCV.2017.74 +[69] Jonathon Shlens. 2014. A tutorial on principal component analysis. arXiv preprint arXiv:1404.1100 (2014). +[70] Karen Simonyan and Andrew Zisserman. 2015. Very deep convolutional networks for large-scale image recognition. +In International Conference on Learning Representations. https://arxiv.org/abs/1409.1556 +[71] Xibin Song, Peng Wang, Dingfu Zhou, Rui Zhu, Chenye Guan, Yuchao Dai, Hao Su, Hongdong Li, and Ruigang Yang. +2019. Apollocar3d: A large 3d car instance understanding benchmark for autonomous driving. In Proceedings of the +IEEE/CVF Conference on Computer Vision and Pattern Recognition. 5452–5462. +[72] J.T. Springenberg, A. Dosovitskiy, T. Brox, and M. Riedmiller. 2015. Striving for Simplicity: The All Convolutional Net. +In ICLR (workshop track). +[73] Stanford Vision Lab. 2015. ImageNet, image database organized according to the WordNet hierarchy. https://www. +image-net.org. Accessed: 2021-12-07. +[74] Helmut Strasser and Christian Weber. 1999. On the Asymptotic Theory of Permutation Statistics (january 1999 ed.). +WorkingPaper 27. SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU +Vienna University of Economics and Business. +[75] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent +Vanhoucke, and Andrew Rabinovich. 2015. Going deeper with convolutions. In Proceedings of the IEEE conference on +computer vision and pattern recognition. 1–9. +[76] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. 2016. Rethinking the inception +architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition. +2818–2826. +[77] Muhammed Talo. 2019. Automated classification of histopathology images using transfer learning. Artificial Intelligence +in Medicine 101 (2019), 101743. +[78] Kaya ter Burg and Heysem Kaya. 2022. Comparing Approaches for Explaining DNN-Based Facial Expression Classifi- +cations. Algorithms 15, 10 (2022), 367. +[79] Robert L Thorndike. 1953. Who belongs in the family? Psychometrika 18, 4 (1953), 267–276. https://doi.org/10.1007/ +BF02289263 +[80] Yuchi Tian, Kexin Pei, Suman Jana, and Baishakhi Ray. 2018. DeepTest: Automated Testing of Deep-Neural-Network- +Driven Autonomous Cars. In Proceedings of the 40th International Conference on Software Engineering (ICSE ’18). +Association for Computing Machinery, New York, NY, USA, 303–314. https://doi.org/10.1145/3180155.3180220 +[81] Udacity. 2017. Udacity Self-driving Car Challenge 2. Accessed: 2022-08-16. +[82] Peking University/Baidu. 2020. Peking University/Baidu - Autonomous Driving. Accessed: 2022-08-16. +, Vol. 1, No. 1, Article . Publication date: February 2023. + +36 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +[83] Graham JG Upton. 1992. Fisher’s exact test. Journal of the Royal Statistical Society: Series A (Statistics in Society) 155, 3 +(1992), 395–402. +[84] Stéfan van der Walt, Johannes L. Schönberger, Juan Nunez-Iglesias, François Boulogne, Joshua D. Warner, Neil Yager, +Emmanuelle Gouillart, and Tony Yu. 2014. scikit-image: image processing in Python. PeerJ 2 (jun 2014), e453. +https://doi.org/10.7717/peerj.453 +[85] Jr. Ward, J. H. 1963. Hierarchical grouping to optimize an objective function. American Statistical Association Journal +58 (1963), 236–244. +[86] Eric W. Weisstein. 2022. Fisher’s Exact Test. https://mathworld.wolfram.com/FishersExactTest.html. +Accessed: +2022-11-15. +[87] Katharina Weitz, Teena Hassan, Ute Schmid, and Jens Garbas. 2018. Towards explaining deep learning networks to +distinguish facial expressions of pain and emotions. In Forum Bildverarbeitung. 197–208. +[88] Claes Wohlin, Per Runeson, Martin Höst, Magnus C. Ohlsson, Björn Regnell, and Anders Wesslén. 2012. Experimentation +in software engineering. Vol. 9783642290. 1–236 pages. https://doi.org/10.1007/978-3-642-29044-2 +[89] S Paul Wright. 1992. Adjusted p-values for simultaneous inference. Biometrics (1992), 1005–1013. +[90] Matthew D. Zeiler and Rob Fergus. 2014. Visualizing and Understanding Convolutional Networks. In Computer Vision +– ECCV 2014, David Fleet, Tomas Pajdla, Bernt Schiele, and Tinne Tuytelaars (Eds.). Springer International Publishing, +Cham, 818–833. +[91] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba. 2016. Learning Deep Features for Discriminative +Localization. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, 2921–2929. https: +//doi.org/10.1109/CVPR.2016.319 +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +37 +A +ADDITIONAL MATERIAL FOR RQ1 +Table 11. Comparing the clusters generated by the different pipelines based on the average of the purity +across root cause clusters. The last column represents the average of averages. +Pipelines +Case Study Subjects +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +1 +HUDD +None +None +K-Means +51.3% +36.6% +40.7% +62.4% +78.9% +39.9% +51.6% +2 +HUDD +None +None +DBSCAN +56.2% +53.4% +43.1% +53.0% +80.6% +63.7% +58.3% +3 +HUDD +None +None +HDBScan +68.5% +61.7% +43.7% +45.4% +51.2% +27.4% +49.6% +4 +HUDD +None +PCA +K-Means +49.1% +56.4% +40.8% +74.8% +79.7% +27.4% +54.7% +5 +HUDD +None +PCA +DBSCAN +43.4% +54.6% +48.7% +48.3% +80.6% +27.4% +50.5% +6 +HUDD +None +PCA +HDBScan +68.5% +61.7% +35.5% +45.4% +74.1% +26.9% +52.0% +7 +HUDD +None +UMAP +K-Means +56.7% +47.6% +42.3% +54.3% +61.1% +32.2% +49.0% +8 +HUDD +None +UMAP +DBSCAN +69.6% +58.7% +68.3% +59.3% +68.7% +53.9% +63.1% +9 +HUDD +None +UMAP +HDBScan +68.5% +61.7% +33.3% +45.4% +74.1% +27.4% +51.7% +10 +LRP +None +None +K-Means +42.9% +36.6% +56.5% +33.8% +82.8% +73.6% +54.4% +11 +LRP +None +None +DBSCAN +39.5% +28.7% +69.8% +50.7% +96.5% +35.4% +53.4% +12 +LRP +None +None +HDBScan +69.0% +58.3% +56.2% +47.8% +42.4% +27.3% +50.2% +13 +LRP +None +PCA +K-Means +54.2% +56.4% +54.9% +35.7% +82.9% +73.6% +59.6% +14 +LRP +None +PCA +DBSCAN +47.2% +20.6% +71.8% +48.9% +79.7% +35.4% +50.6% +15 +LRP +None +PCA +HDBScan +52.5% +58.3% +25.5% +47.8% +46.8% +26.2% +42.8% +16 +LRP +None +UMAP +K-Means +55.9% +47.6% +54.7% +31.8% +67.0% +32.2% +48.2% +17 +LRP +None +UMAP +DBSCAN +67.0% +62.8% +68.9% +49.7% +80.3% +33.5% +60.4% +18 +LRP +None +UMAP +HDBScan +69.0% +58.3% +56.2% +47.8% +51.6% +26.4% +51.5% +19 +VGG-16 +None +None +K-Means +91.7% +92.1% +95.5% +82.5% +97.3% +99.7% +93.2% +20 +VGG-16 +None +None +DBSCAN +87.0% +85.9% +96.7% +57.3% +98.0% +100.0% +87.5% +21 +VGG-16 +None +None +HDBSCAN +52.6% +99.0% +30.7% +73.4% +77.8% +54.5% +64.7% +22 +VGG-16 +None +PCA +K-Means +90.5% +87.6% +58.3% +87.7% +94.2% +92.2% +85.1% +23 +VGG-16 +None +PCA +DBSCAN +90.7% +94.9% +81.0% +71.6% +96.0% +91.8% +87.7% +24 +VGG-16 +None +PCA +HDBSCAN +45.6% +95.1% +56.2% +93.5% +100.0% +76.0% +77.7% +25 +VGG-16 +None +UMAP +K-Means +97.6% +84.4% +93.7% +82.4% +90.3% +97.8% +91.0% +26 +VGG-16 +None +UMAP +DBSCAN +99.0% +93.0% +99.6% +79.7% +98.1% +96.6% +94.3% +27 +VGG-16 +None +UMAP +HDBSCAN +78.0% +96.7% +56.2% +88.9% +44.1% +79.9% +74.0% +28 +VGG-16 +FT +None +K-Means +26.2% +33.9% +15.8% +24.3% +27.9% +25.5% +25.6% +29 +VGG-16 +FT +None +DBSCAN +26.4% +38.5% +18.2% +32.8% +29.6% +25.2% +28.4% +30 +VGG-16 +FT +None +HDBSCAN +23.4% +51.1% +14.3% +48.1% +25.7% +53.2% +36.0% +31 +VGG-16 +FT +PCA +K-Means +25.4% +37.7% +16.7% +24.5% +26.8% +25.8% +26.1% +32 +VGG-16 +FT +PCA +DBSCAN +29.2% +51.4% +23.4% +46.6% +32.3% +29.0% +35.3% +33 +VGG-16 +FT +PCA +HDBSCAN +22.7% +43.7% +13.8% +41.5% +25.3% +23.3% +28.4% +34 +VGG-16 +FT +UMAP +K-Means +26.3% +33.6% +18.0% +25.3% +26.2% +26.2% +25.9% +35 +VGG-16 +FT +UMAP +DBSCAN +45.4% +42.8% +27.0% +39.1% +43.8% +44.0% +40.4% +36 +VGG-16 +FT +UMAP +HDBSCAN +23.5% +40.4% +14.1% +36.6% +22.0% +24.2% +26.8% +37 +ResNet-50 +None +None +K-Means +84.2% +84.6% +74.0% +61.2% +86.0% +83.7% +78.9% +38 +ResNet-50 +None +None +DBSCAN +63.5% +84.6% +87.5% +72.6% +75.5% +72.0% +76.0% +39 +ResNet-50 +None +None +HDBSCAN +96.4% +100.0% +100.0% +78.8% +87.5% +100.0% +93.8% +40 +ResNet-50 +None +PCA +K-Means +67.4% +79.6% +61.3% +53.4% +85.8% +75.6% +70.5% +41 +ResNet-50 +None +PCA +DBSCAN +79.7% +72.9% +51.1% +45.0% +89.8% +80.3% +69.8% +42 +ResNet-50 +None +PCA +HDBSCAN +40.8% +79.6% +56.2% +32.0% +42.5% +49.7% +50.2% +43 +ResNet-50 +None +UMAP +K-Means +99.4% +93.0% +82.3% +79.6% +99.6% +97.7% +91.9% +44 +ResNet-50 +None +UMAP +DBSCAN +100.0% +95.8% +95.8% +79.0% +99.7% +99.3% +94.9% +45 +ResNet-50 +None +UMAP +HDBSCAN +82.6% +87.3% +38.8% +60.0% +30.5% +69.4% +61.4% +46 +ResNet-50 +FT +None +K-Means +26.7% +37.4% +19.3% +30.0% +26.2% +25.6% +27.5% +47 +ResNet-50 +FT +None +DBSCAN +47.2% +40.9% +32.1% +33.7% +35.2% +39.4% +38.1% +, Vol. 1, No. 1, Article . Publication date: February 2023. + +38 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Table 11. Comparing the clusters generated by the different pipelines based on the average of the purity +across root cause clusters. The last column represents the average of averages. +Pipelines +Case Study Subjects +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +48 +ResNet-50 +FT +None +HDBSCAN +55.0% +46.7% +15.4% +45.2% +26.4% +24.9% +35.6% +49 +ResNet-50 +FT +PCA +K-Means +29.5% +37.1% +17.8% +39.5% +26.6% +26.2% +29.5% +50 +ResNet-50 +FT +PCA +DBSCAN +40.1% +45.6% +23.8% +41.5% +39.4% +39.4% +38.3% +51 +ResNet-50 +FT +PCA +HDBSCAN +23.7% +50.7% +15.7% +48.2% +24.6% +23.3% +31.1% +52 +ResNet-50 +FT +UMAP +K-Means +25.5% +34.6% +17.3% +25.7% +27.5% +25.6% +26.0% +53 +ResNet-50 +FT +UMAP +DBSCAN +37.8% +54.8% +35.9% +48.4% +41.7% +50.6% +44.9% +54 +ResNet-50 +FT +UMAP +HDBSCAN +23.9% +44.5% +15.1% +23.3% +23.7% +24.2% +25.8% +55 +Inception-V3 +None +None +K-Means +84.6% +86.0% +95.1% +69.2% +91.2% +87.8% +85.7% +56 +Inception-V3 +None +None +DBSCAN +100.0% +63.2% +80.9% +17.9% +98.4% +60.4% +70.1% +57 +Inception-V3 +None +None +HDBSCAN +62.6% +96.0% +99.8% +77.9% +62.6% +95.6% +82.4% +58 +Inception-V3 +None +PCA +K-Means +66.5% +74.5% +80.9% +68.6% +88.9% +75.7% +75.8% +59 +Inception-V3 +None +PCA +DBSCAN +97.9% +83.8% +86.0% +96.5% +92.0% +82.2% +89.7% +60 +Inception-V3 +None +PCA +HDBSCAN +92.5% +86.1% +53.4% +70.8% +69.4% +34.3% +67.8% +61 +Inception-V3 +None +UMAP +K-Means +94.1% +87.4% +95.0% +67.0% +90.1% +71.3% +84.2% +62 +Inception-V3 +None +UMAP +DBSCAN +93.4% +95.2% +98.1% +76.6% +97.4% +83.1% +90.7% +63 +Inception-V3 +None +UMAP +HDBSCAN +74.0% +83.3% +55.9% +80.1% +65.8% +70.0% +71.5% +64 +Inception-V3 +FT +None +K-Means +26.6% +34.4% +15.6% +23.6% +26.7% +25.2% +25.4% +65 +Inception-V3 +FT +None +DBSCAN +50.0% +38.3% +24.7% +17.0% +51.0% +44.1% +37.5% +66 +Inception-V3 +FT +None +HDBSCAN +49.9% +48.2% +15.4% +44.4% +53.0% +53.8% +44.1% +67 +Inception-V3 +FT +PCA +K-Means +24.4% +31.5% +16.7% +25.6% +27.2% +25.2% +25.1% +68 +Inception-V3 +FT +PCA +DBSCAN +32.1% +50.6% +29.1% +38.5% +35.1% +38.6% +37.3% +69 +Inception-V3 +FT +PCA +HDBSCAN +46.7% +44.3% +15.1% +45.2% +25.0% +22.9% +33.2% +70 +Inception-V3 +FT +UMAP +K-Means +26.1% +31.0% +17.1% +25.8% +25.2% +27.2% +25.4% +71 +Inception-V3 +FT +UMAP +DBSCAN +45.7% +50.4% +23.4% +45.1% +44.4% +52.5% +43.6% +72 +Inception-V3 +FT +UMAP +HDBSCAN +47.2% +26.2% +15.3% +37.8% +24.7% +25.0% +29.4% +73 +Xception +None +None +K-Means +85.2% +90.4% +88.8% +68.4% +95.2% +72.6% +83.4% +74 +Xception +None +None +DBSCAN +60.3% +35.2% +82.6% +34.7% +73.6% +57.0% +57.2% +75 +Xception +None +None +HDBSCAN +88.9% +91.7% +85.6% +71.5% +100.0% +92.2% +88.3% +76 +Xception +None +PCA +K-Means +75.5% +73.3% +67.2% +57.6% +83.6% +44.4% +66.9% +77 +Xception +None +PCA +DBSCAN +78.8% +87.7% +83.6% +71.2% +93.6% +35.2% +75.0% +78 +Xception +None +PCA +HDBSCAN +75.8% +84.5% +47.5% +75.5% +62.0% +32.7% +63.0% +79 +Xception +None +UMAP +K-Means +93.1% +90.2% +89.8% +66.3% +92.6% +66.7% +83.1% +80 +Xception +None +UMAP +DBSCAN +94.7% +92.7% +98.7% +62.4% +99.0% +85.4% +88.8% +81 +Xception +None +UMAP +HDBSCAN +77.9% +86.3% +36.7% +78.1% +99.4% +62.4% +73.5% +82 +Xception +FT +None +K-Means +25.0% +35.6% +17.1% +27.3% +25.4% +26.2% +26.1% +83 +Xception +FT +None +DBSCAN +32.3% +34.9% +22.7% +17.8% +20.2% +55.2% +30.5% +84 +Xception +FT +None +HDBSCAN +34.7% +46.2% +15.9% +33.4% +25.2% +54.0% +34.9% +85 +Xception +FT +PCA +K-Means +27.0% +35.6% +17.0% +26.2% +25.4% +36.8% +28.0% +86 +Xception +FT +PCA +DBSCAN +44.3% +31.1% +21.4% +28.2% +35.4% +35.2% +32.6% +87 +Xception +FT +PCA +HDBSCAN +46.8% +56.3% +14.8% +37.6% +24.9% +26.6% +34.5% +88 +Xception +FT +UMAP +K-Means +26.4% +32.8% +17.1% +28.7% +27.0% +24.6% +26.1% +89 +Xception +FT +UMAP +DBSCAN +36.9% +42.8% +26.7% +50.4% +46.2% +45.5% +41.4% +90 +Xception +FT +UMAP +HDBSCAN +23.1% +26.2% +14.9% +40.9% +24.7% +26.4% +26.1% +91 +AE +None +None +K-Means +40.9% +56.1% +47.0% +41.6% +72.5% +73.0% +55.2% +92 +AE +None +None +DBSCAN +35.2% +60.8% +11.5% +16.9% +20.3% +21.1% +27.6% +93 +AE +None +None +HDBSCAN +36.9% +67.8% +0.0% +67.5% +0.0% +63.0% +39.2% +94 +AE +None +PCA +K-Means +62.5% +55.1% +51.7% +52.9% +60.9% +77.2% +60.0% +95 +AE +None +PCA +DBSCAN +20.4% +73.3% +68.2% +63.6% +89.1% +76.7% +65.2% +96 +AE +None +PCA +HDBSCAN +73.5% +60.8% +25.7% +68.1% +76.1% +27.8% +55.3% +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +39 +Table 11. Comparing the clusters generated by the different pipelines based on the average of the purity +across root cause clusters. The last column represents the average of averages. +Pipelines +Case Study Subjects +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +97 +AE +None +UMAP +K-Means +43.7% +46.2% +36.4% +37.6% +69.2% +66.0% +49.9% +98 +AE +None +UMAP +DBSCAN +65.3% +61.8% +59.0% +51.9% +69.1% +70.6% +62.9% +99 +AE +None +UMAP +HDBSCAN +28.4% +60.5% +45.4% +47.2% +41.6% +41.7% +44.1% +, Vol. 1, No. 1, Article . Publication date: February 2023. + +40 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +B +ADDITIONAL MATERIAL FOR RQ2 +Table 12. Percentage of faulty scenarios covered by the root cause clusters generated for each pipeline. The +last column represents the average of averages. +Pipelines +Case Study Subjects +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +1 +HUDD +None +None +K-Means +0.0% +0.0% +0.0% +20.0% +25.0% +0.0% +7.5% +2 +HUDD +None +None +DBSCAN +25.0% +25.0% +0.0% +80.0% +25.0% +25.0% +30.0% +3 +HUDD +None +None +HDBScan +100.0% +25.0% +0.0% +0.0% +0.0% +0.0% +20.8% +4 +HUDD +None +PCA +K-Means +0.0% +25.0% +0.0% +40.0% +50.0% +0.0% +19.2% +5 +HUDD +None +PCA +DBSCAN +0.0% +25.0% +0.0% +20.0% +50.0% +0.0% +15.8% +6 +HUDD +None +PCA +HDBScan +100.0% +25.0% +0.0% +0.0% +100.0% +0.0% +37.5% +7 +HUDD +None +UMAP +K-Means +25.0% +0.0% +0.0% +0.0% +0.0% +0.0% +4.2% +8 +HUDD +None +UMAP +DBSCAN +100.0% +50.0% +75.0% +60.0% +75.0% +100.0% +76.7% +9 +HUDD +None +UMAP +HDBScan +100.0% +25.0% +0.0% +0.0% +100.0% +0.0% +37.5% +10 +LRP +None +None +K-Means +0.0% +0.0% +12.5% +0.0% +50.0% +25.0% +14.6% +11 +LRP +None +None +DBSCAN +0.0% +0.0% +37.5% +20.0% +50.0% +0.0% +17.9% +12 +LRP +None +None +HDBScan +100.0% +25.0% +12.5% +20.0% +0.0% +0.0% +26.2% +13 +LRP +None +PCA +K-Means +25.0% +25.0% +12.5% +0.0% +50.0% +25.0% +22.9% +14 +LRP +None +PCA +DBSCAN +0.0% +0.0% +12.5% +20.0% +50.0% +0.0% +13.8% +15 +LRP +None +PCA +HDBScan +0.0% +25.0% +0.0% +20.0% +0.0% +0.0% +7.5% +16 +LRP +None +UMAP +K-Means +25.0% +0.0% +25.0% +0.0% +50.0% +0.0% +16.7% +17 +LRP +None +UMAP +DBSCAN +75.0% +100.0% +75.0% +40.0% +100.0% +0.0% +65.0% +18 +LRP +None +UMAP +HDBScan +100.0% +25.0% +12.5% +20.0% +0.0% +0.0% +26.2% +19 +VGG-16 +None +None +K-Means +50.0% +50.0% +87.5% +80.0% +100.0% +100.0% +77.9% +20 +VGG-16 +None +None +DBSCAN +50.0% +50.0% +75.0% +20.0% +100.0% +100.0% +65.8% +21 +VGG-16 +None +None +HDBSCAN +0.0% +75.0% +0.0% +60.0% +50.0% +0.0% +30.8% +22 +VGG-16 +None +PCA +K-Means +50.0% +50.0% +12.5% +60.0% +100.0% +75.0% +57.9% +23 +VGG-16 +None +PCA +DBSCAN +50.0% +50.0% +37.5% +40.0% +75.0% +75.0% +54.6% +24 +VGG-16 +None +PCA +HDBSCAN +0.0% +75.0% +12.5% +100.0% +75.0% +50.0% +52.1% +25 +VGG-16 +None +UMAP +K-Means +75.0% +75.0% +87.5% +80.0% +75.0% +100.0% +82.1% +26 +VGG-16 +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +80.0% +100.0% +100.0% +96.7% +27 +VGG-16 +None +UMAP +HDBSCAN +50.0% +75.0% +12.5% +60.0% +0.0% +50.0% +41.2% +28 +VGG-16 +FT +None +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +29 +VGG-16 +FT +None +DBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +30 +VGG-16 +FT +None +HDBSCAN +0.0% +25.0% +0.0% +20.0% +0.0% +100.0% +24.2% +31 +VGG-16 +FT +PCA +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +32 +VGG-16 +FT +PCA +DBSCAN +0.0% +25.0% +0.0% +20.0% +0.0% +0.0% +7.5% +33 +VGG-16 +FT +PCA +HDBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +34 +VGG-16 +FT +UMAP +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +35 +VGG-16 +FT +UMAP +DBSCAN +25.0% +0.0% +0.0% +0.0% +50.0% +25.0% +16.7% +36 +VGG-16 +FT +UMAP +HDBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +37 +ResNet-50 +None +None +K-Means +50.0% +50.0% +25.0% +40.0% +50.0% +50.0% +44.2% +38 +ResNet-50 +None +None +DBSCAN +25.0% +50.0% +37.5% +40.0% +50.0% +25.0% +37.9% +39 +ResNet-50 +None +None +HDBSCAN +50.0% +100.0% +100.0% +60.0% +75.0% +100.0% +80.8% +40 +ResNet-50 +None +PCA +K-Means +25.0% +50.0% +12.5% +20.0% +50.0% +25.0% +30.4% +41 +ResNet-50 +None +PCA +DBSCAN +50.0% +50.0% +12.5% +0.0% +50.0% +25.0% +31.2% +42 +ResNet-50 +None +PCA +HDBSCAN +0.0% +100.0% +12.5% +0.0% +0.0% +0.0% +18.8% +43 +ResNet-50 +None +UMAP +K-Means +100.0% +75.0% +50.0% +40.0% +100.0% +100.0% +77.5% +44 +ResNet-50 +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +60.0% +100.0% +100.0% +93.3% +45 +ResNet-50 +None +UMAP +HDBSCAN +50.0% +75.0% +0.0% +20.0% +0.0% +25.0% +28.3% +46 +ResNet-50 +FT +None +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +47 +ResNet-50 +FT +None +DBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +41 +Table 12. Percentage of faulty scenarios covered by the root cause clusters generated for each pipeline. The +last column represents the average of averages. +Pipelines +Case Study Subjects +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +48 +ResNet-50 +FT +None +HDBSCAN +50.0% +0.0% +0.0% +0.0% +0.0% +0.0% +8.3% +49 +ResNet-50 +FT +PCA +K-Means +0.0% +0.0% +0.0% +20.0% +0.0% +0.0% +3.3% +50 +ResNet-50 +FT +PCA +DBSCAN +0.0% +0.0% +0.0% +20.0% +0.0% +0.0% +3.3% +51 +ResNet-50 +FT +PCA +HDBSCAN +0.0% +0.0% +0.0% +40.0% +0.0% +0.0% +6.7% +52 +ResNet-50 +FT +UMAP +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +53 +ResNet-50 +FT +UMAP +DBSCAN +0.0% +50.0% +0.0% +40.0% +25.0% +100.0% +35.8% +54 +ResNet-50 +FT +UMAP +HDBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +55 +Inception-V3 +None +None +K-Means +75.0% +75.0% +87.5% +40.0% +100.0% +50.0% +71.2% +56 +Inception-V3 +None +None +DBSCAN +75.0% +25.0% +50.0% +0.0% +100.0% +25.0% +45.8% +57 +Inception-V3 +None +None +HDBSCAN +25.0% +100.0% +87.5% +100.0% +25.0% +100.0% +72.9% +58 +Inception-V3 +None +PCA +K-Means +50.0% +50.0% +37.5% +40.0% +75.0% +25.0% +46.2% +59 +Inception-V3 +None +PCA +DBSCAN +50.0% +75.0% +62.5% +40.0% +75.0% +25.0% +54.6% +60 +Inception-V3 +None +PCA +HDBSCAN +25.0% +75.0% +12.5% +60.0% +25.0% +0.0% +32.9% +61 +Inception-V3 +None +UMAP +K-Means +75.0% +75.0% +87.5% +40.0% +75.0% +50.0% +67.1% +62 +Inception-V3 +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +100.0% +100.0% +100.0% +100.0% +63 +Inception-V3 +None +UMAP +HDBSCAN +25.0% +100.0% +12.5% +100.0% +25.0% +25.0% +47.9% +64 +Inception-V3 +FT +None +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +65 +Inception-V3 +FT +None +DBSCAN +0.0% +0.0% +0.0% +0.0% +25.0% +0.0% +4.2% +66 +Inception-V3 +FT +None +HDBSCAN +75.0% +25.0% +0.0% +40.0% +75.0% +100.0% +52.5% +67 +Inception-V3 +FT +PCA +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +68 +Inception-V3 +FT +PCA +DBSCAN +0.0% +25.0% +0.0% +0.0% +0.0% +0.0% +4.2% +69 +Inception-V3 +FT +PCA +HDBSCAN +25.0% +0.0% +0.0% +20.0% +0.0% +0.0% +7.5% +70 +Inception-V3 +FT +UMAP +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +71 +Inception-V3 +FT +UMAP +DBSCAN +25.0% +50.0% +0.0% +20.0% +25.0% +75.0% +32.5% +72 +Inception-V3 +FT +UMAP +HDBSCAN +50.0% +0.0% +0.0% +0.0% +0.0% +0.0% +8.3% +73 +Xception +None +None +K-Means +50.0% +75.0% +87.5% +40.0% +100.0% +75.0% +71.2% +74 +Xception +None +None +DBSCAN +25.0% +0.0% +37.5% +0.0% +25.0% +25.0% +18.8% +75 +Xception +None +None +HDBSCAN +100.0% +100.0% +62.5% +60.0% +50.0% +100.0% +78.8% +76 +Xception +None +PCA +K-Means +50.0% +50.0% +37.5% +0.0% +75.0% +0.0% +35.4% +77 +Xception +None +PCA +DBSCAN +50.0% +75.0% +50.0% +0.0% +75.0% +0.0% +41.7% +78 +Xception +None +PCA +HDBSCAN +100.0% +50.0% +0.0% +80.0% +25.0% +0.0% +42.5% +79 +Xception +None +UMAP +K-Means +75.0% +75.0% +62.5% +40.0% +100.0% +50.0% +67.1% +80 +Xception +None +UMAP +DBSCAN +100.0% +100.0% +100.0% +40.0% +100.0% +100.0% +90.0% +81 +Xception +None +UMAP +HDBSCAN +50.0% +75.0% +0.0% +60.0% +100.0% +25.0% +51.7% +82 +Xception +FT +None +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +83 +Xception +FT +None +DBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +25.0% +4.2% +84 +Xception +FT +None +HDBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +100.0% +16.7% +85 +Xception +FT +PCA +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +86 +Xception +FT +PCA +DBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +87 +Xception +FT +PCA +HDBSCAN +75.0% +50.0% +0.0% +0.0% +0.0% +0.0% +20.8% +88 +Xception +FT +UMAP +K-Means +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +89 +Xception +FT +UMAP +DBSCAN +0.0% +0.0% +0.0% +60.0% +50.0% +0.0% +18.3% +90 +Xception +FT +UMAP +HDBSCAN +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +0.0% +91 +AE +None +None +K-Means +0.0% +25.0% +12.5% +20.0% +50.0% +50.0% +26.2% +92 +AE +None +None +DBSCAN +0.0% +25.0% +0.0% +0.0% +0.0% +0.0% +4.2% +93 +AE +None +None +HDBSCAN +0.0% +25.0% +0.0% +40.0% +0.0% +25.0% +15.0% +94 +AE +None +PCA +K-Means +0.0% +25.0% +12.5% +0.0% +50.0% +50.0% +22.9% +95 +AE +None +PCA +DBSCAN +0.0% +25.0% +0.0% +20.0% +50.0% +50.0% +24.2% +96 +AE +None +PCA +HDBSCAN +100.0% +50.0% +0.0% +80.0% +50.0% +0.0% +46.7% +, Vol. 1, No. 1, Article . Publication date: February 2023. + +42 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Table 12. Percentage of faulty scenarios covered by the root cause clusters generated for each pipeline. The +last column represents the average of averages. +Pipelines +Case Study Subjects +# +FE +FT +DR +CA +GD +OC +HPD +SVIRO +SAP +CPD +Avg. +97 +AE +None +UMAP +K-Means +0.0% +0.0% +0.0% +0.0% +25.0% +50.0% +12.5% +98 +AE +None +UMAP +DBSCAN +50.0% +50.0% +37.5% +40.0% +50.0% +100.0% +54.6% +99 +AE +None +UMAP +HDBSCAN +0.0% +25.0% +0.0% +20.0% +0.0% +0.0% +7.5% +, Vol. 1, No. 1, Article . Publication date: February 2023. + +DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches +43 +C +ADDITIONAL MATERIAL FOR RQ3 +Table 13. Distribution of faults for the different failure inducing sets for each case study subject. +Case Study +Dataset +Faulty Scenarios +N +B +S +D +M +H +SG +EG +EO +NF +GD +GD_1 +64 +40 +48 +72 +- +- +- +- +- +56 +GD_2 +48 +32 +24 +16 +- +- +- +- +- +64 +GD_3 +24 +64 +16 +40 +- +- +- +- +- +8 +GD_4 +72 +16 +24 +48 +- +- +- +- +- +40 +GD_5 +40 +24 +72 +16 +- +- +- +- +- +8 +GD_6 +56 +32 +48 +16 +- +- +- +- +- +24 +GD_7 +64 +32 +8 +56 +- +- +- +- +- +24 +GD_8 +72 +24 +16 +8 +- +- +- +- +- +64 +GD_9 +40 +64 +48 +16 +- +- +- +- +- +56 +GD_10 +56 +8 +64 +24 +- +- +- +- +- +48 +OC +OC_1 +18 +6 +10 +4 +- +- +- +- +- +8 +OC_2 +4 +18 +12 +2 +- +- +- +- +- +14 +OC_3 +2 +14 +10 +6 +- +- +- +- +- +16 +OC_4 +2 +4 +6 +10 +- +- +- +- +- +8 +OC_5 +6 +4 +16 +18 +- +- +- +- +- +10 +OC_6 +8 +6 +10 +12 +- +- +- +- +- +16 +OC_7 +18 +8 +16 +6 +- +- +- +- +- +2 +OC_8 +16 +18 +14 +10 +- +- +- +- +- +4 +OC_9 +14 +16 +4 +10 +- +- +- +- +- +2 +OC_10 +10 +2 +14 +8 +- +- +- +- +- +18 +HPD +HPD_1 +45 +72 +54 +9 +36 +63 +81 +18 +- +27 +HPD_2 +27 +81 +45 +18 +54 +63 +72 +36 +- +9 +HPD_3 +54 +81 +27 +63 +18 +45 +9 +36 +- +72 +HPD_4 +36 +18 +63 +72 +9 +81 +54 +27 +- +45 +HPD_5 +27 +63 +18 +72 +36 +9 +45 +81 +- +54 +HPD_6 +45 +36 +54 +63 +81 +9 +72 +27 +- +18 +HPD_7 +63 +45 +81 +36 +27 +72 +18 +54 +- +9 +HPD_8 +72 +9 +63 +27 +36 +18 +81 +54 +- +45 +HPD_9 +72 +63 +18 +27 +45 +9 +81 +54 +- +36 +HPD_10 +54 +81 +63 +27 +45 +72 +18 +9 +- +36 +SVIRO +SVIRO_1 +6 +12 +18 +21 +- +- +- +- +24 +15 +SVIRO_2 +9 +24 +6 +12 +- +- +- +- +15 +3 +SVIRO_3 +15 +18 +21 +3 +- +- +- +- +27 +9 +SVIRO_4 +21 +9 +12 +24 +- +- +- +- +6 +27 +SVIRO_5 +27 +21 +3 +9 +- +- +- +- +18 +24 +SVIRO_6 +3 +27 +24 +6 +- +- +- +- +21 +12 +SVIRO_7 +24 +6 +15 +18 +- +- +- +- +3 +21 +SVIRO_8 +15 +3 +27 +24 +- +- +- +- +12 +6 +SVIRO_9 +12 +15 +3 +27 +- +- +- +- +9 +18 +SVIRO_10 +18 +21 +9 +15 +- +- +- +- +21 +18 +CPD +CPD_1 +87 +74 +55 +28 +- +- +- +- +- +44 +CPD_2 +43 +56 +27 +22 +- +- +- +- +- +74 +CPD_3 +6 +32 +4 +62 +- +- +- +- +- +35 +CPD_4 +49 +22 +88 +34 +- +- +- +- +- +5 +CPD_5 +24 +69 +37 +57 +- +- +- +- +- +86 +CPD_6 +13 +69 +58 +54 +- +- +- +- +- +25 +CPD_7 +3 +32 +51 +9 +- +- +- +- +- +59 +CPD_8 +77 +62 +12 +53 +- +- +- +- +- +4 +CPD_9 +85 +27 +78 +30 +- +- +- +- +- +62 +CPD_10 +65 +46 +66 +89 +- +- +- +- +- +40 +, Vol. 1, No. 1, Article . Publication date: February 2023. + +44 +Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand +Table 13. Distribution of faults for the different failure inducing sets for each case study subject. +Case Study +Dataset +Faulty Scenarios +N +B +S +D +M +H +SG +EG +EO +NF +SAP +SAP_1 +22 +33 +54 +48 +- +- +- +- +- +72 +SAP_2 +75 +22 +48 +17 +- +- +- +- +- +57 +SAP_3 +22 +4 +57 +42 +- +- +- +- +- +81 +SAP_4 +74 +21 +40 +36 +- +- +- +- +- +42 +SAP_5 +15 +14 +86 +74 +- +- +- +- +- +51 +SAP_6 +73 +60 +2 +83 +- +- +- +- +- +72 +SAP_7 +58 +57 +47 +83 +- +- +- +- +- +43 +SAP_8 +6 +75 +26 +16 +- +- +- +- +- +70 +SAP_9 +89 +86 +66 +32 +- +- +- +- +- +68 +SAP_10 +67 +77 +14 +4 +- +- +- +- +- +55 +, Vol. 1, No. 1, Article . Publication date: February 2023. + diff --git a/WdFRT4oBgHgl3EQfMjc9/content/tmp_files/load_file.txt b/WdFRT4oBgHgl3EQfMjc9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5cebe00886fd40204aa8190d00ce977437718246 --- /dev/null +++ b/WdFRT4oBgHgl3EQfMjc9/content/tmp_files/load_file.txt @@ -0,0 +1,3850 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf,len=3849 +page_content='DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches MOHAMMED OUALID ATTAOUI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SnT Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' University of Luxembourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Luxembourg HAZEM FAHMY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SnT Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' University of Luxembourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Luxembourg FABRIZIO PASTORE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SnT Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' University of Luxembourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Luxembourg LIONEL BRIAND,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SnT Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' University of Luxembourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Luxembourg and School of EECS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' University of Ottawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Canada The adoption of deep neural networks (DNNs) in safety-critical contexts is often prevented by the lack of effective means to explain their results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' especially when they are erroneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our previous work, we proposed a white-box approach (HUDD) and a black-box approach (SAFE) to automatically characterize DNN failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' They both identify clusters of similar images from a potentially large set of images leading to DNN failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' However, the analysis pipelines for HUDD and SAFE were instantiated in specific ways according to common practices, deferring the analysis of other pipelines to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this paper, we report on an empirical evaluation of 99 different pipelines for root cause analysis of DNN failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' They combine transfer learning, autoencoders, heatmaps of neuron relevance, dimensionality reduction techniques, and different clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Our results show that the best pipeline combines transfer learning, DBSCAN, and UMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It leads to clusters almost exclusively capturing images of the same failure scenario, thus facilitating root cause analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, it generates distinct clusters for each root cause of failure, thus enabling engineers to detect all the unsafe scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Interestingly, these results hold even for failure scenarios that are only observed in a small percentage of the failing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' CCS Concepts: • Software and its engineering → Software defect analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' • Computing methodolo- gies → Machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Additional Key Words and Phrases: DNN Explanation, DNN Functional Safety Analysis, DNN Debugging, Clustering, Transfer Learning ACM Reference Format: Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, 1 (February 2023), 44 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION Deep neural networks (DNNs) have achieved extremely high predictive accuracy in various domains, such as computer vision [3, 64], autonomous driving [42, 80], and natural language processing [18, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Despite their superior performance, the lack of explainability of DNN models remains an issue Authors’ addresses: Mohammed Oualid Attaoui, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, mohammed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='attaoui@uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='lu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Hazem Fahmy, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, hazem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='fahmy@uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='lu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fabrizio Pastore, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, fabrizio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' pastore@uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='lu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Lionel Briand, SnT Centre, University of Luxembourg, JFK 29, Luxembourg, Luxembourg, School of EECS, University of Ottawa, Ottawa, Canada, lionel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='briand@uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' XXXX-XXXX/2023/2-ART $15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='nnnnnnn , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='13506v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SE] 31 Jan 2023 2 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand in many contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' While they can approximate complex and arbitrary functions, studying their structure often provides little or no insight into the underlying prediction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' There seems to be an intrinsic tension between Machine Learning (ML) performance and explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Often the highest-performing methods (for example, Deep Learning) are the least explainable, and the most explainable (for example, decision trees) are the least accurate [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For DNNs to be trustworthy, in many critical contexts where they are used, we must understand why they behave the way they do [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Explanation methods aim at making neural network decisions trustworthy [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Several explanation methods are proposed in the literature (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our work, because of our focus on safety analysis, we focus on explanation methods for root cause analysis, which concerns identifying the underlying reason of a DNN failure (root cause), which is, in our context, an incorrect DNN prediction or classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Root cause analysis techniques based on unsupervised learning have proven their effective- ness [78, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These methods group failure samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', data collected during hardware testing) without requiring diagnostic labels, such that the samples in each cluster share similar root causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Our previous work is the first application of unsupervised learning to perform root cause analysis targeting DNN failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Precisely, we proposed two DNN explanation methods: SAFE (Safety Analysis based on Feature Extraction) [4] and HUDD (Heatmap-based Unsupervised Debugging of DNNs) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' They both process a set of failure-inducing images and generate clusters of similar images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Commonalities across images in each cluster provide information about the root cause of the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For example, applying our approaches to failure-inducing images for a DNN that classifies car seat occupancy may include a cluster of images with child seats containing a bag;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' such cluster may help engineers determine that bags inside child seat are likely to be misclassified and, therefore, the training set should be improved accordingly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', more child seats with objects should be considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Both SAFE and HUDD also support the identification of additional images to be used to retrain the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD and SAFE differ with respect to the kind of data used to perform clustering and the pipeline of steps they rely on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD applies clustering based on internal DNN information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' precisely, for all failure-inducing images, it generates heatmaps capturing the relevance of DNN neurons on the DNN output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, it applies a hierarchical clustering algorithm relying on a distance metric based on the generated heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SAFE is black-box as it does not rely on internal DNN information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It generates clusters based on the visual similarity across failure inducing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To this end it relies on feature extraction based on transfer learning, dimensionality reduction, and the DBSCAN clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SAFE and HUDD rely on a pipeline that has been configured in specific ways according to best practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' However, several variants exist for each component of both approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', different transfer learning models, different clustering algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this paper, we aim to evaluate these pipeline variants for both SAFE and HUDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, we propose an empirical evaluation of 99 alternative configurations for SAFE and HUDD (pipelines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These pipelines were obtained using different combinations of feature extraction methods, clustering algorithms, dimensionality reduction techniques, in addition, we assessed the effect of fine tuning the transfer learning models used by feature extraction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For our empirical evaluation we considered six case study subjects, two of which were provided by our industry partner in the automotive domain, IEE Sensing [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Our subjects’ applications include head pose classification, eye gaze detection, drowsiness detection, steering angle prediction, unattended child detection, and car position detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We present a systematic and extensive evaluation scheme for these pipelines, which entails generating failure causes that resemble realistic scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', poor lighting conditions or camera misconfiguration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since the reason of failure in these scenarios are known a priori, such an , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 3 evaluation scheme enables us to objectively analyze and evaluate the performance and robustness of these pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Our empirical results conclude that the best pipelines support and facilitate the process of functional safety analysis such that they 1) can generate RCCs that group together a very high proportion of images capturing a same root cause (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3%, on average), 2) can capture most of the root causes of failures for all case study subjects (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7%, on average), and 3) are robust to the rarity of failure instances in a data set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', when some causes of failures affect less than 10% of the failure-inducing images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Section 2, we briefly present the main features and limitations of SAFE and HUDD, along with other feature extraction models (Autoen- coders and Backpropagation-based Heatmaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Section 3, we describe the different models and algorithms we use in our evaluated pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Section 4, we present the research questions, the experiment design and results, including a comparison between pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Section 5, we discuss and compare related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, we conclude this paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2 BACKGROUND This section provides an overview of our previous work that inspired this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We focus on clustering methods, heatmap-based DNN Explanations, the HUDD and SAFE DNN explanation methods, and Autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Clustering Clustering is a data analysis method that mines essential information from a dataset by grouping data into several groups called clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In clustering, similar data points are grouped into the same cluster, while non-similar data points are put into different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' There are two main objectives in data clustering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the first objective is to minimize the dissimilarity within the cluster, and the second objective is to maximize the inter-cluster dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD and SAFE rely on hierarchical agglomerative clustering (HAC [63]) and density-based clustering (DBSCAN [19]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In HAC, each observation starts in its own cluster and pairs of clusters are iteratively merged to minimize an objective function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', error sum of squares [85]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DBSCAN works by considering dense regions as clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' it is detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Heatmap-based DNN Explanations Approaches that aim to explain DNN results have been developed in recent years [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Most of these concern the generation of heatmaps that capture the importance of pixels in image predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' They include black-box [13, 60] and white-box approaches [51, 68, 72, 90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Black- box approaches generate heatmaps for the input layer and do not provide insights regarding internal DNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' White-box approaches rely on the backpropagation of the relevance score computed by the DNN [51, 68, 72, 90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this Section, we focus on a white-box technique called Layer-Wise Relevance Propagation (LRP) [51] because it has been integrated into HUDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' LRP was selected because it does not present the shortcomings of other heatmap generation approaches [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' LRP redistributes the relevance scores of neurons in a higher layer to those of the lower layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 1 illustrates how LRP operates on a fully connected network used to classify inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In the forward pass, the DNN receives an input and generates an output (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', classifies the gaze direction as TopLeft) while recording the activations of each neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In the backward pass, LRP generates internal heatmaps for a DNN layer 𝑘, which consists of a matrix with the relevance scores computed for all the neurons of layer 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand TopRight Input Layer Output Layer DNN with LRP Layer k DNN output: Top Right TopCenter TopLeft Layer j wj1k1 wj1k2 wj1k3 I O I Input of DNN Output of LRP O Legend: Neuron connections Image to Classify Heatmap of Input Layer Heatmaps of Internal Layers 7,5 5,3 2,3 O Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Layer-Wise Relevance Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' An example image of HPD subject (on the left) and applied LRP (on the right) showing that the mouth had a large influence on the DNN behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The heatmap in Figure 1 shows that the pupil and part of the eyelid, which are the non-white parts in the heatmap, had a significant effect on the DNN output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Furthermore, the heatmap in Figure 2 shows that the mouth and part of the nose are the input pixels that mostly impacted on the DNN output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A heatmap is a matrix with entries in R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', it is a triple (𝑁, 𝑀, 𝑓 ) where 𝑁, 𝑀 ∈ N and 𝑓 is a map [𝑁] × [𝑀] → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We use the syntax 𝐻 [𝑖, 𝑗]𝐿 𝑥 to refer to an entry in row 𝑖 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', 𝑖 < 𝑁) and column j (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', 𝑗 < 𝑀) of a heatmap 𝐻 computed on layer 𝐿 from an image 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The size of the heatmap matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the number of entries) is 𝑁 · 𝑀, with 𝑁 and 𝑀 are determined by the dimensions of the DNN layer 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For convolution layers, 𝑁 represents the number of neurons in the feature map, whereas 𝑀 represents the number of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For example, the heatmap for the eighth layer , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 5 of AlexNet has size 169 × 256 (convolution layer), while the the heatmap for the tenth layer has size 4096 × 1 (linear layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 Heatmap-based Unsupervised Debugging of DNNs (HUDD) Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Heatmap based clustering Root cause clusters C1 Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Label images Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Identify Unsafe Images Failure-inducing test set images Unsafe Set: improvement set images belonging to the root cause clusters C2 C3 Simulator execution Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Generate new images Collection of field data Improvement set: new images (unlabeled) C1 C2 C3 Labeled Unsafe Set C1 C2 C3 Step 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Retraining Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Functional Safety Analysis Training set images Balanced Labeled Unsafe Set C1 C2 C3 Improved DNN model Step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Bootstrap DNN model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Overview of HUDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Although heatmaps may provide useful information to determine the characteristics of an image that led to an erroneous result from the DNN, they are of limited applicability because, to determine the cause of all DNN errors observed in the test set, engineers may need to visually inspect all the error-inducing images, which is practically infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To overcome such limitations, we recently developed HUDD [20], a technique that facilitates the explanation and removal of the DNN errors observed in a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD generates clusters of images that lead to a DNN error because of the same root cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The root cause is determined by the engineer who visualizes a subset of the images belonging to each cluster and identifies the commonality across each image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', for a Gaze detection DNN, all the images present a closed eye).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To further support DNN debugging, HUDD automatically retrains the DNN by selecting a subset from a pool of unlabeled images that will likely lead to DNN errors because of the same root causes observed in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 3 provides an overview of HUDD, which consists of six steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Step 1, root cause clusters are identified by relying on a hierarchical clustering algorithm applied to heatmaps generated for each failure inducing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Step 2 involves a visual inspection of clustered images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this step, engineers visualize a few representative images for each RCC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the inspection enables the engineers to determine which are the commonalities across the images in each cluster and, therefore, determine the failure root cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Example root causes include the presence of an object inside a child seat (as reported in the Introduction) or a face turned left thus making an eye not visible and causing misclassification in a gaze detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD’s Step 2 supports functional safety analysis because each failure root cause represents a usage scenario in which the DNN is likely to fail, and, based on domain knowledge, engineers can determine the likelihood of each failure scenario, its safety impact, and possible countermeasures, as required by functional safety analysis standards [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For example, objects inside child seats might be very common but they lead to false alarms not hazards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' misclassified gaze may instead instead prevent the system from determining that the driver is not pay attention to the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Countermeasures include the retraining of the DNN, which is supported by HUDD’s Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Step 3, a new set of images, referred to as the improvement set, is provided by the engineers to retrain the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Step 4, HUDD automatically selects a subset of , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 6 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand images from the improvement set called the unsafe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The engineers label the images in the unsafe set in Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, in Step 6, HUDD automatically retrains the model to enhance its prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Heatmap-based Clustering in HUDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Clustering based on heatmaps is a key component of HUDD, an its functioning is useful to understand some of the pipelines considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD relies on LRP to generate an heatmap for every internal layer of the DNN, for each failure- inducing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' However, since distinct DNN layers lead to entries defined on different value ranges [52], to enable the comparison of clustering results across different layers, we generate normalized heatmaps by relying on min-max normalization [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For each DNN layer 𝐿, a distance matrix is constructed using the generated heatmaps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' it captures the distance between every pair of failure-inducing image in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The distance between a pair of images ⟨𝑎,𝑏⟩, at layer 𝐿, is computed as follows: heatmapDistance𝐿(𝑎,𝑏) = EuclideanDistance( ˜𝐻𝐿 𝑎 , ˜𝐻𝐿 𝑏 ) (1) where ˜𝐻𝐿 𝑥 is the heatmap computed for image 𝑥 at layer 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' EuclideanDistance is a function that computes the euclidean distance between two 𝑁 × 𝑀 matrices according to the formula EuclideanDistance(𝐴, 𝐵) = � � � � 𝑁 ∑︁ 𝑖=1 𝑀 ∑︁ 𝑗=1 (𝐴𝑖,𝑗 − 𝐵𝑖,𝑗)2 (2) where 𝐴𝑖,𝑗 and 𝐵𝑖,𝑗 are the values in the cell at row 𝑖 and column 𝑗 of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD applies the HAC clustering algorithm multiple times, once for every DNN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For each DNN layer, HUDD selects the optimal number of clusters using the knee-point method applied to the weighted average intra-cluster distance (WICD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' WICD is defined according to the following formula: WICD(𝐿𝑙) = �|𝐿𝑙 | 𝑗=1 � 𝐼𝐶𝐷(𝐿𝑙,𝐶𝑗) ∗ |𝐶𝑗 | |𝐶 | � |𝐿𝑙 | (3) where 𝐿𝑙 is a specific layer of the DNN, |𝐿𝑙 | is the number of clusters in the layer 𝐿𝑙, 𝐼𝐶𝐷 is the intra-cluster distance for cluster 𝐶𝑖 belonging to layer 𝐿𝑙, |𝐶𝑗 | represents the number of elements in cluster 𝐶𝑗, whereas |𝐶| represents the number of images in all the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Formula 3, ICD(𝐿𝑙,𝐶𝑗) is computed as follows: ICD(𝐿𝑙,𝐶𝑗) = �𝑁𝑗 𝑖=0 heatmapDistance𝐿𝑙 (𝑝𝑎 𝑖 , 𝑝𝑏 𝑖 ) 𝑁𝑗 (4) where 𝑝𝑖 is a unique pair of images in cluster 𝐶𝑗, and 𝑁𝑗 is the total number of pairs it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The superscripts 𝑎 and 𝑏 refer to the two images of the pair to which the distance formula is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD then select the layer 𝐿𝑚 with the minimal WICD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' By definition, the clusters generated for layer 𝐿𝑚 are the ones that maximize cohesion and we therefore anticipate that they will group together images that exhibit similar characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our study, we rely on HUDD as a feature extraction method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' precisely, we use the heatmaps generated by the layer selected by HUDD as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 Safety Analysis based on Feature Extraction (SAFE) SAFE is based on a combination of a transfer learning-based feature extraction method, a clustering algorithm, and a dimensionality reduction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The workflow of SAFE matches HUDD’s, except for Step 1 and Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In SAFE’s Step 1 RCCs are identified by relying on non-convex clustering (DBSCAN) applied to features extracted from failure-inducing images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD, instead, , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 7 applies hierarchical clustering to heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Step 4, SAFE selects the improvement step using a procedure that relies on DBSCAN’s outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The pipelines evaluated in this paper had been inspired by the pipeline implemented in SAFE’s Step 1, which consists of three stages (see Figure 4): Feature Extraction, Dimensionality Reduction, and Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this paper we investigate variants of the SAFE pipeline using different combinations of these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Additionally, we introduce a fine-tuning stage where we fine-tune the pre- trained transfer learning models to generate more domain-specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Excluding clustering, which was introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1, the components of SAFE’s pipeline are briefly described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Transfer Learning and Feature Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To maximize the accuracy of image-processing DNNs in a cost-effective way, engineers often rely on the transfer learning approach, which consists of transferring knowledge from a generic domain, usually ImageNet [73], to another specific domain, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', safety analysis, in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In other terms, we try to exploit what has been learned in one task and generalize it to another task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Researchers have demonstrated the efficiency of transfer learning from ImageNet to other domains [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Transfer learning-based Feature Extraction is an efficient method to transform unstructured data into structured raw data to be exploited by any machine learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this method, the features are extracted from images using a pre-trained CNN model [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The standard CNN architecture comprises three types of layers: convolutional layers, pooling layers, and fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The convolutional layer is considered the primary building block of a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This layer extracts relevant features from input images during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Convolutional and pooling layers are stacked to form a hierarchical feature extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The CNN model receives an input image of size (224, 224, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This image is then passed through the network’s layers to generate a vector of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The feature extraction process, for each image, generates raw data represented by a 2𝐷 matrix (denoted as 𝑋) formalized below: 𝑋 = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 𝑥11 𝑥12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 𝑥1𝑚 𝑙1 𝑥21 𝑥22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 𝑥2𝑚 𝑙2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 𝑥𝑘1 𝑥𝑘2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 𝑥𝑘𝑚 𝑙𝑘 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,𝑙𝑖 ∈ {𝐶1,𝐶2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=',𝐶𝑐} (5) where 𝐶𝑖 represent the class categories, 𝑐 is the number of categories, 𝑚 = 𝑁 × 𝑁 is the number of features, and 𝑘 is the size of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SAFE uses the VGG16 model pre-trained on the ImageNet dataset as a feature extraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Dimensionality Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Dimensionality reduction aims at approximating data in high- dimensional vector spaces [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It is important in our context since we extract a high number of Features Extraction Detection of root causes Failure inducing images Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Data Preprocessing Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Features Extraction Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Clustering Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Root cause clusters Step1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Dimensionality Reduction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Generation of root cause clusters with SAFE , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 211 C12 1m 12 21 C22 C2m li e{l, l2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='., le], i e [1, c Ck1 k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' CkmPC 1 PC 2 X PCA PC2 王 中 国 PC18 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand features from the images (512 to 2048).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In SAFE, we used the Principal Component Analysis (PCA) dimensionality reduction method to reduce the number of features from 2048 to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5 Autoencoders Autoencoders (AE) are unsupervised artificial neural networks that learn how to compress and encode the data before reconstructing it from the compressed encoded version to a representation that resembles the original input as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' AEs can extract features that can be used to improve downstream tasks, such as clustering or supervised learning, that benefit from dimension- ality reduction and higher-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In other words, AEs try to learn an approximation to the identity function and, by placing various constraints on the network’s architecture and activation functions, they extract useful representations [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 5 illustrates the neural network architecture of a simple AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It consist of four main components: Encoder: learns how to compress the input data and reduce its dimensions into an encoded representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Bottleneck: contains the encoded representation of the input data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the extracted features vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Decoder: reconstructs the input data from the encoded version (retrieved from the Bottleneck) such that it resembles the original input data as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Reconstruction Loss: the difference between the Encoder’s input and the reconstructed version (the Decoder’s output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The objective is to minimize such loss during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The objective of an AE’s training process is to minimize its reconstruction loss, measured as either the mean-squared error or the cross-entropy loss between original inputs and its constructed inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' x Encoder Decoder x’ Bottleneck Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Autoencoder Architecture 3 THE PROPOSED PIPELINES This section presents the different pipelines that can be used to implement variants of SAFE and HUDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The evaluated pipelines differ from the original SAFE and HUDD variants with respect to four components: Feature Extraction, Dimensionality Reduction, Clustering, and Fine-Tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Each pipeline is a combination of a feature extraction method (FE), a dimensionality reduction technique (DR), and a clustering algorithm (CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' When feature extraction is based on transfer learning, we distinguish between models that are fine-tuned and not fine-tuned (FT/NoFT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' feature extraction approaches not based on transfer learning cannot be fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We refer to each pipeline with the pattern FE/{FT,NoFT}/DR/CA, with each keyword being replaced with the name of the selected method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We depict in Figure 6 all the pipelines evaluated in our study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the different components are described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 9 PCA UMAP NONE 4x Transfer Learning Models LRP HUDD Fine-tuning No Fine-tuning DBSCAN HDBSCAN K-Means 99 Pipelines Feature Extraction Method Fine-tuning Dimensionality Reduction Technique Clustering Algorithm AE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines evaluated in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Feature Extraction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Feature Extraction based on Transfer Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Several DNN architectures to extract features based on transfer learning have been proposed: Inception-V3 [75], VGGNet [70], ResNet-50 [33], and Xception [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These DNNs were trained on ImageNet [14], which is a dataset with more than 14 millions annotated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The number of extracted features depends on the selected DNN architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Inception-V3, VGGNet-16, ResNet-50, and Xception generate 2048, 512, 2048, and 2048 features, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' They are described in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' VGG-16: VGG-16 is a Convolution Neural Network (CNN) architecture and the winner of the ILSVR (Imagenet) competition in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' VGG-16 focuses on convolution layers of 3 × 3 filters with a stride of 1 and always uses the same padding and maxpooling layer of 2 × 2 with a stride of 2 instead of having a large number of hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It follows this arrangement of convolution and max pool layers consistently throughout the whole architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' VGG-16 has two fully connected layers followed by a softmax layer as an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The network has an image input size of 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ResNet-50: ResNet [33] is a CNN based on residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This architecture aims to solve vanishing gradient problems in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' During the backpropagation process, the gradient diminishes dramatically in deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Small values of gradients prevent the weights from changing their values, which slows the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To solve this issue, ResNet introduces residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These building blocks present skip connections between the previous convolutional layer’s input and the current convolutional layer’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Similar to VGG-16, the network has an image input size of 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Inception-V3: Inception-V3 is a refined version of Inception [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This network proposes additional variants of Inception blocks to reduce the number of multiplications in the convolu- tion and minimize computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These variants are based on two factorizations: factorization into smaller convolutions and factorization into asymmetric convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The network has an image input size of 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Xception: Xception is a pre-trained CNN that is 71 layers deep and can classify images into 1, 000 different classes such as animals, objects, and humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This allowed the model to learn various feature representations for a wide range of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The Xception’s input size is 299 × 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fine-tuning is a typical strategy for extracting features using transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Specifically, fine-tuning relies on taking a model that has already been trained for a particular task 𝐴 as a starting point and then removing, adding, freezing, or unfreezing some layers to improve , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 10 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand the performance for a similar task 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It aims to benefit from the knowledge gained from a source task and generalize it to a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fine-tuning consists of freezing the shallow layers (close to the input), which learn more generic features (edges, shapes, and textures), and retraining the deeper layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', we let the DNN algorithm update the weights of the layers close to the output), which learn more specific features from the input data[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To fine-tune a pretrained DNN model, we follow four steps: (1) Create a new model whose layers (along with their weights) are cloned from the pre-trained model, except for the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' (2) Add a new fully connected output layer with a number of outputs equal to the number of classes in the target dataset, and initialize its weights with random values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' (3) Freeze shallow layers in the network, which are responsible for the feature extraction process (to guarantee that all the important features, previously learned by the pre-trained model, are not eliminated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' (4) Start training the new model on the target dataset, where the weights of all the non-frozen layers will keep updating using the backpropagation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For the termination criterion, we use 100 epochs or until the loss stops improving (whichever criterion is met first).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 Feature Extraction based on Autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5, an Encoder plus a Decoder make up an autoencoder (AE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The input is compressed by the Encoder, and the Decoder reconstructs the input using the Encoder’s compressed version (at the Bottleneck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since AEs extracts only the few input features necessary to aid the reconstruction of the output, the encoder might ignore other features which are not prioritized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For example in case of face images, the AE can discard the color of the skin because it is a non-prioritized feature to the AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' However, the encoder often learns useful properties of the data [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The model can then receive input data from any domain, and a fixed-length feature vector obtained at the Bottleneck can be used for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such a vector offers a compressed version of the input data representation containing sufficient information about this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 Feature Extraction based on Heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our work, we rely on heatmaps as an additional method for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since heatmaps represent the relevance of each neuron on DNN outputs, failure-inducing inputs sharing the same underlying cause should show similar heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Precisely, we rely on two different methodologies for extracting features using heatmaps, we refer to them as LRP and HUDD, according to the name of the technique driving feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' LRP and HUDD have been introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Feature extraction based on LRP, which generates heatmaps for internal layers but does not integrate a mechanism to select the most informative layer, considers the heatmap computed by the LRP technique for the input layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Feature extraction based on HUDD, instead, considers the heatmap generated for the DNN internal layer selected by HUDD as the best for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Dimensionality Reduction Several dimensionality reduction techniques exist in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this paper, we rely on two state-of-the-art techniques: Principal Component Analysis (PCA) [57] and Uniform Manifold Approximation and Projection (UMAP) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' PCA is used for its simplicity of implementation and because it doesn’t require much time and memory resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' UMAP is used for its effectiveness when applied before clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' UMAP groups data points based on relative proximity, which optimizes the clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' PCA and UMAP are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Principal Component Analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To reduce dimensionality, PCA creates a 2-dimensional matrix of variables and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Then, for this matrix, it constructs a variable space with a , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 11 dimension corresponding to the number of variables available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, it projects each data point onto the first few maximum variance directions in the variable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This procedure allows PCA to obtain a lower-dimensional data representation while maximizing data variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The first principal component can equivalently be defined as the direction that maximizes the variance of the projected data [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our context, we reduce the features for all our evaluated pipelines to 10 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We empirically obtained this number in a preliminary investigation conducted with one of our case study subjects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', HPD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Precisely, we executed a clustering algorithm (K-means) multiple times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' each execution was performed with a set of features obtained by applying PCA with a different number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We evaluated all the clustering solutions using the Silhouette Index [66] (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3) and chose the number of components yielding the highest index value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Uniform Manifold Approximation and Projection (UMAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Uniform Manifold Approximation and Projection (UMAP) is a dimension reduction technique that can be used for visualization but also for general non-linear dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' UMAP is fast, and scaling well in terms of both dataset size and dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The main limitation of UMAP is that it doesn’t preserve the density of the data, which is, instead, better preserved by PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' First, UMAP forms a weighted graph representation between each pair of data points, where the edge weights are the probability of two data points being connected to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This graph is obtained by extending a radius outward each data point such that two data points are connected if their radii overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' However, since an underestimation of such a radius can lead to the generation of small, isolated clusters, and its overestimation can lead to connecting all data points together, UMAP selects such a radius locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The radius selection is performed based on the distance from each data point to its ’𝑛 − 𝑡ℎ’ neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, UMAP decreases the likelihood of two data points getting connected past the first neighbor (as the radius grows larger), which preserves the balance between the high-dimensional and low-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Once the high-dimensional graph is constructed, UMAP optimizes the layout of a low-dimensional representation to be as similar as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The general idea is to initialize the low-dimensional data points and then move them around until they form clusters that have the same structure as the high-dimensional data, preserving the connectedness of the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' UMAP calculates Similarity Scores (distances) in the high dimensional graph to help identify the clustered points and tries to preserve that clustering in the low dimensional graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since UMAP can keep the structure of the data, even in a 2-dimensional space, we reduce the number of features to 2 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 Clustering algorithms In this study, we rely on three well-known clustering algorithms, K-means [46], DBSCAN [19], and HDBSCAN [48] described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These three clustering algorithms were chosen after preliminary experiments including also the Hierarchical Agglomerative Clustering (HAC) [53] and the Mean Shift algorithm [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' When generating clusters for one of our subjects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', HPD, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1), HAC and Mean Shift yielded much lower values of the Silhouette Index than the DBSCAN, HDB- SCAN, and K-means algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' therefore, we discarded HAC and Mean Shift from our selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since a clustering algorithm may require the manual selection of parameters’ values, such as the number of clusters (K-means) or the minimum distance between data points (DBSCAN), we rely on an internal evaluation metric (the Silhouette Index [66]) and the knee-point method [67] to automate the selection of such values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The Silhouette Index is a standard practice in cluster analysis that maximizes cohesion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', how closely related objects are in a cluster) and separation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', how well-separated a cluster is from other clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 12 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Approximating the optimal number of clusters 𝐾 using the Knee-point method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this case the optimal 𝐾 is equal to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The knee-point method automates the elbow method heuristics [79] by fitting a spline to the raw data using univariate interpolation, normalizing min/max values of the fitted data, and selecting the knee-points at which the curve most significantly deviates from the straight line segment that connects the first and last data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We rely on the knee-point method to automatically select the optimal number of clusters for the K-means algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 K-means: K-means is a well-known clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It takes a number 𝐾 as input and divides the data into 𝐾 clusters based on the distance calculated from the data points to the center of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This algorithm’s main function is to minimize the distance between the data points and their cluster center as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In the original K-means algorithm, the number of clusters (𝐾) is set manually, which can affect the quality of the clusters since we don’t have any prior knowledge of the data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', in our context, engineers cannot know in advance how many root causes of failures should be identified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To select an optimal value of 𝐾, we rely on the knee-point method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Precisely, we cluster the data with different values of 𝐾 (in our evaluation, we consider the range [5 − 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For each clustering result, we compute the within-cluster sum of squared errors (SSD), which is the sum of the distances of each point to its cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We then apply the knee-point approach to these SSDs and their respective 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 7 shows an example of 𝐾-approximation using the knee-point method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 DBSCAN:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DBSCAN (Density-Based Spatial Clustering of Applications with Noise) [19], is an algorithm that defines clusters using local density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It can be divided into four steps: (1) The 𝜖-neighborhood of a data point is determined as the set of data points that are at most 𝜖 distant from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' (2) If a data point has a number of neighbors, above a configurable threshold (called MinPts), it is then considered a core point, and a high-density area has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 14000 12000 10000 8000 SSE 6000 4000 2000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 kDNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 13 (3) Since core points can be in each other’s neighborhoods, a cluster consists of the set of core points that can be reached through their 𝜖-neighborhoods and all the data points in these 𝜖-neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' (4) Any data point that is not a core point and does not have a core point in its neighborhood is considered noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To obtain clusters using DBSCAN, we need to select two configuration parameters: (1) the distance threshold, 𝜖, to determine the 𝜖-neighborhood of each data point, and (2) the minimum number of neighbors, MinPts, needed for a data point to be considered a core point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For the identification of the values for 𝜖 and MinPts, we rely on the same strategy integrated in SAFE, described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We determine the optimal value for 𝜖 by first computing the Euclidean distance from each data point to its closest neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Then, we identify the optimal 𝜖 value as the knee-point of the curve obtained by considering those distances in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To select an optimal MinPts value, we execute DBSCAN multiple times with varying 𝑀𝑖𝑛𝑃𝑡𝑠 values and with an 𝜖 equal to the optimal value determined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We then select the clustering configuration that corresponds to the highest Silhouette Index value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 HDBSCAN:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HDBSCAN (Hierarchical Density-Based Spatial Clustering of Applications with Noise) is an extension of DBSCAN to solve its main limitation: selecting a global 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DBSCAN uses a single global 𝜖 value to determine the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' When the clusters have varying densities, using one global value can lead to a suboptimal partitioning of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Instead, HDBSCAN overcomes such a limitation by relying on different 𝜖 values for each cluster, thus finding clusters of varying densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HDBSCAN first builds a hierarchy using varying 𝜖 to figure out which clusters end up merging together and in which order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Based on the hierarchy of the clusters, HDBSCAN selects the most persisting clusters as final clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Cluster persistence represents how long a cluster stays without splitting when decreasing the value of 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' After selecting a cluster, all its descendants are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 8 shows an example of the clusters’ hierarchy found by HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The 𝑦-axis represents the values of 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Vertical bars represent clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the color and width of each vertical bar depict the size of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We can notice that certain clusters split after the value of 𝜖 is increased, while others persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HDBSCAN decides which subclusters to select based on their persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The persistence of a subcluster is captured by the length of the colored vertical bars in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HDBSCAN selects the clusters having the highest persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The unselected data points are considered noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our example, only 6 clusters are selected (circled bars);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' they are the longest vertical bars in the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 14 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Example clusters selected by HDBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4 EMPIRICAL EVALUATION In this Section, we aim to evaluate the pipelines presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A pipeline leads to the generation of clusters of images that are visually inspected by safety engineers to determine the root cause captured by each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We assume that a root cause can be described in terms of the commonalities across the images in a cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' each root cause is thus a distinct scenario in which the DNN may fail (hereafter, failure scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The pipeline that best support such process should be the one requiring minimal effort towards accurate identification of root causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, the best pipeline is the one that generates clusters having a high proportion of similar images (to facilitate the identification of the root cause, based on analyzing similarities across images in a cluster), enable the detection of all the root causes of failures, and be robust to the rarity of a particular root cause (to avoid ignoring infrequent but unsafe failure causes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Based on the above, we defined three research questions to drive our empirical evaluation: RQ1 Which pipeline generates root cause clusters with the highest purity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We define a pure cluster as one that contains only images representing the same failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such clusters are expected to be easier to interpret;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' indeed, the engineer should more easily determine the root cause of failures if all the images share the same characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, the best pipeline is the one that leads to clusters with the highest degree of purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The purity of a cluster is computed as the maximum proportion of images belonging to a same failure scenario in this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ2 Which pipelines generate root cause clusters covering more failure scenarios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This research question investigates to which extent the different pipelines miss failure failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ideally, all failure scenarios should be captured by one or more clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We say that a failure scenario is covered by a cluster if a majority of the images in the cluster belong to the scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' indeed, commonalities shared by most of the images in a cluster should be noticed by engineers during visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We aim to determine which pipeline maximizes such coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 0 2000 20 1500 40 of points 3 imber 60 80 500 100DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 15 RQ3 How is the quality of root cause clusters generated affected by infrequent failure scenarios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Some failure scenarios may be infrequent but are nevertheless important to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ideally, a pipeline should be able to produce high-quality clusters even when a small number of images belong to one or more failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In this research question, we vary the number of images belonging to failure scenariosand study how the effectiveness of pipelines (purity and coverage of the generated clusters) is affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To perform our empirical evaluation, we have implemented the transfer learning models using Tensorflow [1] and Keras [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The clustering algorithms and the dimensionality reduction methods were implemented using the Scikit-Learn library [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' All the experiments were carried out on an Intel Core i9 processor running macOS with 32GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Additionally, in our experiments, we relied on the LRP implementation provided by LRP authors [50] for well-known types of layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', MaxPooling, AvgPooling, Linear, and Convolutional layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, we extended the implementation of LRP to include DNN models implemented in PyTorch [62], Tensorflow, and Keras libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Subjects of the study To evaluate our pipelines, we consider four different DNNs that process synthetic images in the automotive domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These DNNs support gaze detection, drowsiness detection, headpose detection, and unattended child detection, which are subjects of ongoing innovation projects at IEE Sensing, our industry partner developing sensing components for automotive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Additionally, we consider two DNNs that process real-world images to support autonomous driving: steering angle prediction and car position detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The gaze detection DNN (GD) performs gaze tracking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' it can be used to determine a driver’s focus and attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It divides gaze directions into eight categories: TopLeft, TopCenter, TopRight, MiddleLeft, MiddleRight, BottomLeft, BottomCenter, and BottomRight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The drowsiness detection DNN (OC) has the same architecture as the gaze detection DNN and relies on the same dataset, except that it predicts whether the driver’s eyes are open or closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The head-pose detection DNN (HPD) is an important cue for scene interpretation and computer remote control, such as in driver assistance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It determines the pose of a driver’s head in an image based on nine categories: straight, rotated left, rotated left, rotated top left, rotated bottom right, rotated right, rotated top right, tilted, and headed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The unattended child detection DNN is trained with the Synthetic dataset for Vehicle Interior Rear seat Occupancy detection (SVIRO) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SVIRO is a dataset generated by IEE that represents scenes in the passenger compartment of ten different vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The dataset has been used by IEE to train DNNs performing rear seat occupancy detection using a camera system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We use it to train a DNN for unattended child detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We consider a seat empty when there is an object, an empty infant/child seat, or nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We consider the presence of a child/infant as a class and the presence of an adult as another class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' As a result, we have labelled the dataset with three classes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', empty seats, children/infants not accompanied by adults, and presence of an adult).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For Steering Angle Prediction (SAP), we rely on the pre-trained Autumn DNN model [61], which follows the DAVE-2 architecture [6] provided by NVIDIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It is a DNN to automate steering commands of self-driving vehicles [81];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' it predicts the angle at which the wheel should be turned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It has been trained on a dataset of road images captured by a dashboard camera mounted in the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Car Position Detection (CPD) DNNs are used by most Advanced-Driver Assistance Systems (ADAS) to predict the positions of nearby cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We rely on the CenterNet DNN [17], which is an accurate DNN used by most competition-winning approaches for object detection [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' It has been trained on images from the ApolloScape dataset [35] collected using a dashboard camera to estimate the absolute position of vehicles with respect to the ego-vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 16 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Case Study Systems DNN Data Training Test Failure # # # # # # # # # # Source Set Size Set Size (Ac- curacy) inducing images M1 N2 H3 B5 SG6 EG7 EO8 S9 D10 NF11 GD UnityEyes 61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='063 132,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='630 (96%) 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='371 80 80 80 80 80 OC UnityEyes 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='704 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='232 (88%) 506 20 20 20 20 20 HPD Blender 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='013 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='825 (44%) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='580 90 90 90 90 90 90 90 90 90 SVIRO Blender 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='489 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='427 (74%) 884 30 30 30 30 30 30 SAP Autopilot [8] 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='808 45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='406 (84%) 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='169 90 90 90 90 90 CPD Apollo [35] 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='208 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='996 (91%) 444 90 90 90 90 90 1 Mask 2 Noise 3 Hand 5 Blurriness 6 SunGlasses 7 EyeGlasses 8 Everyday Object 9 Scaling 10 Darkness 11 No Injected Fault For each subject DNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' we apply our pipelines to a set of failure-inducing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such sets consist of (1) images belonging to a provided test set and leading to a DNN failure and (2) test set images that were not leading to a DNN failure but had been modified to cause a DNN failure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the latter are images with injected root causes of failures and are described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In classifier DNNs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', OC, GD, HPD, and SVIRO) a failure occurs in the presence of an image being incorrectly classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For SAP and CPD, which are regression DNNs, we set a threshold to determine DNN failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For SAP, we observe a DNN failure when the squared error between the predicted and the true angle is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 radian (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3◦), which is deemed to be an acceptable error in related work [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For CPD, since it tackles a multi-object detection problem, we report a DNN failure when the result contains at least one false positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the distance between the predicted position of the car and the ground truth is above 10 pixels [71]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Table 1, we provide details about the case study subjects used to evaluate our pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For each subject, we report the source of the data set (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the simulator used to generate the data), the training and test set sizes, the accuracy of the DNN on the original test set, the number of failure-inducing images and the number of images for each injected root cause (they are detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We fine-tune the pipelines relying on transfer learning using the test sets of the respective case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We use the resulting fine-tuned model to extract the features from the failure-inducing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We train on the test sets because the number of images in each set is sufficient for the model to learn the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Also, we train the autoencoders on the training set, and use the test set of the respective case study to validate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The termination criterion is 50 epochs unless we reach an early stopping point (the model stops improving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' After training, we use only the encoder part to extract the features from the images in the failure inducing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Injected Failure Scenarios To assess the ability of different pipelines to generate clusters that are pure and cover all the root causes of failures, we need to know the root causes of failures in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since such root causes may vary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', lack of sufficient illumination, presence of a shadow in a specific part of the image) and it is not possible to objectively demonstrate that a failure cause has been correctly captured by a cluster (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', some readers may not agree that certain images show lack of sufficient illumination), to avoid introducing bias and subjectivity in our results, we modify a subset of the provided test set images so that they will fail because of known root causes of failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In total, we considered nine different root causes to be injected in our test set images and refer to them as injected failure scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', failure scenarios with injected root causes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We derive an image belonging to an injected failure scenario by modifying a test set image according to the specific root cause we aim to inject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' for example, by covering the mouth of a , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 17 person with a mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To ensure that a modified image leads to a DNN failure because of the injected root cause, we modify only test set images that, before modification, lead to a correct DNN output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 9 illustrates the different injected failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Below, we describe the nine root causes considered in our study: Hand: The presence of a hand blocking the full view of the driver’s face could affect the DNN result, leading it to mispredict the driver’s head direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We simulate a hand that is partially covering the face appearing in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Mask: Similar to Hand, the presence of a mask covering the nose or the mouth may affect a DNN that recognizes the driver’s head pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Using image key points, we drew the shape of a white mask to simulate a mask covering the nose and the mouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Sunglasses: As for the Mask, we use the eyes’ key points to draw sunglasses covering the driver’s eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Eyeglasses: Different from the Sunglasses, we draw glasses with the eyes being still visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Noise: A noisy image is one that contains random perturbations of colors or brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This failure scenario has been considered in related work to evaluate DNN robustness against adversarial attacks [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In real-world automotive systems, such a failure scenario resembles a defective camera or a high signal-to-noise ratio (SNR) in the communication channel between different electronic control units (ECUs), resulting in a noisy input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We use the Scikit-Image library [84] to add Gaussian Noise, a statistical noise with a probability density function equal to a normal distribution, also known as Gaussian Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Blurriness: As for Noise, this failure scenario was used to evaluate DNN robustness [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We use the Pillow library [11] to add blurriness to images using a radius of 30 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Darkness: Once again, this failure scenario was used in related work to evaluate DNN robustness [59] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In practice, poor lighting conditions could make the DNN fail because it cannot clearly recognize what is depicted in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We use the Pillow library [11] to decrease the brightness of images by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' we selected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 because it is the lowest value introducing failures in our subject results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Scaling: Such a failure scenario mimics the situation where a camera is misconfigured, leading to rescaled images being fed to the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We reduce the size of an image by a value based on the image size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', large 1200px × 1200px images are scaled by 400px, small 320px × 320px images by 70px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' and insert a black background using the Pillow library [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Everyday Object: For the SVIRO dataset, we introduce, in the car’s rear seat, an object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', a washing machine or a handbag) never observed in the training set, thus simulating the effect of an incomplete training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 18 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Mask Noise Blurriness Darkness Hand Sunglasses Eyeglasses Scaling No injected fault Everyday object Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Injected failure scenarios in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For regression DNNs (SAP and CPD), we randomly selected 90 images to be copied and modified for each failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For classifier DNNs, for each failure scenario, we randomly selected 10 images for each class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Please note that in addition to the injected failure scenarios explained above, our DNNs are affected by other natural failure causes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', borderline images that are misclassified because they are very similar to the ones belonging to another class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such cases are observed with any machine learning model since it is usually not possible to achieve perfect accuracy through training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We refer to these images as belonging to natural failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our analysis, we include a number of images belonging to natural failure scenarios equal to those with injected failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is because natural failure scenarios are usually observed with any DNN and, therefore, should be considered when generating RCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 RQ1: Which pipeline generates root cause clusters with the highest purity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Design and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A pure cluster includes only images presenting the same root cause (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', common cause leading to a DNN failure);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' for example, a hand covering a person’s mouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pure clusters simplify root cause analysis because they should make it easier for an engineer to determine the commonality across images and therefore the cause of failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since the likely root cause of the failure in our injected failure scenarios is known, we focus on these scenarios to respond to RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For each RCC, we compute the proportion of images belonging to each injected failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, we measure the purity 𝑃 of a cluster 𝐶 (hereafter, 𝑃𝐶) as the highest proportion of images belonging to one injected failure scenario 𝑓 ∈ 𝐹 assigned to cluster 𝐶, where 𝐹 is the set of all failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 𝑃𝐶 is computed as follows: P𝐶 = max 𝑓 ∈𝐹 �𝐶𝑓 |𝐶| � (6) The proportion of a failure scenario 𝑓 in a cluster 𝐶 is computed as the number of images belonging to 𝑓 assigned to cluster 𝐶 (𝐶𝑓 ), divided by the size of cluster 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 19 Clusters that do not include any image belonging to an injected failure scenario are assumed to capture root causes due to natural failure scenarios and, consequently, are excluded from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We study the purity distribution across RCCs generated for the different case study subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since, ideally, we would like to obtain pure clusters, the best pipeline is the one that maximizes the average purity across the generated RCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 10 depicts a regression tree illustrating how the different components of a pipeline (feature extraction methods, fine-tuning, dimensionality reduction techniques and clustering algorithms) determine the purity of the clusters generated by a pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We use the Conditional Inference Tree (CTree) algorithm [34] to generate this decision tree with a maximum depth set to 4 (we have four components in a pipeline) and a minimum split set to 10 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the weight of a node to be considered for splitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The dataset used to build the tree consists of the components of each pipeline as attributes, and the purity of the generated clusters as the predicted outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The dataset size is equal to 99, the number of pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Each node of the tree represents a feature of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Leaves (terminal nodes) depict box plots representing distributions of the average purity across RCCs generated by the pipelines belonging to each leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Each point in the box plot is the average purity of one pipeline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the average of the purity of all the RCCs generated across all case study subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To split a node, the CTree algorithm first identifies the feature with the highest association (covariance) with the response variable (purity, in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Precisely, it relies on a permutation test of independence (null hypothesis) between any of the features and the response [74];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the feature with the lowest significant p-value is then selected (alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='05, in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Once a feature has been selected, a binary split is then performed by identifying the value that maximizes the test statistics across the two subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since we are in the presence of multiple hypothesis (assume 𝑚, for each node), to prevent a Type I error, for each feature 𝑗, CTree computes its Bonferroni-adjusted [89] 𝑝-value𝑗 as adjusted 𝑝-value𝑗 = 1 − (1 − 𝑝-value𝑗)𝑚 In Figure 10, we notice that the pipelines with fine-tuned models (Node 3 and 4) generate lower- purity clusters than those without any fine-tuning (Node 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is because these models were fine-tuned on a test set that did not include any injected root cause (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', only natural failure scenarios);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' recall that fine tuning is performed with labeled images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', training set) and since our injected root causes capture scenarios not foreseen at training time, it would be unrealistic to consider such scenarios for fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fine-tuning a model on a set of images means that it will learn all the features of those images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, clustering based on fine-tuned models will generate clusters based on the features observed during training, excluding injected features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e, the injected root causes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' As a result, in our experiments, images are clustered based only on their natural fault (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', borderline class) instead of the injected faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The pipelines using non-fine-tuned transfer learning models as a feature extraction method (Node 7) generate purer clusters (min = 50%, median = 80%, max = 96%) than the pipelines using an autoencoder model, HUDD, or LRP (Node 6) (min = 50%, median = 65%, max = 70%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The purpose of the Autoencoder model is to provide a condensed representation of the image to be used for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is done by ignoring the features that the model considers insignificant and only keeping the features that help the encoder reconstruct the image accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, since the autoencoder is trained on the training set, the injected faults are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Given that clustering is based on the condensed representation, the generated clusters are less pure than the ones generated by the pipelines with transfer learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 20 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand As for HUDD and LRP, it seems that their main limitation is that heatmaps cannot capture the presence of root causes affecting all the pixels in an image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the result of noise, blurriness, darkness, scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Heatmaps mainly capture which pixels of the image drive the DNN output, thus leading clustering to group images where the same pixels affected the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For instance, the DNN’s response to a blurred image with a shadow on the mouth could be different from that of another blurred image with a shadow on the eyes, thus leading to different clusters for these images although they represent the same injected failure scenario (blurriness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, we notice that the pipelines using HDBSCAN and DBSCAN (Node 3) as a clustering algorithm yield purer clusters (min = 25%, median = 40%, max = 80%) than those using K-means (Node 4, min = 22%, median = 27%, max = 29%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is because K-means faces difficulty dealing with non-convex clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A cluster is convex if, for every pair of points belonging to it, it also includes every point on the straight line segment between them [41], which gives the cluster a spherical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Nevertheless, in many practical cases, the data leads to clusters with arbitrary, non-convex shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such clusters, however, cannot be appropriately detected by a centroid-based algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', K-means), as they are not designed for arbitrary-shaped clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DBSCAN and HDBSCAN are density-based clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' They consider high-density regions as clusters (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The root cause clusters generated by DBSCAN and HDBSCAN are arbitrary-shaped and more homogeneous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', clusters with higher within-cluster similarity) with very similar images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In contrast, a convex cluster generated by K-means tends to be less dense and can group rather dissimilar images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' As a result, a convex cluster is less pure than a non-convex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Decision Tree illustrating how the different features of a pipeline determine the average purity of root-cause clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We report the significance of these results in Table 2, including the values of the Vargha and Delaney’s ˆ𝐴12 effect size and the 𝑝-values resulting from performing a Mann-Whitney U-test to compare the average purity of the pipelines using transfer learning models (Node 7 in the decision tree) and the pipelines represented by the other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Typically, an ˆ𝐴12 effect size above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='56 is considered practically significant with higher thresholds for medium (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='64) and large (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='71) effects [39], thus suggesting the effect sizes between the pipelines using transfer learning models and other pipelines are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, 𝑝-values suggest these differences are statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' finetuning p< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='001 NoFT 5 clusteringalgs transfermodel p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='002 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='023 Dbscan,HDBSCAN K-means AE, HUDD, LRP InceptionV3, ResNet50, VGG16, Xception Node 3 (n = 24) Node 4 (n = 12) Node 6 (n = 27) Node 7 (n = 36) 100 - 100 - 100 - 100 - 0 oo 80 80 80 - 80 60 - 60 - 09 09 40 - 40 - 40 - 40 -DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 21 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ1: Pipelines with a purity greater than 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' purity across RCCs # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 19 VGG-16 NO None K-Means 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25 VGG-16 NO UMAP K-Means 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 26 VGG-16 NO UMAP DBSCAN 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 39 ResNet-50 NO None HDBSCAN 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 43 ResNet-50 NO UMAP K-Means 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 44 ResNet-50 NO UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 62 Inception-V3 NO UMAP DBSCAN 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 𝐹𝐸 Feature Extraction 𝐹𝑇 Fine-tuning 𝐷𝑅 Dimensionality Reduction 𝐶𝐴 Clustering Algorithm Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ1: p-values and and effect size values when comparing the results of the pipelines with the best purity of clusters (according to the decision tree) to the other pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Node 3 Node 4 Node 6 p-value 7e-11 2e-7 5e-6 ˆ𝐴12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80 Finally, in Table 3, we report the pipelines that generated clusters with an average purity above 90% across all case study subjects, along with the purity obtained for each subject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the complete results obtained for all pipelines appear in Appendix A, Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' An average purity of 100% means that all the clusters generated by the pipeline are pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Interestingly, all the pipelines in Table 3 belong to Node 7 in Figure 10, thus confirming our main finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Five of these seven best pipelines, rely on UMAP, without fine-tuning but with a transfer learning model, which is therefore our suggestion to perform root cause analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The best result is obtained with ResNet-50 combined with UMAP and DBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 RQ2: Which pipelines generate root cause clusters covering more failure scenarios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Design and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This research question investigates the extent to which our pipelines identify all failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We compare pipelines in terms of the percentage of injected failure scenarios being covered by at least one RCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A failure scenario is covered by an RCC if it enables the engineer to determine the root cause of the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Precisely, when images belonging to a failure scenario 𝑓 represent a sufficiently large share of images in a cluster 𝐶, it is easier for an engineer to notice that 𝑓 is a likely root cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, we assume that an injected failure scenario 𝑓 is covered by a cluster 𝐶 if it contains at least 90% of images with 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since this threshold is relatively high, our results can be considered conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Given that our injected failure scenarios are represented by the same number of images in the failure-inducing test set, every failure scenario has the same likelihood of being observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, we expect to obtain RCCs corresponding to each failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 11 shows a decision tree illustrating how the different components of a pipeline determine the coverage of failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' As for RQ1, we used the Conditional Inference Tree CTree algorithm to generate this decision tree (with the same parameter settings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Each leaf node depicts a box plot with the distribution of the percentages of failure scenarios covered by the set of pipelines that include the components listed in the decision nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For instance, Node 9 provides the distribution of the percentage of failure scenarios covered by the , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 22 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand RCCs generated by pipelines using UMAP as a dimensionality reduction technique and non-fine- tuned transfer learning models as feature extraction methods (12 pipelines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ideally, the root-cause clusters generated by a pipeline should cover 100% of the failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The decision tree in Figure 11 confirms RQ1 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The pipelines without fine-tuning (Nodes 6, 8 and 9) outperform the pipelines with fine-tuning (Nodes 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The pipelines with transfer learning models (Nodes 8 and 9) generate clusters that cover more failure scenarios than those generated by the pipelines using HUDD, LRP, and AE (Node 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Also, the pipelines using the DBSCAN and HDBSCAN clustering algorithms (Node 3) yield better results than the ones using K-means (Node 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, the decision tree in Figure 11 gives us more insights into which dimensionality reduction method is more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We notice that the root-cause clusters generated by the pipelines using UMAP (Node 9) lead to a better distribution (min = 45%, median = 85%, max = 100%) than the pipelines using PCA or not using any dimensionality reduction (Node 8, min = 25%, median = 55%, max = 90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is because UMAP yields a better separation of the clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', less clusters overlap) compared to PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' When using UMAP, all the data points converge towards their closest neighbor (the most similar data point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, neighboring data points in higher dimensions end up in the same neighborhood in lower dimensions, resulting in a compact and well-separated clusters where it is easier for the clustering algorithms to distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Decision Tree illustrating how the different features of a pipeline determine the coverage of the failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=') We report the significance of these results in Table 4, including the values of the Vargha and Delaney’s ˆ𝐴12 effect size and the 𝑝-values resulting from performing a Mann-Whitney U-test to compare the percentages of covered failure scenarios resulting from the pipelines using UMAP (Node 9 in the decision tree in Figure 11), and the other pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Table 4 shows that the 𝑝-values, when comparing the pipelines using UMAP to the other pipelines, are always below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This implies that in all the cases, differences are statistically significant with large effect sizes (above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' finetuning p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='001 NoFT 2 5 clusteringalgs transfermodel p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='009 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='013 AE, HUDD, LRP InceptionV3, ResNet50, VGG16, Xception 7 dimreduction Dbscan, HDBSCAN K-means p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='031 None, PCA UMAP Node 3 (n = 24) Node 4 (n = 12) Node 6 (n = 27) Node 8 (n = 24) Node 9 (n = 12) 100 100 100 100 100 80 - 80 80 - 80 - 80 - 60 60 60 60 60 40 - 40 40 - 40 - 40 - 20 20 20 20 20 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 23 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ2: Pipelines with a coverage greater than 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Percentage of covered faulty scenarios # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 26 VGG-16 None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 44 ResNet-50 None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 62 Inception-V3 None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80 Xception None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 𝐹𝐸 Feature Extraction 𝐹𝑇 Fine-tuning 𝐷𝑅 Dimensionality Reduction 𝐶𝐴 Clustering Algorithm Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ2: p-values and and effect size values when comparing the results of the pipelines with the best coverage of the faulty scenarios (according to the decision tree) to the other pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Node 3 Node 4 Node 6 Node 8 p-value 1e-5 1e-5 4e-5 8e-3 ˆ𝐴12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='77 In Table 5, we report the pipelines that generated clusters covering at least 90% of the failure scenarios across all case study subjects, along with the coverage obtained for each case study subject (complete results for all the pipelines are reported in Appendix B, Table 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' If the coverage is equal to 100%, all the failure scenarios are covered by the RCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Unsurprisingly, the pipelines in Table 5 belong to Node 7 in Figure 11: they rely on a non-fine-tuned transfer learning model for feature extraction, and UMAP for dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, they all use DBSCAN for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These pipelines consistently yielded the best results for all individual case studies (confirming the results obtained in RQ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such findings are further supported by the results in Table 11 and Table 12, where we notice that the combination of UMAP with DBSCAN always achieves higher purity and coverage (in bold) than its alternatives, regardless of the used feature extraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5 RQ3: How is the quality of root cause clusters generated affected by infrequent failure scenarios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Design and measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We study the effect of infrequent failure scenarios on the quality of the RCCs generated by the pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We consider a failure scenario infrequent when it is observed in a low proportion of the images in the failure-inducing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To be practically useful, a good pipeline should be able to generate root-cause clusters even for infrequent failure scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' indeed, in safety-critical contexts, infrequent failure scenarios may lead to hazards and thus should be detected when testing the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For instance, if only five out of hundred failure-inducing images belong to a failure scenario and we have three failure scenarios in total, a robust pipeline should still generate an RCC containing only the images of the infrequent failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We generate 10 different failure-inducing sets for each case study subject (a total of 60 failure- inducing sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To construct a failure-inducing set, for each root cause that might affect the case study (see Table 1, Page 16), we generate a number 𝑛 of images affected by the injected root cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We randomly select a number 𝑛 that is lower than the number of images selected for the same root cause in RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, for classifier DNNs, we select a value higher than the number of classes of the corresponding case study (we enforce one root cause of failures for one image per class, at least);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' for regression DNNs, we select a value above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since 𝑛 is randomly selected (uniform distribution), we obtained failure-inducing sets containing failure scenarios whose number vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In addition, we also include a randomly selected number of images belonging to natural failure , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 24 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand scenarios, to mimic what happens in practice (see RQ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The number of images belonging to natural failure scenarios varies between two and the total number of injected failure scenario images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In total, we generated 60 failure-inducing sets (10 × 6 subject DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For each failure-inducing set, we randomly selected the number of images representing a failure scenario (injected or natural scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such number should be higher than the number of classes (to ensure that there is at least one scenario for each class) and lower than the total number of images representing a failure scenario generated in RQ1 (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In the case of regression DNNs, the minimum number of images representing a failure scenario is set to 2 since a cluster is formed by grouping at least two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For instance, the number of images representing a failure scenario for each failure- inducing set of the HPD case study (9 classes) is randomly selected between 9 and 90 (see Table 13, Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since we aim to study the effect of infrequent failure scenarios on the quality of the generated RCCs, we categorize our 290 failure scenarios into infrequent and frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Infrequent failure scenarios are the ones that include a proportion of injected images that is lower than the median proportion in all the generated failure-inducing sets (equals to 18% in our study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For example, noise is frequent in the dataset GD_1 (64 > 18) but infrequent in the dataset OC_2 (4 < 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We consider only the best pipelines resulting from the experiments in RQ1 and RQ2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', having purity or coverage above 90% as shown in Tables 3 and 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' they are pipeline 26 (VGG16/DB- SCAN/UMAP/NoFT), 44 (ResNet50/DBSCAN/UMAP/NoFT), 62 (InceptionV3/DBSCAN/UMAP/NoFT), 19 (VGG16/K-means/None/NoFT), 25 (VGG16/K-means/UMAP/NoFT), 39 (ResNet50/HDBSCAN/None/NoFT), 43 (ResNet50/K-means/UMAP/NoFT), and 80 (Xception/DBSCAN/UMAP/NoFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The first three pipelines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', 26, 44, 62) were the best for both RQ1 and RQ2, the next four (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', 19, 25, 39, 43) were selected based on RQ1 results while the latter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', 80) based those of RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We compute the purity and coverage of the RCCs generated by each of these pipelines, following the same procedures adopted for RQ1 and RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We then compare the distribution of purity and coverage for infrequent and frequent failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The most robust pipelines are the ones being affected the least, in terms of purity and coverage, by infrequent failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Figure 12, for each selected pipeline, we report the average purity across all the RCCs1 with the injected failure scenarios having a certain frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The 𝑥-axis reports the proportion of images for failure scenarios whereas the 𝑦-axis reports the average purity of the RCCs associated to each failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 12 shows that when the frequency of the failure scenarios is below the median (infrequent scenario), the cluster purity obtained by pipelines tends to significantly lower and decrease rapidly as the frequency decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is expected because when a failure scenario is infrequent, the clustering algorithm tends to either cluster its images as noise or distribute them over the other clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For density-based clustering algorithms, images belonging to infrequent scenarios may not become core points when the identification of a core point requires more data points in their neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In such case, images belonging to infrequent scenarios will be either labeled as noise points or border points (belonging to other clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The same is true for K-means, where these points are usually spread across other clusters because they cannot form a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To strengthen our findings, in Table 6, we report the results when comparing the purity of the selected pipelines for frequent and infrequent failure scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' further, we report the Vargha and Delaney’s ˆ𝐴12 effect size and the 𝑝-values resulting from performing a Mann-Whitney U-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We notice that for all pipelines, the difference between frequent and infrequent scenarios are significant 1As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The red vertical line represents the median frequency of failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We say that an RCC is associated with (or captures) an injected failure scenario 𝑓 when the majority of the images in the cluster belong to scenario 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 25 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ3: p-values and effect size values when comparing the purity of the best pipelines with the frequent and infrequent failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines 26 44 62 39 80 19 25 43 Average Purity for infrequent failure scenarios 94% 87% 91% 79% 87% 76% 70% 65% Average Purity for frequent fail- ure scenarios 100% 100% 100% 92% 99% 96% 96% 93% p-value 4e-6 2e-10 1e-6 2e-9 8e-9 8e-5 2e-10 3e-14 ˆ𝐴12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='75 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ3: p-values and effect size values when comparing the best pipeline in Table 6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', Pipeline 26, VGG16/Dbscan/UMAP/NoFT) to the other pipelines based on the average purity of the clusters associated to infrequent failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines 44 62 39 80 19 25 43 p-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='51 4e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='006 3e-12 2e-14 4e-21 ˆ𝐴12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='77 (p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' However, the effect sizes for Pipelines 26, 62, 45, and 80 are small, while they are medium for Pipelines 19 and 44, which indicates that pipelines including DBSCAN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', Pipelines 26, 62, 45, and 80) are much more robust to infrequent scenarios than others (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the difference between frequent and infrequent scenarios is less pronounced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Actually, the pipelines using DBSCAN fare better than the rest also in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Indeed, almost all the injected failure scenarios with frequency above 18% have 100% purity (see Figure 12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' further for infrequent failure scenarios they include less data points below 100% than the other pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is because DBSCAN tends to find clusters with different sizes if these clusters are dense enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' K-means, instead, derives clusters that are of similar size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, we notice that the purity of the clusters generated by Pipeline 26 (VGG16/Dbscan/UM- AP/NoFT), for infrequent failure scenarios, is higher (average is 94%) than the purity of the clusters generated by the other pipelines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' differences are significant (see Table 7), thus suggesting Pipeline 26 might be the best choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Concerning coverage, Figure 13 shows, for each pipeline, histograms with the average coverage obtained for failure scenarios having proportions of failure inducing images within specific ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In general, we observe that coverage is higher for frequent scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is due to the correlation between pure clusters and coverage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the less pure the generated clusters, the fewer failure scenarios they cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' When the failure scenarios are infrequent, their images are distributed over the other clusters, reducing their purity and, thus, reducing the probability of these scenarios being covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To demonstrate the significance of the difference between coverage results obtained with frequent and infrequent scenarios, we apply the Fisher’s Exact test2 to compare the coverage of frequent and infrequent scenarios for the clusters generated by the selected pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We report the 𝑝- values resulting from the Fisher’s Exact test in Table 8 and observe that differences are statistically significant thus indicating that pipelines perform better with frequent failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, Figure 13 shows that pipeline 62 (InceptionV3/DBSCAN/UMAP/NoFT) is the one perform- ing best with the least frequent scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', range 0-5%) but no pipeline fares well in that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2The Fisher’s Exact test [83] is a statistical test used to determine if there is a non-random association between two categorical variables [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 26 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ3: Fisher exact test values when comparing the coverage of the lowly represented and highly represented faulty scenarios by the clusters generated by the best pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines 26 44 62 39 80 19 25 43 Average Coverage for infre- quent failure scenarios 85% 71% 82% 66% 73% 51% 46% 34% Average Coverage for frequent failure scenarios 100% 98% 99% 86% 98% 86% 87% 77% Fisher’s Exact test 1e-5 1e-5 1e-5 1e-5 1e-5 2e-4 1e-5 1e-5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Purity of the clusters associated with frequent and infrequent failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The x-axis captures the frequency of a failure scenario (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', proportion of failure-inducing images for a failure scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Each data point is the average of all the RCCs associated to one distinct failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The red vertical line represents the median frequency of failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipeline 26 (VGG16/DBSCAN/UMAP/NoFT) is the one performing best with infrequent scenarios in the range 5% to 20%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' indeed, it is the only pipeline providing an average coverage above 90% for that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To further demonstrate the significance of the difference in performance between Pipeline 26 and the other pipelines, we apply Fisher’s exact test to the coverage obtained for infrequent scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We report the 𝑝-values resulting from this test in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We notice that all the 𝑝-values are below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='05 except when Pipeline 26 is compared to Pipeline 62;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' indeed, the results of these two pipelines are similar as visible in Figure 13), even though Pipeline 26 performs slightly better on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In conclusion, infrequent failure scenarios affect both purity and coverage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' however, some pipelines fare better than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Our results suggest that the pipeline (26) relying on a non-fine- tuned VGG16 model, with UMAP and DBSCAN (Pipeline 26) is the best choice because it yields , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='VGG16/Dbscan/UMAP/NoFT(26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='ResNet50/Dbscan/UMAP/NoFT (44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='InceptionV3/Dbscan/UMAP/NoFT(62) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='ResNet50/HDBSCAN/None/NoFT (39) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Xception/Dbscan/UMAP/NoFT (80) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='VGG16/K-means/None/NoFT(19) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='VGG16/K-means/UMAP/NoFT (25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='ResNet50/K-means/UMAP/NoFT(43) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%08 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%08 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='50% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='%09DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Comparing the percentage of coverage across different ranges of proportions of failure scenarios in each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RQ3: Fisher exact test values when comparing the best pipeline "VGG16/Dbscan/UMAP/NoFT" to the other pipelines based on the coverage of the infrequent failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines 44 62 39 80 19 25 43 Fisher’s Exact test 4e-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='55 1e-5 2e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='018 1e-5 1e-5 significantly higher purity and coverage than the other pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipeline 26 is also less negatively affected by infrequent failure scenarios since coverage is above 90% when the frequency is above 5%, which is not the case for all the other pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 Discussion The results of RQ1 and RQ2 show that there is a family of pipelines leading to higher purity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', they simplify the identification of root causes) and coverage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', they enable the identification of all root causes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such pipelines rely on transfer learning, UMAP for dimensionality reduction, DBSCAN for clustering, and are not using fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Among such pipelines, considering that it is reasonable to expect unsafe scenarios to be infrequent, based on the results of RQ3, we suggest to use the pipeline relying on VGG16 (Pipeline 26) as transfer learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In our study, we focused on effectiveness, not cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' indeed, our main purpose is to identify the pipeline that generates clusters that do not confuse the end-user (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', they are pure) and is likely to help identify all the root causes of failures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', they have high coverage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In contrast, cost is related to the number of clusters being inspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' However, our root cause analysis toolset [21] includes the generation of animated gifs, one for each cluster, thus enabling the quick visualization of all the images in a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' With such toolset we conjecture that the number of clusters’s images does not strongly impact cost as all the images are, anyway, quickly visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' What is important, instead, is the purity of clusters as with low purity the end-user will not find it easy to determine commonalities among images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Nevertheless, to further discuss cost, we measure the number of clusters to be inspected for each pipeline considering the dataset used for RQ1 and RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We count only clusters capturing the injected failure scenarios since for the others we cannot precisely determine what is the expected number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A lower number of clusters should indicate lower cost and, since a number , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='VGG16/Dbscan/UMAP/NoFT(26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Inception_V3/Dbscan/UMAP/NoFT (62) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='ResNet50/Dbscan/UMAP/NoFT (44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='ResNet50/HDBSCAN/None/NoFT (39) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='erage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion of faulty scenarios in a set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion of faulty scenarios in a set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion offaulty scenarios in a set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion of faulty scenarios in a set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Xception/Dbscan/UMAP/NoFT (80) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='VGG16/K-means/None/NoFT(19) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='VGG16/K-means/UMAP/NoFT(25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='ResNet50/K-means/UMAP/NoFT(43) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Cover ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion of faulty scenarios in a set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion of faulty scenarios in a set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion of faulty scenarios in a set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Ranges of the proportion of faulty scenarios in a set28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Mohammed Oualid Attaoui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Hazem Fahmy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fabrizio Pastore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' and Lionel Briand Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Examples of clusters generated by pipeline 26 for the HPD case study subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' of clusters higher than the number of failure scenarios to be discovered implies the presence of redundant clusters, we compute the degree of redundancy as: redundancy ratio = number of clusters covered failure scenarios Finally, to discuss how well each pipeline improves current practice in industry, we estimate the degree of savings with respect to the such practice, which entails the visual inspection of all images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To do so, we assume that inspecting a single cluster using animated gifs is as inexpensive as visualizing one single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Indeed, though clusters involve several images, through animation, they actually make it easier to quickly identify commonalities rather than guessing root causes from a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Figure 14 shows four example clusters where all the images present a commonality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', the root cause of the DNN failure) that is easy to determine when visualizing all the images in a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, we estimate savings as: savings = 1 − number of clusters number of images Table 10 shows our results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' it reports the number of RCCs generated for each case study DNN and across all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Further, it reports the percentage and number of failure scenarios covered by each pipeline (used to compute redundancy and providing information about the effectiveness of a pipeline), along with redundancy ratio and savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We report only the results for the best pipelines , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Cluster 1 (images with glasses) Cluster 2 (images with noise) Cluster 3 (images with mask) Cluster 4 (images with scaling)DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 29 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The number of redundant clusters generated by the best pipelines for each case study subject and across all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last columns represent the number and the percentage of failure scenarios covered by the pipelines, the redundancy ratio, and the savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' pipelines Number of generated clusters Covered failure scenarios (percentage %) Redundancy ratio Savings GD HPD OC SVIRO CPD SAP TOTAL VGG16/K-means/None/NoFT 3 5 3 3 3 2 19 17 (59%) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='99 VGG16/K-means/UMAP/NoFT 4 8 2 4 3 3 24 20 (69%) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='99 ResNet50/K-means/UMAP/NoFT 3 4 3 2 2 4 18 15 (52%) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='20 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='99 VGG16/Dbscan/UMAP/NoFT 26 77 13 8 13 37 174 28 (97%) 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='21 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='91 ResNet50/Dbscan/UMAP/NoFT 42 51 5 10 27 44 179 27 (93%) 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='91 Inception_V3/Dbscan/UMAP/NoFT 28 60 7 9 15 42 161 29 (100%) 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='55 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='92 Xception/Dbscan/UMAP/NoFT 33 30 9 2 9 14 97 25 (86%) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='88 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='95 ResNet50/HDBSCAN/None/NoFT 14 171 15 7 74 3 284 24 (83%) 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='83 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='86 identified when addressing RQ1 to RQ2 because there is no reason to select pipelines that do not achieve high purity and coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The number of clusters generated by the selected pipelines ranges between 18 and 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The pipelines leading to the lowest number of clusters are the ones including K-means: ResNet50/K- means/UMAP/NoFT (18), VGG16/K-means/None/NoFT (19), and VGG16/K-means/UMAP/NoFT (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines with DBSCAN and HDBSCAN lead to a much higher number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To discuss the practical impact of such a high number of clusters, we focus on the redundancy ratio, which ranges between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' the redundancy ratio indicates that the pipeline with the highest number of clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', ResNet50/HDBSCAN/None/NoFT), on average, presents 11 redundant clusters for each identified failure scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Given that, in the presence of pure clusters, understanding the scenario captured by one pipeline is quick with animated gifs, we consider that inspecting 11 redundant clusters per fault has a limited impact on cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, if we focus on savings, we can observe that respect to current practice, all the pipelines except (ResNet50/HDBSCAN/Only/NoFT) lead to savings above 90%, thus showing that their adoption is highly beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Although the pipelines including K-means lead to the lowest cost, their coverage is particularly low for infrequent scenarios (see Table 8, with coverage below 35% for the range [0-5], and below 60% for the range [5-10]), which is bound to be a common situation in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since pipelines leading to a small number of clusters can be highly ineffective in realistic safety-critical contexts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', when some failure causes are infrequent), assuming that redundant clusters are easy to manage, we conclude that the best choice are the pipelines that maximize purity and coverage, as discussed above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', Pipeline 26, VGG16/DBSCAN/UMAP/NoFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A possible tradeoff is Pipeline 80 (Xception/DBSCAN/UMAP/NoFT), which is among the best performing for RQ3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', coverage above 40% for the range [0-5], and above 70% for the range [5-10]) and leads to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 redundant clusters only, on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7 Threats to validity We discuss internal, conclusion, construct, and external validity according to conventional prac- tices [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1 Internal validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Since 72 of our 99 pipelines use a Transfer Learning pre-trained model to extract the features, a possible internal threat is that this model can negatively affect our results if inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Indeed, clustering relies on the similarity computed on the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To mitigate this threat, we visually inspected the clusters to check their consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Having consistent clusters means the features extracted by the models contain enough information to cluster the images based on their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Another potential threat might be that the dataset (with the injected faults) was created with the proposed approach in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Therefore, there might be a risk of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To mitigate this risk, all the , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 30 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand methods used in our pipelines (feature extraction methods, clustering algorithms, dimensionality reduction techniques) are independent of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These methods do not require any a priori knowledge on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We also publish our data to further mitigate this risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' All the experiments can be reproduced with any injected faulty scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 External validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To alleviate the threats related to the choice of the case study DNNs, we use six well-studied datasets with diverse complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Four out of six subject DNNs implement tasks motivated by IEE business needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' These DNNs address problems that are quite common in the automotive industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The other two DNNs are also related to the automotive industry and were used in many Kaggle challenges [56, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Although our pipelines were only tested on case study DNNs related to the automotive industry, we believe they will perform well with other data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This is because the models used for the feature extraction were pre-trained on ImageNet, which means that the model can capture features related to 1, 000 classes, including humans, animals, and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' As for AE, it can learn the aspects of any data set during training and provide high-quality clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, for HUDD and LRP, the extraction of heatmap-based features is performed on well-known layer types that are part of any DNN model, regardless of the task at hand (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', they can be extended to DNNs that were not studied in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 Construct validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The construct considered in our work is effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We measure the effectiveness through complementary indicators as follows: For RQ1, we evaluate the effectiveness of our pipelines by computing the purity of the generated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The purity of a cluster is measured as the maximum proportion of images representing one faulty scenario in this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For RQ2, we evaluate the effectiveness of our pipelines based on the coverage of the injected faulty scenarios by the root cause clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A faulty scenario is covered by a cluster if at least 90% of the images in this cluster represent such faulty scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finally, for RQ3, we consider both the purity and the coverage to measure the robustness of the top-performing pipelines to rare faulty scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 Conclusion validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Conclusion validity addresses threats that impact the ability to conclude appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To mitigate such threats and to avoid violating parametric assumptions in our statistical analysis, we rely on a non-parametric test and effect size measure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', Mann Whitney U-test and the Vargha and Delaney’s ˆ𝐴12 statistics, respectively) to assess the statistical significance of differences in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Additionally, we applied the Fisher’s exact test when comparing coverage results related to different distributions of faulty scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', RQ3), which is commonly used in similar contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' All results were reported based on both purity and coverage parameters, and six datasets were analyzed during our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8 Data Availability All our implementations, the failure-inducing sets, the generated root-cause clusters and the data generated to address our research questions are available online [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 5 RELATED WORK Our paper is related to the literature on DNN debugging and applications of transfer learning to perform root cause analysis [55, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Heatmap-based approaches [13, 51, 60, 68, 72, 90, 91] explain the DNN’s prediction of an image by highlighting which region of that image influenced the most the DNN output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' For example, Grad-CAM generates a heatmap from the gradient flowing into the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The heatmap is then , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 31 superposed on the original image to highlight the regions of the image that activated the DNN and influenced the decision [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The main limitation of these approaches is that they require the inspection of all the heatmaps generated for the images processed by the DNN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', error- inducing images) and, different from our pipelines, do not provide engineers with guidance for their inspection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', one cluster for each failure root cause).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SHAP (SHapley Additive exPlanation) [45] generates explanations by calculating the contribution of each feature to predictions, thus explaining what features are the most important for each prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In the case of an image CNN, SHAP considers a group of pixels as a feature and calculates their contribution to the decision made by the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Like heatmap-based approaches, SHAP does not provide guidance for the investigation of multiple failure-inducing images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DeepJanus [65] helps identify misbehaviors in a Deep Learning system by finding a set of pairs of inputs that are similar to each other and that trigger different behaviors of the Deep Learning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This set of pairs represents the border between the input regions where the DNN behaves as expected or fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Different from our work, DeepJanus characterizes the behaviour of a DNN that can be tested with a simulator but cannot provide explanations for failures observed with real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Some DNN testing approaches explain the input regions where DNN errors are observed by relying on decision trees constructed using the simulator parameters used to generate test input images [2, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Although decision trees are an effective mean to provide explanations for failures detected during simulator-based testing, they cannot be applied to provide explanations for failures observed with real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To overcome such a limitation, we have recently developed SEDE [22], a technique that applies HUDD to failure-inducing real-world images to generate root cause clusters and then relies on evolutionary algorithms to drive the generation, for each RCC, of similar images using simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The simulator parameter values used to generate such images are then fed into PART [24], a tree-based rule learning algorithm to characterize each RCCs in terms of simulator parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', it generates expressions that constrain simulator parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The work in this paper is complementary to SEDE since the latter can be applied to clusters generated with the best pipeline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', Pipeline 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [55] combine Transfer Learning with clustering to find root causes of hardware failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The proposed method uses different clustering algorithms (K-means [47], decision tree clustering [44], hierarchical clustering [40]) on hardware test data to cluster failures likely due to the same causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Different from their work, we aim to explain failures in DNNs that process images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', our feature space is much larger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ter Burg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [78] explain DNNs based on a transfer learning model that has been fine-tuned to detect geometric shapes connecting face landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Such shapes are treated as features and the contribution of each feature is computed by relying on SHAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The output should help end-users determine what influenced the DNN output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Unfortunately, similar to heatmap-based approaches, this approach does not support the explanation of multiple failures but require engineers to process them one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' To conclude, our previous works (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', HUDD [20] and SAFE [4]) have been the first to apply clustering algorithms to white-box and black-box feature extraction approaches to explain failure causes in DNN-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' This study is the first to systematically assess and compare the performance of alternative white-box and black-box feature extraction approaches, dimensionality reduction techniques, and clustering algorithms using a wide variety of practical, realistic failure scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we presented an large-scale empirical evaluation of 99 different pipelines for root cause analysis of DNN failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Our pipelines receive as input a set of images leading to DNN , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 32 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand failures and generate as output cluster of images sharing similar characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' As demonstrated by our previous work, by visualizing the images in each cluster, an engineer can notice commonalities across the images in each cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' such commonalities represent the root causes of failures, help characterize failure scenarios and, thus, support engineers in improving the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', by selecting additional similar images to retrain the DNN or by introducing countermeasures in the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We considered 99 pipelines resulting from the combination of five methods for feature extraction, two techniques for dimensionality reduction and three clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Our methods for feature extraction include white-box (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', heatmap generation techniques) and black-box approaches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', fine-tuned and non-finetuned transfer learning models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Additionally, we rely on PCA and UMAP for dimensionality reduction and K-means, DBSCAN, and HDBSCAN for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' We evaluated our pipelines in terms of clusters’ purity and coverage of failures based on failure scenarios widely varying in terms of frequency, thus analyzing the impact of rare scenarios on our best pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Based on six case study subjects in the automotive domain, our results show that the best results are obtained with pipelines relying on VGG-16 as transfer learning model, not using fine tuning, leveraging UMAP as a dimensionality reduction technique, and using DBSCAN as clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' When the failure scenarios are equally distributed, the best pipeline achieved a purity of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', almost all the images in RCCs present the same failure scenario) and a coverage of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The same pipeline also performs well with rare failure scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' indeed, when images belonging to failure scenarios represent between 5 and 10% of the total number of images, it still can cover 90% of the failure scenarios with a cluster purity above 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ACKNOWLEDGMENTS This project has received funding from IEE Luxembourg, Luxembourg’s National Research Fund (FNR) under grant BRIDGES2020/IS/14711346/FUNTASY, and NSERC of Canada under the Dis- covery and CRC programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Authors would like to thank Thomas Stifter from IEE for his valuable support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The experiments presented in this paper were carried out using the HPC facilities of the University of Luxembourg (see http://hpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='lu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' REFERENCES [1] Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Corrado,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Andy Davis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Jeffrey Dean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Matthieu Devin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Sanjay Ghemawat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ian Goodfellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Andrew Harp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Geoffrey Irving,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Michael Isard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Yangqing Jia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Rafal Jozefowicz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Lukasz Kaiser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Manjunath Kudlur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Josh Levenberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Dandelion Mané,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Rajat Monga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Sherry Moore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Derek Murray,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Chris Olah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Mike Schuster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Jonathon Shlens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Benoit Steiner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ilya Sutskever,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Kunal Talwar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Paul Tucker,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Vincent Vanhoucke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Vijay Vasudevan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fernanda Viégas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Oriol Vinyals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pete Warden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Martin Wattenberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Martin Wicke,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Yuan Yu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' and Xiaoqiang Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/ Software available from tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [2] Raja Ben Abdessalem, Shiva Nejati, Lionel C Briand, and Thomas Stifter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Testing vision-based control systems using learnable evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2018 IEEE/ACM 40th International Conference on Software Engineering (ICSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE, 1016–1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [3] Mohammed Oualid Attaoui, Nassima Dif, Hanene Azzag, and Mustapha Lebbah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Regions of interest selection in histopathological images using subspace and multi-objective stream clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The Visual Computer (2022), 1–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [4] Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Black-Box Safety Analysis and Retraining of DNNs Based on Feature Extraction and Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' (jul 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1145/3550271 [5] Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Anal- ysis: an Empirical Evaluation of Clustering-based Approaches - Replicability package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='com/projects/ DNN_Explanation_for_Safety_Analysis_an_Empirical_Evaluation_of_Clustering-based_Approaches/157973 [6] Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' End to end learning for self-driving cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' arXiv preprint arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='07316 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 33 [7] Diogo V Carvalho, Eduardo M Pereira, and Jaime S Cardoso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Machine learning interpretability: A survey on methods and metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Electronics 8, 8 (2019), 832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [8] Sully Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Autopilot-Tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='com/SullyChen/Autopilot-TensorFlow [9] François Chollet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Xception: Deep learning with depthwise separable convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1251–1258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [10] Francois Chollet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='com/fchollet/keras [11] Alex Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pillow (PIL Fork) Documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://buildmedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/media/pdf/pillow/latest/ pillow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='pdf [12] Steve Dias Da Cruz, Oliver Wasenmüller, Hans-Peter Beise, Thomas Stifter, and Didier Stricker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SVIRO: Synthetic Vehicle Interior Rear Seat Occupancy Dataset and Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2020 IEEE Winter Conference on Applications of Computer Vision (WACV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE, 962–971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1109/WACV45572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9093315 [13] Piotr Dabkowski and Yarin Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Real Time Image Saliency for Black Box Classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the 31st International Conference on Neural Information Processing Systems (NIPS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Curran Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', Red Hook, NY, USA, 6970–6979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [14] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Imagenet: A large-scale hierarchical image database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2009 IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ieee, 248–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [15] Nassima Dif, Mohammed Oualid Attaoui, Zakaria Elberrichi, Mustapha Lebbah, and Hanene Azzag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Transfer learning from synthetic labels for histopathological images classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Applied Intelligence (2021), 1–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [16] Raghav Bali Dipanjan Sarkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Transfer Learning in Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Manning Publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 69–76 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://livebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='com/book/transfer-learning-in-action/chapter-1/v-1/69 (visited 2022-02-19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [17] Kaiwen Duan, Song Bai, Lingxi Xie, Honggang Qi, Qingming Huang, and Qi Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' CenterNet: Keypoint Triplets for Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2019 IEEE/CVF International Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE Computer Society, 6568–6577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [18] Ekin EKİNCİ, Hidayet TAKCI, and Sultan ALAGÖZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Poet Classification Using ANN and DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Electronic Letters on Science and Engineering 18, 1 (2022), 10–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [19] Martin Ester, Hans-Peter Kriegel, Jörg Sander, Xiaowei Xu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A density-based algorithm for discovering clusters in large spatial databases with noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='. In Kdd, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 226–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [20] Hazem Fahmy, Fabrizio Pastore, Mojtaba Bagherzadeh, and Lionel Briand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Supporting Deep Neural Network Safety Analysis and Retraining Through Heatmap-Based Unsupervised Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE Transactions on Reliability (2021), 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1109/TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3074750 [21] Hazem Fahmy, Fabrizio Pastore, and Lionel Briand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' HUDD: A tool to debug DNNs for safety analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2022 IEEE/ACM 44th International Conference on Software Engineering: Companion Proceedings (ICSE-Companion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 100–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1145/3510454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3516858 [22] Hazem Fahmy, Fabrizio Pastore, Lionel Briand, and Thomas Stifter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Simulator-Based Explanation and Debugging of Hazard-Triggering Events in DNN-Based Safety-Critical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Methodol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' (oct 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1145/3569935 [23] Florent Forest, Mustapha Lebbah, Hanene Azzag, and Jérôme Lacaille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Deep embedded self-organizing maps for joint representation learning and topology-preserving clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Neural Computing and Applications 33, 24 (2021), 17439–17469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [24] Eibe Frank and Ian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Witten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Generating Accurate Rule Sets Without Global Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Fifteenth International Conference on Machine Learning, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Shavlik (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Morgan Kaufmann, 144–151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [25] Keinosuke Fukunaga and Larry Hostetler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The estimation of the gradient of a density function, with applications in pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE Transactions on information theory 21, 1 (1975), 32–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [26] Rafael Garcia, Alexandru C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Telea, Bruno Castro da Silva, Jim Torresen, and Joao Luiz Dihl Comba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A task-and- technique centered survey on visual analytics for deep learning model engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Computers and Graphics 77 (2018), 30 – 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='cag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='018 [27] Leilani H Gilpin, David Bau, Ben Z Yuan, Ayesha Bajwa, Michael Specter, and Lalana Kagal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Explaining explanations: An overview of interpretability of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2018 IEEE 5th International Conference on data science and advanced analytics (DSAA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE, 80–89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [28] Ian Goodfellow, Yoshua Bengio, and Aaron Courville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='deeplearningbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [29] Alexander N Gorban and Andrei Y Zinovyev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Principal graphs and manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Handbook of research on machine learning applications and trends: algorithms, methods, and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IGI Global, 28–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [30] David Gunning and David Aha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DARPA’s explainable artificial intelligence (XAI) program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' AI magazine 40, 2 (2019), 44–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [31] Jiawei Han, Micheline Kamber, and Jian Pei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Data Mining: Concepts and Techniques (3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Morgan Kaufmann Publishers Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=', San Francisco, CA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 34 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand [32] Fitash Ul Haq, Donghwan Shin, Lionel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Briand, Thomas Stifter, and Jun Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Automatic Test Suite Generation for Key-Points Detection DNNs Using Many-Objective Search (Experience Paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the 30th ACM SIGSOFT International Symposium on Software Testing and Analysis (ISSTA 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 91–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1145/3460319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3464802 [33] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [34] Torsten Hothorn, Kurt Hornik, and Achim Zeileis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ctree: Conditional inference trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The comprehensive R archive network 8 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [35] Xinyu Huang, Peng Wang, Xinjing Cheng, Dingfu Zhou, Qichuan Geng, and Ruigang Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The ApolloScape Open Dataset for Autonomous Driving and Its Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence 42, 10 (oct 2020), 2702–2719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1109/tpami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2926463 [36] IEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEE Sensing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='iee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [37] International Organization for Standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ISO, ISO-24765-2017, Systems and software engineering - Vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [38] International Organization for Standardization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ISO, ISO26262-1:2018, Road vehicles: Functional safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [39] Barbara Kitchenham, Lech Madeyski, David Budgen, Jacky Keung, Pearl Brereton, Stuart Charters, Shirley Gibbs, and Amnart Pohthong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Robust Statistical Methods for Empirical Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Empirical Software Engineering 22, 2 (2017), 579–630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1007/s10664-016-9437-5 [40] Hans-Friedrich Köhn and Lawrence J Hubert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Hierarchical cluster analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Wiley StatsRef: Statistics Reference Online (2014), 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [41] Hans-Peter Kriegel, Peer Kröger, Jörg Sander, and Arthur Zimek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Density-based clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 1, 3 (2011), 231–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [42] Zhong Li, Minxue Pan, Tian Zhang, and Xuandong Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Testing DNN-based Autonomous Driving Systems under Critical Environmental Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' PMLR, 6471–6482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [43] Bin Liang, Hongcheng Li, Miaoqiang Su, Xirong Li, Wenchang Shi, and Xiaofeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Detecting Adversarial Image Examples in Deep Neural Networks with Adaptive Noise Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE Transactions on Dependable and Secure Computing 18, 1 (jan 2021), 72–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1109/tdsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2874243 [44] Bing Liu, Yiyuan Xia, and Philip S Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Clustering through decision tree construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the ninth international conference on Information and knowledge management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 20–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [45] Scott M Lundberg and Su-In Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A unified approach to interpreting model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Advances in neural information processing systems 30 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [46] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' MacQueen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Some methods for classification and analysis of multivariate observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Proceedings of BSMSP (1967), 281–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [47] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' MacQueen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Some Methods for Classification and Analysis of Multivariate Observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability - Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Le Cam and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Neyman (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' University of California Press, Berkeley, CA, USA, 281–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [48] Leland McInnes, John Healy, and Steve Astels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' hdbscan: Hierarchical density based clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Journal of Open Source Software 2, 11 (2017), 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [49] Leland McInnes, John Healy, Nathaniel Saul, and Lukas Grossberger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' UMAP: Uniform Manifold Approximation and Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The Journal of Open Source Software 3, 29 (2018), 861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [50] Gregoire Montavon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' WIFS 2017 Tutorial on Methods for Understanding DNNs and their Predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' http: //heatmapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/wifs2017/ [51] Grégoire Montavon, Alexander Binder, Sebastian Lapuschkin, Wojciech Samek, and Klaus Robert Müller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Layer- Wise Relevance Propagation: An Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Springer International Publishing, Cham, 193–209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1007/ 978-3-030-28954-6_10 [52] Gregoire Montavon, Sebastian Lapuschkin, Alexander Binder, Wojciech Samek, and Klaus-Robert Muller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ex- plaining nonlinear classification decisions with deep Taylor decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pattern Recognition 65 (2017), 211 – 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='patcog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='008 [53] Fionn Murtagh and Pierre Legendre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ward’s Hierarchical Agglomerative Clustering Method: Which Algorithms Implement Ward’s Criterion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Journal of Classification 31, 3 (01 Oct 2014), 274–295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1007/s00357- 014-9161-z [54] A Kousar Nikhath, MD Abdul Rab, N Venkata Bharadwaja, L Goutham Reddy, K Saicharan, C Venkat Manas Reddy, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' An Intelligent College Enquiry Bot using NLP and Deep Learning based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2022 International Conference for Advancement in Technology (ICONAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [55] Renjian Pan, Xin Li, and Krishnendu Chakrabarty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Unsupervised Root-Cause Analysis with Transfer Learning for Integrated Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2021 IEEE 39th VLSI Test Symposium (VTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [56] ADVAY PATIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Car Object Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='com/code/advaypatil/car-object-detection , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 35 [57] Karl Pearson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' LIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' On lines and planes of closest fit to systems of points in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The London, Edinburgh, and Dublin philosophical magazine and journal of science 2, 11 (1901), 559–572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [58] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pedregosa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Varoquaux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Gramfort, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Michel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Thirion, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Grisel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Blondel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Prettenhofer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Weiss, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Dubourg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Vanderplas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Passos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Cournapeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Brucher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Perrot, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Duchesnay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Scikit-learn: Machine Learning in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Journal of Machine Learning Research 12 (2011), 2825–2830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [59] Kexin Pei, Yinzhi Cao, Junfeng Yang, and Suman Jana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Deepxplore: Automated whitebox testing of deep learning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In proceedings of the 26th Symposium on Operating Systems Principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [60] Vitali Petsiuk, Abir Das, and Kate Saenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' RISE: Randomized Input Sampling for Explanation of Black-box Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the British Machine Vision Conference (BMVC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [61] Autumn pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Udacity Self-Driving Car Challenge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='com/udacity/self-driving- car/tree/master/steering-models/community-models/autumn [62] PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' PyTorch DNN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org [63] Sizhe Rao, Minghui Wang, Cuixia Tian, Xiangqiao Ao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A Hierarchical Tree-Based Syslog Clustering Scheme for Network Diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2021 17th International Conference on Network and Service Management (CNSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE, 28–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [64] Satti RG Reddy, GP Varma, and Rajya Lakshmi Davuluri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Deep Neural Network (DNN) Mechanism for Identifica- tion of Diseased and Healthy Plant Leaf Images Using Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Annals of Data Science (2022), 1–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [65] Vincenzo Riccio and Paolo Tonella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Model-based exploration of the frontier of behaviours for deep learning system testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 876–888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [66] Peter J Rousseeuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Silhouettes: a graphical aid to the interpretation and validation of cluster analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Journal of computational and applied mathematics 20 (1987), 53–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [67] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Satopaa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Albrecht, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Irwin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Raghavan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Finding a "Kneedle" in a Haystack: Detecting Knee Points in System Behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2011 31st International Conference on Distributed Computing Systems Workshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 166–171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [68] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Selvaraju, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Cogswell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Das, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Vedantam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Parikh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Grad-CAM: Visual Explanations from Deep Networks via Gradient-Based Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2017 IEEE International Conference on Computer Vision (ICCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 618–626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1109/ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='74 [69] Jonathon Shlens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' A tutorial on principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' arXiv preprint arXiv:1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1100 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [70] Karen Simonyan and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Very deep convolutional networks for large-scale image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/abs/1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1556 [71] Xibin Song, Peng Wang, Dingfu Zhou, Rui Zhu, Chenye Guan, Yuchao Dai, Hao Su, Hongdong Li, and Ruigang Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Apollocar3d: A large 3d car instance understanding benchmark for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 5452–5462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [72] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Springenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Dosovitskiy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Brox, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Riedmiller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Striving for Simplicity: The All Convolutional Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In ICLR (workshop track).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [73] Stanford Vision Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ImageNet, image database organized according to the WordNet hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' image-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Accessed: 2021-12-07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [74] Helmut Strasser and Christian Weber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' On the Asymptotic Theory of Permutation Statistics (january 1999 ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' WorkingPaper 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [75] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Going deeper with convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [76] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Rethinking the inception architecture for computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2818–2826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [77] Muhammed Talo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Automated classification of histopathology images using transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Artificial Intelligence in Medicine 101 (2019), 101743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [78] Kaya ter Burg and Heysem Kaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Comparing Approaches for Explaining DNN-Based Facial Expression Classifi- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Algorithms 15, 10 (2022), 367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [79] Robert L Thorndike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Who belongs in the family?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Psychometrika 18, 4 (1953), 267–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1007/ BF02289263 [80] Yuchi Tian, Kexin Pei, Suman Jana, and Baishakhi Ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DeepTest: Automated Testing of Deep-Neural-Network- Driven Autonomous Cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Proceedings of the 40th International Conference on Software Engineering (ICSE ’18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 303–314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1145/3180155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3180220 [81] Udacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Udacity Self-driving Car Challenge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Accessed: 2022-08-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [82] Peking University/Baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Peking University/Baidu - Autonomous Driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Accessed: 2022-08-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 36 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand [83] Graham JG Upton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fisher’s exact test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series A (Statistics in Society) 155, 3 (1992), 395–402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [84] Stéfan van der Walt, Johannes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Schönberger, Juan Nunez-Iglesias, François Boulogne, Joshua D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Warner, Neil Yager, Emmanuelle Gouillart, and Tony Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' scikit-image: image processing in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' PeerJ 2 (jun 2014), e453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7717/peerj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='453 [85] Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ward, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Hierarchical grouping to optimize an objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' American Statistical Association Journal 58 (1963), 236–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [86] Eric W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Weisstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Fisher’s Exact Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://mathworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='com/FishersExactTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Accessed: 2022-11-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [87] Katharina Weitz, Teena Hassan, Ute Schmid, and Jens Garbas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Towards explaining deep learning networks to distinguish facial expressions of pain and emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Forum Bildverarbeitung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 197–208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [88] Claes Wohlin, Per Runeson, Martin Höst, Magnus C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Ohlsson, Björn Regnell, and Anders Wesslén.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Experimentation in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 9783642290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1–236 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1007/978-3-642-29044-2 [89] S Paul Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Adjusted p-values for simultaneous inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Biometrics (1992), 1005–1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [90] Matthew D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Zeiler and Rob Fergus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Visualizing and Understanding Convolutional Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In Computer Vision – ECCV 2014, David Fleet, Tomas Pajdla, Bernt Schiele, and Tinne Tuytelaars (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Springer International Publishing, Cham, 818–833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' [91] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Khosla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Lapedriza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Oliva, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Learning Deep Features for Discriminative Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' IEEE, 2921–2929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1109/CVPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='319 , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 37 A ADDITIONAL MATERIAL FOR RQ1 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Comparing the clusters generated by the different pipelines based on the average of the purity across root cause clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Case Study Subjects # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1 HUDD None None K-Means 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 2 HUDD None None DBSCAN 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 3 HUDD None None HDBScan 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 4 HUDD None PCA K-Means 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 5 HUDD None PCA DBSCAN 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 6 HUDD None PCA HDBScan 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 7 HUDD None UMAP K-Means 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 8 HUDD None UMAP DBSCAN 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 9 HUDD None UMAP HDBScan 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 10 LRP None None K-Means 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 11 LRP None None DBSCAN 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 12 LRP None None HDBScan 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 13 LRP None PCA K-Means 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 14 LRP None PCA DBSCAN 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 15 LRP None PCA HDBScan 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 16 LRP None UMAP K-Means 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 17 LRP None UMAP DBSCAN 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 18 LRP None UMAP HDBScan 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 19 VGG-16 None None K-Means 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 20 VGG-16 None None DBSCAN 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 21 VGG-16 None None HDBSCAN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 22 VGG-16 None PCA K-Means 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 23 VGG-16 None PCA DBSCAN 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 24 VGG-16 None PCA HDBSCAN 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 25 VGG-16 None UMAP K-Means 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 26 VGG-16 None UMAP DBSCAN 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 27 VGG-16 None UMAP HDBSCAN 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 28 VGG-16 FT None K-Means 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 29 VGG-16 FT None DBSCAN 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 30 VGG-16 FT None HDBSCAN 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 31 VGG-16 FT PCA K-Means 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 32 VGG-16 FT PCA DBSCAN 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 33 VGG-16 FT PCA HDBSCAN 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 34 VGG-16 FT UMAP K-Means 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 35 VGG-16 FT UMAP DBSCAN 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 36 VGG-16 FT UMAP HDBSCAN 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 37 ResNet-50 None None K-Means 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 38 ResNet-50 None None DBSCAN 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 39 ResNet-50 None None HDBSCAN 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 40 ResNet-50 None PCA K-Means 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 41 ResNet-50 None PCA DBSCAN 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 42 ResNet-50 None PCA HDBSCAN 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 43 ResNet-50 None UMAP K-Means 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 44 ResNet-50 None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 45 ResNet-50 None UMAP HDBSCAN 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 46 ResNet-50 FT None K-Means 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 47 ResNet-50 FT None DBSCAN 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 38 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Comparing the clusters generated by the different pipelines based on the average of the purity across root cause clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Case Study Subjects # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 48 ResNet-50 FT None HDBSCAN 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 49 ResNet-50 FT PCA K-Means 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 50 ResNet-50 FT PCA DBSCAN 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 51 ResNet-50 FT PCA HDBSCAN 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 52 ResNet-50 FT UMAP K-Means 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 53 ResNet-50 FT UMAP DBSCAN 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 54 ResNet-50 FT UMAP HDBSCAN 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 55 Inception-V3 None None K-Means 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 56 Inception-V3 None None DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 57 Inception-V3 None None HDBSCAN 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 58 Inception-V3 None PCA K-Means 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 59 Inception-V3 None PCA DBSCAN 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 60 Inception-V3 None PCA HDBSCAN 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 61 Inception-V3 None UMAP K-Means 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 62 Inception-V3 None UMAP DBSCAN 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 63 Inception-V3 None UMAP HDBSCAN 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 64 Inception-V3 FT None K-Means 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 65 Inception-V3 FT None DBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 66 Inception-V3 FT None HDBSCAN 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 67 Inception-V3 FT PCA K-Means 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 68 Inception-V3 FT PCA DBSCAN 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 69 Inception-V3 FT PCA HDBSCAN 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 70 Inception-V3 FT UMAP K-Means 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 71 Inception-V3 FT UMAP DBSCAN 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 72 Inception-V3 FT UMAP HDBSCAN 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 73 Xception None None K-Means 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 74 Xception None None DBSCAN 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 75 Xception None None HDBSCAN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 76 Xception None PCA K-Means 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 77 Xception None PCA DBSCAN 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 78 Xception None PCA HDBSCAN 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 79 Xception None UMAP K-Means 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 80 Xception None UMAP DBSCAN 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 81 Xception None UMAP HDBSCAN 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 82 Xception FT None K-Means 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 83 Xception FT None DBSCAN 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 84 Xception FT None HDBSCAN 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 85 Xception FT PCA K-Means 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 86 Xception FT PCA DBSCAN 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 87 Xception FT PCA HDBSCAN 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 88 Xception FT UMAP K-Means 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 89 Xception FT UMAP DBSCAN 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 90 Xception FT UMAP HDBSCAN 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 91 AE None None K-Means 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 92 AE None None DBSCAN 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 93 AE None None HDBSCAN 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 94 AE None PCA K-Means 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 95 AE None PCA DBSCAN 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 96 AE None PCA HDBSCAN 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 39 Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Comparing the clusters generated by the different pipelines based on the average of the purity across root cause clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Case Study Subjects # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 97 AE None UMAP K-Means 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 98 AE None UMAP DBSCAN 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 99 AE None UMAP HDBSCAN 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 40 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand B ADDITIONAL MATERIAL FOR RQ2 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Percentage of faulty scenarios covered by the root cause clusters generated for each pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Case Study Subjects # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1 HUDD None None K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 2 HUDD None None DBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 3 HUDD None None HDBScan 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 4 HUDD None PCA K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 5 HUDD None PCA DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 6 HUDD None PCA HDBScan 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 7 HUDD None UMAP K-Means 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 8 HUDD None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 9 HUDD None UMAP HDBScan 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 10 LRP None None K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 11 LRP None None DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 12 LRP None None HDBScan 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 13 LRP None PCA K-Means 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 14 LRP None PCA DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 15 LRP None PCA HDBScan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 16 LRP None UMAP K-Means 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 17 LRP None UMAP DBSCAN 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 18 LRP None UMAP HDBScan 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 19 VGG-16 None None K-Means 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 20 VGG-16 None None DBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 21 VGG-16 None None HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 22 VGG-16 None PCA K-Means 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 23 VGG-16 None PCA DBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 24 VGG-16 None PCA HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 25 VGG-16 None UMAP K-Means 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 26 VGG-16 None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 27 VGG-16 None UMAP HDBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 28 VGG-16 FT None K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 29 VGG-16 FT None DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 30 VGG-16 FT None HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 31 VGG-16 FT PCA K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 32 VGG-16 FT PCA DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 33 VGG-16 FT PCA HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 34 VGG-16 FT UMAP K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 35 VGG-16 FT UMAP DBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 36 VGG-16 FT UMAP HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37 ResNet-50 None None K-Means 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 38 ResNet-50 None None DBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 39 ResNet-50 None None HDBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 40 ResNet-50 None PCA K-Means 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 41 ResNet-50 None PCA DBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 42 ResNet-50 None PCA HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 43 ResNet-50 None UMAP K-Means 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 44 ResNet-50 None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 45 ResNet-50 None UMAP HDBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 46 ResNet-50 FT None K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 47 ResNet-50 FT None DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 41 Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Percentage of faulty scenarios covered by the root cause clusters generated for each pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Case Study Subjects # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 48 ResNet-50 FT None HDBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 49 ResNet-50 FT PCA K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 50 ResNet-50 FT PCA DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 51 ResNet-50 FT PCA HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 52 ResNet-50 FT UMAP K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 53 ResNet-50 FT UMAP DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 54 ResNet-50 FT UMAP HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 55 Inception-V3 None None K-Means 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 56 Inception-V3 None None DBSCAN 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 57 Inception-V3 None None HDBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 58 Inception-V3 None PCA K-Means 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 59 Inception-V3 None PCA DBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 60 Inception-V3 None PCA HDBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 61 Inception-V3 None UMAP K-Means 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 62 Inception-V3 None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 63 Inception-V3 None UMAP HDBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 64 Inception-V3 FT None K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 65 Inception-V3 FT None DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 66 Inception-V3 FT None HDBSCAN 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 67 Inception-V3 FT PCA K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 68 Inception-V3 FT PCA DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 69 Inception-V3 FT PCA HDBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 70 Inception-V3 FT UMAP K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 71 Inception-V3 FT UMAP DBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 72 Inception-V3 FT UMAP HDBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 73 Xception None None K-Means 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 74 Xception None None DBSCAN 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 75 Xception None None HDBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 76 Xception None PCA K-Means 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4% 77 Xception None PCA DBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 78 Xception None PCA HDBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 79 Xception None UMAP K-Means 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='1% 80 Xception None UMAP DBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 81 Xception None UMAP HDBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 82 Xception FT None K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 83 Xception FT None DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 84 Xception FT None HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% 85 Xception FT PCA K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 86 Xception FT PCA DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 87 Xception FT PCA HDBSCAN 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8% 88 Xception FT UMAP K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 89 Xception FT UMAP DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3% 90 Xception FT UMAP HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 91 AE None None K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 92 AE None None DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 93 AE None None HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 94 AE None PCA K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9% 95 AE None PCA DBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2% 96 AE None PCA HDBSCAN 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='7% , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 42 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Percentage of faulty scenarios covered by the root cause clusters generated for each pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' The last column represents the average of averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Pipelines Case Study Subjects # FE FT DR CA GD OC HPD SVIRO SAP CPD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 97 AE None UMAP K-Means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 98 AE None UMAP DBSCAN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6% 99 AE None UMAP HDBSCAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='0% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='5% , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' DNN Explanation for Safety Analysis: an Empirical Evaluation of Clustering-based Approaches 43 C ADDITIONAL MATERIAL FOR RQ3 Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Distribution of faults for the different failure inducing sets for each case study subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Case Study ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='Faulty Scenarios ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='EG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='EO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='NF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='48 40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='56 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8 64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='GD_10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='OC_10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='8 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='HPD_10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='21 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='SVIRO_10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='15 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='87 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='28 44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='22 74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='62 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='88 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='34 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='69 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='57 86 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='69 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='54 25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='9 59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='53 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='85 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='78 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='30 62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='CPD_10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content='89 40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 44 Mohammed Oualid Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand Table 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Distribution of faults for the different failure inducing sets for each case study subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Case Study Dataset Faulty Scenarios N B S D M H SG EG EO NF SAP SAP_1 22 33 54 48 72 SAP_2 75 22 48 17 57 SAP_3 22 4 57 42 81 SAP_4 74 21 40 36 42 SAP_5 15 14 86 74 51 SAP_6 73 60 2 83 72 SAP_7 58 57 47 83 43 SAP_8 6 75 26 16 70 SAP_9 89 86 66 32 68 SAP_10 67 77 14 4 55 , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} +page_content=' Publication date: February 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFRT4oBgHgl3EQfMjc9/content/2301.13506v1.pdf'} diff --git a/XtE1T4oBgHgl3EQfvwXS/content/tmp_files/2301.03404v1.pdf.txt b/XtE1T4oBgHgl3EQfvwXS/content/tmp_files/2301.03404v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7db67705de2bde6d99174aa8b35b7faaa54c345c --- /dev/null +++ b/XtE1T4oBgHgl3EQfvwXS/content/tmp_files/2301.03404v1.pdf.txt @@ -0,0 +1,526 @@ +CSRCZ: A Dataset About Corporate Social Responsibility in Czech +Republic +Xhesilda Vogli +Department of Management +Faculty of Economics and Management +Czech University of Life Sciences +vogli@pef.czu.cz +Erion Çano +Digital Philology +Data Mining and Machine Learning +University of Vienna, Austria +erion.cano@univie.ac.at +Abstract +As stakeholders’ pressure on corporates for +disclosing their corporate social responsibility +operations grows, it is crucial to understand +how efficient corporate disclosure systems are +in bridging the gap between corporate social +responsibility reports and their actual practice. +Meanwhile, research on corporate social re- +sponsibility is still not aligned with the recent +data-driven strategies, and little public data are +available. This paper aims to describe CSRCZ, +a newly created dataset based on disclosure re- +ports from the websites of 1 000 companies +that operate in Czech Republic. +Each com- +pany was analyzed based on three main param- +eters: company size, company industry, and +company initiatives. We describe the content +of the dataset as well as its potential use for +future research. We believe that CSRCZ has +implications for further research, since it is the +first publicly available dataset of its kind. +1 +Introduction +Corporate Social Responsibility (CSR) has evolved +from a “why” in the early 1950s (Carroll and +Brown, 2018) to a “must” in recent years. Gen- +erally, CSR is considered a self-regulating business +model which helps companies to contribute to so- +cietal goals and be socially accountable to them- +selves and the public. It is highly influenced by the +legal context (LIANG and RENNEBOOG, 2017) +and the socio-political context (Tilt, 2016) of the +countries where the companies operate. Globally, +more and more companies are engaging in CSR +initiatives. They are therefore providing more so- +cial information to the public. As a result, CSR +disclosure has grown to be one of the main study +directions for researchers of this field (Goyal et al., +2015; Halkosa and Skouloudis, 2016). +While reaching adequate standards of sustain- +ability disclosure or reporting is desirable, there are +several obstacles to overcome. Sustainability re- +porting is optional, in contrast, to strictly regulated +financial reporting, and it is consequently charac- +terized by a lack of uniformity (Braam and Peeters, +2018; Bhattacharyya and Cummings, 2015). Prior +studies have been generally focused on the fac- +tors that drive the disclosure of these initiatives, the +given information, the mode of communication and +their impact on the company’s performance and im- +age (Gonçalves and Gaio, 2023; Benoit-Moreau +and Parguel, 2011). These factors that may in- +fluence CSR disclosure reports of a company are +usually classified as: (i) internal, such as company +size, industry sector, financial performance, and +corporate governance; (ii) external, such as country +of origin, stakeholders, media, or social and politi- +cal environment (Fifka, 2013; Morhardt, 2009). +Considering the limited research that is avail- +able, a few studies also try to investigate the pos- +sibility that “country” can influence CSR initia- +tives and disclosure levels (Kansal et al., 2014; +Fufa and Roba, 2021; Khan et al., 2021). On one +hand, deeper correlations between other factors and +the CSR initiatives of companies are mostly miss- +ing. On the other hand, most of the studies (e.g., +those cited above) are methodologically “conserva- +tive” and do not exploit data-driven approaches that +have surged in the last decade (Pugna et al., 2022; +Abuimara et al., 2022; Çano, 2018). This trend +towards data-driven research is mostly conducted +using English language resources (e.g., datasets) +which are the most numerous on the internet. There +are still several studies and resources in Czech or +other languages becoming common and available +(Çano and Bojar, 2019; Sestino and Mauro, 2022). +In this paper, we try to foster data-driven re- +search about CSR by creating and describing +CSRCZ, a freely available dataset containing pub- +lic information of 1 000 companies operating in +the Czech Republic.1 In the following sections, +we present the information retrieval process steps +that were followed. We also describe the available +1https://zenodo.org/record/7495802 +arXiv:2301.03404v1 [econ.GN] 5 Jan 2023 + +Attribute +Content Type +Company Name +String +Number of employees +Integer +Has a CSR page +Binary +Industry Sector +String +Size of company +Categorical +Initiatives +String +Website +URL +Table 1: Data attributes and their respective types. +data fields (especially those related to CSR), their +characteristic values, and some relevant statistics. +Finally, we discuss potential utilization of CSRCZ +content in the context of future CSR research. +2 +Dataset Content +The sources for constructing the CSRCZ dataset +were collected from the public websites of 1 000 +companies currently operating in the Czech Repub- +lic. Initially, the websites of those companies were +retrieved by jobs.cz. Each website was analyzed +and only the information relating to CSR was col- +lected. The relevant attributes that were considered +are presented in Table 1. +Company Name represents the official name of +the company as it is registered in the Czech Re- +public. It is saved as a text string. Number of +employees is an integer that includes the total num- +ber of full-time employees, part-time employees, +seasonal workers, and partners. Has a CSR page is +a binary value with ‘1’ indicating that this company +includes in its website some page with information +regarding CSR policies or practices, and ‘0’ indi- +cating that it does not. Industry Sector is a string +describing the market segment of the company or +the type of activity it mostly performs. +Size of company is a categorical variable that de- +scribes the size of the company. Any company with +fewer than 10 employees is considered as ‘Micro’. +Those with up to 50 employees are ‘Small’ compa- +nies. The companies are considered ‘Medium’ if +they have 51 up to 250 employees. Any company +with 251 or more employees is ‘Large’. Initia- +tives is probably the most important attribute with +respect to the CSR analysis. It is a long string +describing any CSR-related policies, practices or +initiatives that the company outlines. Finally, Web- +site is the URL from which the information was +retrieved. +Size +Number +Percent +Unknown +1 +0.1 +Micro +125 +12.5 +Small +214 +21.4 +Medium +330 +33 +Large +330 +33 +Table 2: Size statistics of the selected companies +3 +Dataset Statistics +In the following sections, CSRCZ content is dis- +cussed in detail. The characteristics values of the +respective fields are analyzed and presented in a +tabular format. The codes for deriving the statistics +are available online.2 +3.1 +Size and Employees +The size of a company is an important factor that +is usually related to the capacities that a company +has to implement goals and practices in fulfilment +of its CSR strategy. One way to determine the size +of a company is by using the number of its employ- +ees, same as we described in Section 2. This is +obviously a simplistic approach, since other factors +like different types of assets the company owns (un- +fortunately, this type of information is not always +public) do also indicate how big it is. +We inspected the collected data and found that +most of the companies are large or medium, with +each category representing 33 % of the instances. +There are also 214 small companies which make +up 21.4 % of the total. There are also 125 compa- +nies (representing 12.5 % of the total) which are +considered to be very small or “Micro”. For one +of the sampled companies, it was not possible to +determine its size. The full statistics are presented +in Table 2 and depicted in Figure 1. +We also checked the number of employees for +each size category. Specifically, we found the min- +imum, maximum and average number of employ- +ees in the ‘Micro’, ‘Small’, ‘Medium’, and the +‘Large’ companies in CSRCZ data. In the case +of ‘Micro’ companies, there are at least 5 and at +most 9 employees, with an average of 6.54. The +same statistics for the case of ‘Small’ companies +are 10, 49 and 31.97 respectively. Companies of +a ‘Medium’ size have an average of 169.62 em- +ployees. Finally, the ‘Large’ companies do have +a maximum of 10000 employees (the biggest in +2https://github.com/erionc/csrcz-stats + +Figure 1: Size distribution of the selected companies. +Company +Min +Max +Avg +Micro +5 +9 +6.54 +Small +10 +49 +31.97 +Medium +50 +249 +169.62 +Large +299 +10000 +1635.58 +Table 3: Minimum, maximum and average number of +employees for each company category. +CSRCZ) with an average of 1635.58. The statistics +are summarized in Table 3 and depicted in Figure 3. +Figure 2: Average number of employees in each com- +pany size category. +3.2 +Industry Sector +The industry sector is an interesting attribute since +it could shed light on important trends that relate +to the CSR initiatives and the different sectors the +companies operate. According to GICS (Global +Industry Classification Standard), eleven industry +sectors represent the majority of industry types +nowadays.3 +3https://www.msci.com/our-solutions/ +indexes/gics +Sector +Number +Percent +Unknown +607 +60.7 +Communication Services +17 +1.7 +Consumer Discretionary +91 +9.1 +Consumer Staples +31 +3.1 +Energy +15 +1.5 +Financials +28 +2.8 +Health Care +16 +1.6 +Industrials +111 +11.1 +Information Technology +56 +5.6 +Materials +25 +2.5 +Real estate +3 +0.3 +Utilities +0 +0 +Table 4: Sector statistics of the selected companies +Communication Services is an industry that in- +cludes media and entertainment or any of the +telecommunication services. +Consumer Discretionary involves the retail in- +dustry, hotels, restaurants, leisure, and house- +hold durables. +Consumer Staples is an industry category that +groups all food products, beverages, and to- +bacco. +Energy includes oil, gas, consumable fuels, and +energy services. +Financials is a category grouping all banking ser- +vices, capital markets, and insurance services. +Health Care involves health care providers and +pharmaceuticals. +Industrials includes transportation services such +as airlines, marine, road & rail and all services +related to it. +Information Technology involves IT services, +software, technology hardware, storage, and +peripherals. +Materials includes all industry sectors that pro- +duce chemicals, construction materials, pack- +aging, metals, and mining. +Real estate includes real estate investment trusts +and real estate services. +Utilities includes electric, gas, and water utilities +services. + +33.0% +33.0% +30 +25 +21.4% +20 +15 +12.5% +10 +5 - +0 +0.1% +Unknown +Micro +Small +Medium +Large1635.58 +1600 +1400 +1200 +1000 +800 +600 +400 +200 +169.62 +6.54 +31.97 +0 +Micro +Small +Medium +LargeCSR Initiatives +Min +Max +Avg +Characters +0 +32023 +1218.01 +Tokens +0 +4870 +191.73 +Table 5: Minimum, maximum and average number of +characters and tokens for each CSR initiative. +We explored the data and identified the num- +ber and percentage of the companies belonging +to each of the above listed industry sectors. The +gathered statistics are summarized in Table 4. Un- +fortunately, this indicator is not available for many +of the data records. Among the available sectors +we found, ‘Industrials’ is the most popular, with +111 companies or 11.1 % of the total. The sec- +tor ‘Consumer Dicretionary’ comes next with 91 +companies. ‘Information Technology’, ‘Cosumer +Staples’ and ‘Financials’ are also common, with +56, 31 and 28 records each. The most unpopular +sectors are ‘Real estate’ and ‘Utilities’, with 3 and +0 companies. +3.3 +CSR Initiatives +The most important record attribute of the CSRCZ +dataset is probably ‘Initiatives’, where the CSR +mission, goals and practices of the companies are +summarized. This information usually comes as +a sequence of sentences, or sometimes as a few +paragraphs. A trivial statistical evaluation here is +to check its length in characters or tokens, despite +the fact that a short or long ‘Initiatives’ text in the +website does not necessarily mean that the CSR +commitment of a company is low or high. +We used NLTK word tokenizer to tokenize the +texts.4 Unfortunately, a high number of the sam- +pled companies (more specifically 610 which have +0 length of characters and tokens) have not pro- +vided such a description in their websites. The +longest CSR initiatives texts have 32023 charac- +ters and 4870 tokens. The average length of this +attribute is about 1218 characters and 191 tokens. +These statistics are summarized in Table 5. +4 +Discussion +Despite the fact that information is broadly avail- +able for a lot of organizations, many companies +regularly fail to present the CSR data in a consis- +tent way and assorted according to a framework. +As the attention towards CSR is raising and the +community becoming more watchful, the need for +4https://www.nltk.org/ +a standardized definition and CSR framework has +been rising. The need for applying data-driven +methodologies and providing structured datasets is +also in rise. +The purpose of this work is to foster data-driven +CSR research by providing and describing CSRCZ, +a recently created dataset. We believe that using +CSRCZ can provide a better view of the current +understanding of CSR in companies that operate +in the Czech Republic and in a global context as +well. Various correlations between internal and +external company factors and its CSR initiatives +can be found. Those findings could be used to de- +velop further frameworks and management strate- +gies in order to better communicate CSR initiatives +to stakeholders being those external or internal. +References +Tareq Abuimara, Brodie W Hobson, Burak Gunay, and +William O’Brien. 2022. +A data-driven workflow +to improve energy efficient operation of commer- +cial buildings: A review with real-world examples. +Building Services Engineering Research and Tech- +nology, 43(4):517–534. +Florence Benoit-Moreau and Béatrice Parguel. 2011. +Building brand equity with environmental commu- +nication: an empirical investigation in france. Eu- +romed Journal of Business, 6:100–116. +Asit Bhattacharyya and Lorne Cummings. 2015. Mea- +suring corporate environmental performance – stake- +holder engagement evaluation. +Business Strategy +and the Environment, 24(2):309–325. +Geert Braam and Roy Peeters. 2018. Corporate sustain- +ability performance and assurance on sustainability +reports: Diffusion of accounting practices in the +realm of sustainable development. +Corporate So- +cial Responsibility and Environmental Management, +25(2):164–181. +Erion Çano. 2018. Text-based Sentiment Analysis and +Music Emotion Recognition. Ph.D. thesis, Computer +Engineering, Politecnico di Torino, Turin, Italy. +Erion Çano and Ondˇrej Bojar. 2019. Sentiment anal- +ysis of czech texts: +An algorithmic survey. +In +Proceedings of the 11th International Conference +on Agents and Artificial Intelligence - Volume 2: +NLPinAI, pages 973–979. INSTICC, SciTePress. +A.B. Carroll and J.A. Brown. 2018. +Corporate so- +cial responsibility: A review of current concepts, re- +search, and issues. International Journal of Corpo- +rate Social Responsibility (JCSR), 2(2):39–69. +Matthias S. Fifka. 2013. Corporate responsibility re- +porting and its determinants in comparative perspec- +tive – a review of the empirical literature and a meta- + +analysis. +Business Strategy and The Environment, +22:1–35. +Tolossa Fufa and Yadessa Roba. 2021. Internal and ex- +ternal determinants of corporate social responsibility +practices in multinational enterprise subsidiaries in +developing countries: evidence from ethiopia. Fu- +ture Business Journal, 7. +Tiago Cruz Gonçalves and Cristina Gaio. 2023. Cor- +porate sustainability disclosure and media visibil- +ity: Mixed method evidence from the tourism sector. +Journal of Business Research, 155:113447. +Praveen Goyal, Zillur Rahman, and Absar Ahmad +Kazmi. 2015. Identification and prioritization of cor- +porate sustainability practices using analytical hier- +archical process. Journal of Modelling in Manage- +ment, 10(1):23–49. +George Halkosa and Antonis Skouloudis. 2016. +Ex- +ploring the current status and key determinants of +corporate disclosure on climate change: Evidence +from the greek business sector. Environmental Sci- +ence & Policy, 56(1):22–31. +Monika Kansal, Mahesh Joshi, and Gurdip Singh Batra. +2014. Determinants of corporate social responsibil- +ity disclosures: Evidence from india. Advances in +Accounting, 30(1):217–229. +Majid Khan, James Lockhart, and Ralph Bathurst. +2021. The institutional analysis of csr: Learnings +from an emerging country. Emerging Markets Re- +view, 46:100752. Corporate Social Responsibility +in Emerging Markets. +HAO LIANG and LUC RENNEBOOG. 2017. On the +foundations of corporate social responsibility. The +Journal of Finance, 72(2):853–910. +J. Emil Morhardt. 2009. Corporate social responsibil- +ity and sustainability reporting on the internet. Busi- +ness Strategy and The Environment, 19:436–452. +Irina Bogdana Pugna, Dana Maria Boldeanu, Mirela +Gheorghe, Gabriel Cozgarea, and Adrian Nicolae +Cozgarea. 2022. Management perspectives towards +the data-driven organization in the energy sector. +Energies, 15(16). +Andrea Sestino and Andrea De Mauro. 2022. Leverag- +ing artificial intelligence in business: Implications, +applications and methods. Technology Analysis & +Strategic Management, 34(1):16–29. +Carol A. Tilt. 2016. Corporate social responsibility re- +search: The importance of context. +International +Journal of Corporate Social Responsibility (JCSR), +1(2):1–9. + diff --git a/XtE1T4oBgHgl3EQfvwXS/content/tmp_files/load_file.txt b/XtE1T4oBgHgl3EQfvwXS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..27761c233b655029a8bb16cc64eed38b1e3665a8 --- /dev/null +++ b/XtE1T4oBgHgl3EQfvwXS/content/tmp_files/load_file.txt @@ -0,0 +1,260 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf,len=259 +page_content='CSRCZ: A Dataset About Corporate Social Responsibility in Czech Republic Xhesilda Vogli Department of Management Faculty of Economics and Management Czech University of Life Sciences vogli@pef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='czu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='cz Erion Çano Digital Philology Data Mining and Machine Learning University of Vienna, Austria erion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='cano@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='at Abstract As stakeholders’ pressure on corporates for disclosing their corporate social responsibility operations grows, it is crucial to understand how efficient corporate disclosure systems are in bridging the gap between corporate social responsibility reports and their actual practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Meanwhile, research on corporate social re- sponsibility is still not aligned with the recent data-driven strategies, and little public data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' This paper aims to describe CSRCZ, a newly created dataset based on disclosure re- ports from the websites of 1 000 companies that operate in Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Each com- pany was analyzed based on three main param- eters: company size, company industry, and company initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We describe the content of the dataset as well as its potential use for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We believe that CSRCZ has implications for further research, since it is the first publicly available dataset of its kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 1 Introduction Corporate Social Responsibility (CSR) has evolved from a “why” in the early 1950s (Carroll and Brown, 2018) to a “must” in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Gen- erally, CSR is considered a self-regulating business model which helps companies to contribute to so- cietal goals and be socially accountable to them- selves and the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' It is highly influenced by the legal context (LIANG and RENNEBOOG, 2017) and the socio-political context (Tilt, 2016) of the countries where the companies operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Globally, more and more companies are engaging in CSR initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' They are therefore providing more so- cial information to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' As a result, CSR disclosure has grown to be one of the main study directions for researchers of this field (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Halkosa and Skouloudis, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' While reaching adequate standards of sustain- ability disclosure or reporting is desirable, there are several obstacles to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Sustainability re- porting is optional, in contrast, to strictly regulated financial reporting, and it is consequently charac- terized by a lack of uniformity (Braam and Peeters, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Bhattacharyya and Cummings, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Prior studies have been generally focused on the fac- tors that drive the disclosure of these initiatives, the given information, the mode of communication and their impact on the company’s performance and im- age (Gonçalves and Gaio, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Benoit-Moreau and Parguel, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' These factors that may in- fluence CSR disclosure reports of a company are usually classified as: (i) internal, such as company size, industry sector, financial performance, and corporate governance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' (ii) external, such as country of origin, stakeholders, media, or social and politi- cal environment (Fifka, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Morhardt, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Considering the limited research that is avail- able, a few studies also try to investigate the pos- sibility that “country” can influence CSR initia- tives and disclosure levels (Kansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Fufa and Roba, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' On one hand, deeper correlations between other factors and the CSR initiatives of companies are mostly miss- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' On the other hand, most of the studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=', those cited above) are methodologically “conserva- tive” and do not exploit data-driven approaches that have surged in the last decade (Pugna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Abuimara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Çano, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' This trend towards data-driven research is mostly conducted using English language resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=', datasets) which are the most numerous on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' There are still several studies and resources in Czech or other languages becoming common and available (Çano and Bojar, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Sestino and Mauro, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' In this paper, we try to foster data-driven re- search about CSR by creating and describing CSRCZ, a freely available dataset containing pub- lic information of 1 000 companies operating in the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1 In the following sections, we present the information retrieval process steps that were followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We also describe the available 1https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='org/record/7495802 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='03404v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='GN] 5 Jan 2023 Attribute Content Type Company Name String Number of employees Integer Has a CSR page Binary Industry Sector String Size of company Categorical Initiatives String Website URL Table 1: Data attributes and their respective types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' data fields (especially those related to CSR), their characteristic values, and some relevant statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Finally, we discuss potential utilization of CSRCZ content in the context of future CSR research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2 Dataset Content The sources for constructing the CSRCZ dataset were collected from the public websites of 1 000 companies currently operating in the Czech Repub- lic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Initially, the websites of those companies were retrieved by jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='cz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Each website was analyzed and only the information relating to CSR was col- lected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The relevant attributes that were considered are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Company Name represents the official name of the company as it is registered in the Czech Re- public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' It is saved as a text string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Number of employees is an integer that includes the total num- ber of full-time employees, part-time employees, seasonal workers, and partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Has a CSR page is a binary value with ‘1’ indicating that this company includes in its website some page with information regarding CSR policies or practices, and ‘0’ indi- cating that it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Industry Sector is a string describing the market segment of the company or the type of activity it mostly performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Size of company is a categorical variable that de- scribes the size of the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Any company with fewer than 10 employees is considered as ‘Micro’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Those with up to 50 employees are ‘Small’ compa- nies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The companies are considered ‘Medium’ if they have 51 up to 250 employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Any company with 251 or more employees is ‘Large’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Initia- tives is probably the most important attribute with respect to the CSR analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' It is a long string describing any CSR-related policies, practices or initiatives that the company outlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Finally, Web- site is the URL from which the information was retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Size Number Percent Unknown 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1 Micro 125 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='5 Small 214 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='4 Medium 330 33 Large 330 33 Table 2: Size statistics of the selected companies 3 Dataset Statistics In the following sections, CSRCZ content is dis- cussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The characteristics values of the respective fields are analyzed and presented in a tabular format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The codes for deriving the statistics are available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1 Size and Employees The size of a company is an important factor that is usually related to the capacities that a company has to implement goals and practices in fulfilment of its CSR strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' One way to determine the size of a company is by using the number of its employ- ees, same as we described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' This is obviously a simplistic approach, since other factors like different types of assets the company owns (un- fortunately, this type of information is not always public) do also indicate how big it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We inspected the collected data and found that most of the companies are large or medium, with each category representing 33 % of the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' There are also 214 small companies which make up 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='4 % of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' There are also 125 compa- nies (representing 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='5 % of the total) which are considered to be very small or “Micro”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' For one of the sampled companies, it was not possible to determine its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The full statistics are presented in Table 2 and depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We also checked the number of employees for each size category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Specifically, we found the min- imum, maximum and average number of employ- ees in the ‘Micro’, ‘Small’, ‘Medium’, and the ‘Large’ companies in CSRCZ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' In the case of ‘Micro’ companies, there are at least 5 and at most 9 employees, with an average of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The same statistics for the case of ‘Small’ companies are 10, 49 and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='97 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Companies of a ‘Medium’ size have an average of 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='62 em- ployees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Finally, the ‘Large’ companies do have a maximum of 10000 employees (the biggest in 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='com/erionc/csrcz-stats Figure 1: Size distribution of the selected companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Company Min Max Avg Micro 5 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='54 Small 10 49 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='97 Medium 50 249 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='62 Large 299 10000 1635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='58 Table 3: Minimum, maximum and average number of employees for each company category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' CSRCZ) with an average of 1635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The statistics are summarized in Table 3 and depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Figure 2: Average number of employees in each com- pany size category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='2 Industry Sector The industry sector is an interesting attribute since it could shed light on important trends that relate to the CSR initiatives and the different sectors the companies operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' According to GICS (Global Industry Classification Standard), eleven industry sectors represent the majority of industry types nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='3 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='msci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='com/our-solutions/ indexes/gics Sector Number Percent Unknown 607 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='7 Communication Services 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='7 Consumer Discretionary 91 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1 Consumer Staples 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1 Energy 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='5 Financials 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='8 Health Care 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='6 Industrials 111 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1 Information Technology 56 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='6 Materials 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='5 Real estate 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='3 Utilities 0 0 Table 4: Sector statistics of the selected companies Communication Services is an industry that in- cludes media and entertainment or any of the telecommunication services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Consumer Discretionary involves the retail in- dustry, hotels, restaurants, leisure, and house- hold durables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Consumer Staples is an industry category that groups all food products, beverages, and to- bacco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Energy includes oil, gas, consumable fuels, and energy services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Financials is a category grouping all banking ser- vices, capital markets, and insurance services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Health Care involves health care providers and pharmaceuticals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Industrials includes transportation services such as airlines, marine, road & rail and all services related to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Information Technology involves IT services, software, technology hardware, storage, and peripherals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Materials includes all industry sectors that pro- duce chemicals, construction materials, pack- aging, metals, and mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Real estate includes real estate investment trusts and real estate services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Utilities includes electric, gas, and water utilities services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='0% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='0% 30 25 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='4% 20 15 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='5% 10 5 - 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1% Unknown Micro Small Medium Large1635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='58 1600 1400 1200 1000 800 600 400 200 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='54 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='97 0 Micro Small Medium LargeCSR Initiatives Min Max Avg Characters 0 32023 1218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='01 Tokens 0 4870 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='73 Table 5: Minimum, maximum and average number of characters and tokens for each CSR initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We explored the data and identified the num- ber and percentage of the companies belonging to each of the above listed industry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The gathered statistics are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Un- fortunately, this indicator is not available for many of the data records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Among the available sectors we found, ‘Industrials’ is the most popular, with 111 companies or 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='1 % of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The sec- tor ‘Consumer Dicretionary’ comes next with 91 companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' ‘Information Technology’, ‘Cosumer Staples’ and ‘Financials’ are also common, with 56, 31 and 28 records each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The most unpopular sectors are ‘Real estate’ and ‘Utilities’, with 3 and 0 companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='3 CSR Initiatives The most important record attribute of the CSRCZ dataset is probably ‘Initiatives’, where the CSR mission, goals and practices of the companies are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' This information usually comes as a sequence of sentences, or sometimes as a few paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' A trivial statistical evaluation here is to check its length in characters or tokens, despite the fact that a short or long ‘Initiatives’ text in the website does not necessarily mean that the CSR commitment of a company is low or high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We used NLTK word tokenizer to tokenize the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='4 Unfortunately, a high number of the sam- pled companies (more specifically 610 which have 0 length of characters and tokens) have not pro- vided such a description in their websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The longest CSR initiatives texts have 32023 charac- ters and 4870 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The average length of this attribute is about 1218 characters and 191 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' These statistics are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 4 Discussion Despite the fact that information is broadly avail- able for a lot of organizations, many companies regularly fail to present the CSR data in a consis- tent way and assorted according to a framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' As the attention towards CSR is raising and the community becoming more watchful, the need for 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='nltk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='org/ a standardized definition and CSR framework has been rising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The need for applying data-driven methodologies and providing structured datasets is also in rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The purpose of this work is to foster data-driven CSR research by providing and describing CSRCZ, a recently created dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' We believe that using CSRCZ can provide a better view of the current understanding of CSR in companies that operate in the Czech Republic and in a global context as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Various correlations between internal and external company factors and its CSR initiatives can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Those findings could be used to de- velop further frameworks and management strate- gies in order to better communicate CSR initiatives to stakeholders being those external or internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' References Tareq Abuimara, Brodie W Hobson, Burak Gunay, and William O’Brien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' A data-driven workflow to improve energy efficient operation of commer- cial buildings: A review with real-world examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Building Services Engineering Research and Tech- nology, 43(4):517–534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Florence Benoit-Moreau and Béatrice Parguel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Building brand equity with environmental commu- nication: an empirical investigation in france.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Eu- romed Journal of Business, 6:100–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Asit Bhattacharyya and Lorne Cummings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Mea- suring corporate environmental performance – stake- holder engagement evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Business Strategy and the Environment, 24(2):309–325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Geert Braam and Roy Peeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Corporate sustain- ability performance and assurance on sustainability reports: Diffusion of accounting practices in the realm of sustainable development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Corporate So- cial Responsibility and Environmental Management, 25(2):164–181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Erion Çano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Text-based Sentiment Analysis and Music Emotion Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' thesis, Computer Engineering, Politecnico di Torino, Turin, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Erion Çano and Ondˇrej Bojar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Sentiment anal- ysis of czech texts: An algorithmic survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' In Proceedings of the 11th International Conference on Agents and Artificial Intelligence - Volume 2: NLPinAI, pages 973–979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' INSTICC, SciTePress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Carroll and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Corporate so- cial responsibility: A review of current concepts, re- search, and issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' International Journal of Corpo- rate Social Responsibility (JCSR), 2(2):39–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Matthias S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Fifka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Corporate responsibility re- porting and its determinants in comparative perspec- tive – a review of the empirical literature and a meta- analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Business Strategy and The Environment, 22:1–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Tolossa Fufa and Yadessa Roba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Internal and ex- ternal determinants of corporate social responsibility practices in multinational enterprise subsidiaries in developing countries: evidence from ethiopia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Fu- ture Business Journal, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Tiago Cruz Gonçalves and Cristina Gaio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Cor- porate sustainability disclosure and media visibil- ity: Mixed method evidence from the tourism sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Journal of Business Research, 155:113447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Praveen Goyal, Zillur Rahman, and Absar Ahmad Kazmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Identification and prioritization of cor- porate sustainability practices using analytical hier- archical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Journal of Modelling in Manage- ment, 10(1):23–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' George Halkosa and Antonis Skouloudis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Ex- ploring the current status and key determinants of corporate disclosure on climate change: Evidence from the greek business sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Environmental Sci- ence & Policy, 56(1):22–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Monika Kansal, Mahesh Joshi, and Gurdip Singh Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Determinants of corporate social responsibil- ity disclosures: Evidence from india.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Advances in Accounting, 30(1):217–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Majid Khan, James Lockhart, and Ralph Bathurst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The institutional analysis of csr: Learnings from an emerging country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Emerging Markets Re- view, 46:100752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Corporate Social Responsibility in Emerging Markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' HAO LIANG and LUC RENNEBOOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' On the foundations of corporate social responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' The Journal of Finance, 72(2):853–910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Emil Morhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Corporate social responsibil- ity and sustainability reporting on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Busi- ness Strategy and The Environment, 19:436–452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Irina Bogdana Pugna, Dana Maria Boldeanu, Mirela Gheorghe, Gabriel Cozgarea, and Adrian Nicolae Cozgarea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Management perspectives towards the data-driven organization in the energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Energies, 15(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Andrea Sestino and Andrea De Mauro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Leverag- ing artificial intelligence in business: Implications, applications and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Technology Analysis & Strategic Management, 34(1):16–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Carol A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' Corporate social responsibility re- search: The importance of context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} +page_content=' International Journal of Corporate Social Responsibility (JCSR), 1(2):1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtE1T4oBgHgl3EQfvwXS/content/2301.03404v1.pdf'} diff --git a/Z9FAT4oBgHgl3EQf4R6K/content/tmp_files/2301.08725v1.pdf.txt b/Z9FAT4oBgHgl3EQf4R6K/content/tmp_files/2301.08725v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..668bdd0fd8d2bbf37e26bf503bf8773f35e96b78 --- /dev/null +++ b/Z9FAT4oBgHgl3EQf4R6K/content/tmp_files/2301.08725v1.pdf.txt @@ -0,0 +1,882 @@ + TITLE + +Using Gamma Functions in the Mathematical Formulation of the +Impact Crater Size-Age Frequency Distribution on Earth and Mars. + Author: William F Bruckman + Abstract +A review of a mathematical formulation that describes the number of impact craters as +function of diameter and time of formation is presented, where the use of Gamma +functions is emphasized. The application of this formalism for the description of the +impact crater data of Planets Earth and Mars is also discussed. + +1. Introduction + When solving differential or integral equations an ideal outcome is to express the +solution in terms of elementary or special functions. In that case the mathematical and +physical interpretation of the solutions is clarified. Moreover, with the use of algebraic +computing, the comparison of the prediction of theoretical models with the observational +data is greatly facilitated. + This paper will consider work in reference1 (Earth and Mars Crater Size Frequency +Distribution and Impact Rates: Theoretical and Observational Analysis; William +Bruckman, Abraham Ruiz, and Elio Ramos; Arxiv: 1212.3273), which presented a +theoretical formulation describing impact crater data on Earth and Mars, giving the +number of craters as functions of diameter, and time of formation, successfully +reproducing the observations. The revision will emphasize the presentation of the +solutions of the models in terms of Gamma functions. +2. General Considerations + Impact craters, of a given diameter 𝐷, are formed at a certain rate 𝛷, and are also +depleted, as they get older, by a variety of processes, at a rate proportional to their +already existing number of craters, 𝑁. Hence, the number of craters eliminated in the +time interval dt can be express as 𝐶𝑁𝑑𝑡, where 𝐶 is a parameter representing the rate of +elimination per crater. On the other hand, in this time interval we also have that the +number of craters produced by impacts is 𝛷𝑑𝑡, and thus the net change in the number +of crater numbers, 𝑑𝑁, is given by + + 𝑑𝑁 = 𝛷𝑑𝑡 − 𝐶𝑁𝑑𝑡 = ( +𝛷 +𝐶 − 𝑁 )𝐶𝑑𝑡 . (1) +This equation is expected to represent well the observational data if the number of +craters is large enough to justify the assumptions that analytical mathematical continuity +is a good approximation to the discrete and probabilistic nature of the problem. + We see from Eq. (1) that 𝑁= constant implies that + 𝑁 = +𝛷 +𝐶 = 𝛷𝜏𝑚, (2) + 𝜏𝑚 ≡ 1/𝐶. (3) +In this situation (saturation) the number of craters produced by impacts is equal to the +number of craters eliminated. The dimension of 𝜏𝑚 is time, and we will see later in this +section that this time is related to the concept of “craters mean life.” + Equation (1) was integrated in reference (1) to obtain + 𝑁(𝐷, 0, 𝜏) = ∫ {𝛷(𝐷, 𝜏`) +𝜏 +0 +𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏`. (4) + 𝐶̅ ≡ +∫ 𝐶𝑑𝜏` +𝜏 +0 +𝜏 + , (5) +where 𝐶̅ is the time average of 𝐶, and 𝑁(𝐷, 0, 𝜏) defined in Eq. (4) denotes the number +of craters of diameter 𝐷, per bin size, observed at the present time (𝜏` = 0), with age +younger than 𝜏 . Accordingly, defining the term “per bin”, we have that the integral + 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫ +𝑁(𝐷, 0, 𝜏)𝑑𝐷 +𝐷𝑓 +𝐷𝑖 +, (6) + +gives the total number of craters with diameters in the interval between 𝐷𝑖 and 𝐷𝑓, +observed at the present time, with age of formation younger than 𝜏. Also, 𝛷(𝐷, 𝜏) is the +rate of meteorite impacts, per bin, forming craters of diameter 𝐷 at time 𝜏, so that 𝛷𝐶: + 𝛷𝐶(𝐷𝑖, 𝐷𝑓, 𝜏) = ∫ +{𝛷(𝐷, 𝜏) +𝐷𝑓 +𝐷𝑖 +}𝑑𝐷, (7) +Is the cumulative impact rate of formation of craters with diameters in the interval +between 𝐷𝑖 and 𝐷𝑓. For instance, if 𝐷𝑓 → ∞, which is of common use, the above integral +is the total cumulative impact rate of formation of craters with diameters larger than 𝐷𝑖. + Equations (6) and (4) can be generalized so that the lower 𝜏 limit is different from +zero: + 𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) ≡ ∫ +𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷 +𝐷𝑓 +𝐷𝑖 + , (8) + +where + 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) = ∫ {𝛷(𝐷, 𝜏`) +𝜏𝑓 +𝜏𝑖 +𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏`. (9) +Thus, Eqs. (8) and (9) refer to craters with ages between 𝜏𝑖 and 𝜏𝑓. + Further discussion and applications of Eq. (8) to the Earth’s crater record will be +continued in Section 4. For the planet Mars, however, we will be applying Eq. (4) in the +next section, but now continue its interpretation below. + Since the quantity 𝛷(𝐷, 𝜏`)𝑑𝜏`. Is the number of craters formed at time 𝜏`, during +the interval 𝑑𝜏`, and the integrand in Eq. (4): 𝛷(𝐷, 𝜏`)𝑑𝜏`𝐸𝑥𝑝[−𝐶̅𝜏`], is the number of +these craters, of age 𝜏` , that remain at the present time, then the expression 𝐸𝑥𝑝[−𝐶̅𝜏`] +represents the fraction of these formed craters that survive after the time 𝜏`. It is then +usual to call the inverse of 𝐶̅ “the mean life”: 𝜏𝑚𝑒𝑎𝑛, + 𝜏𝑚𝑒𝑎𝑛 ≡ +1 +𝐶̅ ; 1/𝜏𝑚𝑒𝑎𝑛 = 𝐶̅ ≡ +∫ 𝐶𝑑𝜏` +𝜏 +0 +𝜏 += +∫ (1 +𝜏𝑚)𝑑𝜏` +𝜏 +0 +𝜏 +. (10) +Thus, in this context 𝜏𝑚𝑒𝑎𝑛 can be viewed as the mean life of craters of diameter 𝐷. +Also, this interpretation suggests thinking of 𝛷 as a probability of impact, rather than an +impact flux, thus emphasizing the statistical nature of the impacts of asteroids and +comets. Conversely, if we start with the definition of 𝐸𝑥𝑝[−𝐶̅𝜏`] as the fraction of craters +surviving after the interval 𝜏` from their formation, then we can construct Eq. (4) to +represent the sum of all the contributions, to the present number, for all times 𝜏` +younger than 𝜏, and then find that 𝑁 satisfies the differential equation implied in Eq. (1). + Consider the following definition: + 𝑇(𝐷, 𝜏, ) ≡ ∫ 𝐶𝑑𝜏` +𝜏 +0 + = 𝐶̅ 𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛 . (11) +Hence 𝑇 is a dimensionless time that measures the numbers of mean-life in an interval +𝜏. From Eq. (11) it follows that + 𝑑𝑇/𝑑𝜏 = 𝐶(𝐷, 𝜏) , (12) +where 𝐷 is considered here as a constant parameter. Since crater elimination is a +decay process, where 𝐶 is strictly positive, we have + 𝑑𝑇/𝑑𝜏 > 0 . (13) +Consequently, the function 𝑇(𝐷, 𝜏, ) can be inverted to express 𝜏 as a function of 𝑇 and +𝐷: 𝜏(𝐷, 𝑇). Likewise, 𝐶 and 𝛷 are each expressible as functions of 𝑇 and 𝐷. We can +then rewrite Eq. (4), using Eqs. (11), (12) and (3), in the form + + 𝑁(𝐷, 0, 𝜏) = ∫ {𝛷(𝐷, 𝜏`) +𝜏 +0 +𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏` = ∫ {( +𝛷 +𝐶) +𝑇 +0 +𝐸𝑥𝑝[−𝑇`]}𝑑𝑇` = + ∫ {(𝛷𝜏𝑚) +𝑇 +0 +𝐸𝑥𝑝[−𝑇`]}𝑑𝑇`. (14) +where, in the right-hand side of Eq. (14), +𝛷 +𝐶 = 𝛷𝜏𝑚 is considered now a function of 𝑇, +and the parameter 𝐷. For instance, if 𝛷𝜏𝑚 is a sum like + 𝛷𝜏𝑚 = 𝛴𝑎𝑠𝑇𝑠 , 𝑎𝑠 and 𝑠 are independent of 𝑇, (15) +then we have, from Eq. (14), + 𝑁(𝐷, 0, 𝑇) = 𝛴𝑎𝑠 ∫ {𝑇`𝑠 +𝑇 +0 +𝐸𝑥𝑝[−𝑇`]}𝑑𝑇` = 𝛴𝑎𝑠 𝛾(𝑠 + 1, 𝑇) , (16) +where the lower incomplete gamma function notation was used above: + 𝛾(𝑠 + 1, 𝑇) = ∫ {𝑇`𝑠 +𝑇 +0 +𝐸𝑥𝑝[−𝑇`]}𝑑𝑇`. (17) +If 𝑠 is a whole number, as in a Taylor-Maclaurin series, we can also write + 𝛾(𝑠 + 1, 𝑇) = 𝑠! (1 - 𝑒−𝑇 ∑ +𝑇𝑘 +𝑠 +𝑘=0 +/𝑘! ). (18) +This is our first encounter with the use of gamma functions expressing the number of +craters as a function of diameter and age. We will see further use of gamma functions +when considering applications to Earth’s impact crater data in Section (4). We will now +focus our attention on applications of Eq. (14) to the planet Mars. +3. Applications to the Crater-Size Frequency Distribution of Mars + It was discussed in Section 2 that the product +𝛷 +𝐶 = 𝛷𝜏𝑚 represents the value of 𝑁 +when the production and the elimination of craters are equal and 𝑑𝑁 = 0. Then in a +steady state situation we will have 𝑁 = 𝛷𝜏𝑚 = constant. However, in general, 𝛷𝜏𝑚 +could depend on time, since both 𝛷 and 𝜏𝑚 could depend on time. On the other hand, +since 𝐶 is by definition the rate of crater elimination per number of craters, we have +then that 𝜏𝑚 ≡ 1/𝐶 is strongly influenced by the elimination of old craters due to impacts +forming new craters. Therefore, an increase or decrease of 𝛷 would be correlated with +an increase or decrease of 𝐶. Consequently, if obliterations by impacts are important, +the changes in time of +𝛷 +𝐶 = 𝛷𝜏𝑚 are smoothed out relative to the individual changes in +time of 𝛷, 𝐶,or 𝜏𝑚. In such a heuristic and realistic situation, a model in which it is +assumed that +𝛷 +𝐶 = 𝛷𝜏𝑚 is constant should be a good representation of the +observations. In this case, Eq. (14) becomes + 𝑁 = 𝛷𝜏𝑚(1 - 𝑒−𝑇) (19) + + With the above simple model we were able to represent (reference 1) remarkably +well the pioneering Mars crater database catalog of Barlow (1988), as illustrated in Fig. +(1). Also, Fig. 2 compares the model with the more recent Mars data catalog of Robbins +and Hynek (2012), and also the model is in very good agreement with observations +(Bruckman 2019). The values of 𝛷𝜏𝑚 and 𝑇 for Barlow’s model are + 𝛷𝜏𝑚 = +1.43𝑥105 +𝐷1.8 +, (20) + 𝑇 = 𝐶̅𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛 = +2.48𝑥104 +𝐷2.5 +, (21) +and then Eq. (19) becomes + 𝑁 = 𝛷𝜏𝑚(1 - 𝑒−𝑇) = +1.43𝑥105 +𝐷1.8 +(1 − 𝐸𝑥𝑝[− +2.48𝑥104 +𝐷2.5 +]), (22) +where the unit of 𝐷 is kilometers. It can also be shown (Appendix A), using the +assumption that 𝛷𝜏𝑚 is independent of 𝑇, that + 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏 +𝜏 +0 += 𝛷𝜏𝑚𝑇 = +3.55𝑥109 +𝐷4.3 +, (23) +where 𝛷̅ is the time average of 𝛷, and 𝛷̅𝜏 is the total number of craters, of diameter 𝐷, +per bin, created over the total time of production 𝜏 . The corresponding expression for +the number of craters created with diameters in the interval between 𝐷𝑖 and 𝐷𝑓 is then : + 𝜏𝛷̅𝐶(𝐷𝑖, 𝐷𝑓, 𝜏) = ∫ +𝛷̅𝜏 +𝐷𝑓 +𝐷𝑖 +𝑑𝐷. = (3.55/3.3)109( +1 +𝐷𝑖3.3 - +1 +𝐷𝑓3.3) . (24) +It is common to take the upper limit 𝐷𝑓 to be infinite to obtain the total number of craters +produced larger than 𝐷𝑖 : + 𝜏𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = 1.076𝑥109( +1 +𝐷𝑖3.3) , (25) +or + 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = (1.076𝑥109/𝜏)( +1 +𝐷𝑖3.3) , (26) +where 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) is the time average of the cumulative impact rate for the formation of +craters larger than 𝐷𝑖. For instance, it is interesting to note that for 𝐷𝑖 = 1 km, +approximately 109 such impacts were produced. Therefore, assuming that the total time +of crater production 𝜏 was 3000 to 4000 million years, we get an average of one +impact, making craters larger than 1 km, approximately every three to four years. Since +the energy associated to impacts with a diameter of 1 km is close to one megaton, this + +result is of concern for explorations of Mars, assuming that the present impact flux +average is comparable to that given by Eq. (26). + It is expected that also the corresponding impact rate for Earth has, similar to +Mars, a crater diameter dependency of the form +1 +𝐷𝑖3.3, and indeed, we found that for our +planet such a relation is consistent with the observations (Appendix B). + Let us continue our analysis of the implications of the above model, by looking at +Eq. (21), rewritten in the form + +𝜏𝑚𝑒𝑎𝑛 +𝜏 += 𝐷2.5/2.48𝑥104 . (27) +An interesting interpretation of the above equation (Reference 1) is that it represents a +proportionality relation between the mean life, of a crater of diameter 𝐷, and the initial +volume of this crater. This conclusion is based on observations on Mars that +established that the initial depths of pristine craters are proportional to 𝐷𝑘/2, with 𝑘 ≈ 1, +and, consequently, the expected initial volumes for these craters are proportional to +𝐷2𝐷𝑘/2 ≈ 𝐷2.5. For instance, Garvin (2002) gives 𝑘 ≈ 0.98, while Boyce et al. (2007) +give 𝑘 ≈ 1.04. Furthermore, from the application to Earth of the above formalism, to be +discussed in the next section, it was concluded that craters in our planet also have their +mean-life proportional to ≈ 𝐷2.5. Thus, we have that the relation 𝜏𝑚𝑒𝑎𝑛 proportional to +the crater initial volume is not only intuitively appealing, but also helps us understand +why we have similar 𝐷 exponents in the 𝜏𝑚𝑒𝑎𝑛 for Earth and Mars, notwithstanding +these planets contrasting geological evolutions. + + + + + + +FIGURE (1): Log-Log plot of number of craters per bin, 𝑁(𝐷) 𝑣𝑠 𝐷 based on Barlow’s Mars catalog +(1988). The number 𝑁(𝐷) is calculated by counting the number of craters in a bin ∆𝐷 = 𝐷𝑅 − 𝐷𝐿, and +then dividing this number by the bin size. The point is placed at the mathematical average of 𝐷 in the +bin: (𝐷𝑅 + 𝐷𝐿)/2. The bin size is ∆𝐷 = (√2 − 1)𝐷𝐿, so that +𝐷𝑅 +𝐷𝐿 = √2. ). The curve is from the model +implied by Eq. (22). We see that the theoretical curve shown differs significantly from the observed data +for 𝐷 less than about 8𝑘𝑚. However, according to Barlow, the empirical data undercounts the actual +crater population for 𝐷 less than 8𝑘𝑚. However, more recent Mars crater data by Robbins et al. (2012) +was used to update the observations, yielding similar results to the model in Figure 1, but extending the +range to craters with diameters down to 1 km (see Fig. 2). + + +FIGURE (2): Log-Log plot of 𝑵(𝐷), 𝑣𝑠 𝐷(km), based on the Mars catalog of Robbins et al (2012), +(Bruckman (2019)). Bin size is ∆𝐷 = ( 21/6 − 1)𝐷𝐿. Note that for 𝐷 > ~300 𝑘𝑚, the data points are +above the curve of the analytic model. However, we expect that the analytical model will be less +reliable when the number of craters in a given bin is so small that statistical continuous models break +down. Moreover, another source of discrepancy could be that these very large craters were being +formed at high proportions at older times 𝜏, thus perhaps belonging to the so-called heavy +bombardment era, characterized by a much higher impact flux. + +10 +20 +50 +100 +200 +500 +D +0.1 +1 +10 +100 +1000 +N + +Log[N(D)] +3 E +2 E +1上 +1.0 +1.5 +2.0 +Log[P] 4. Applications to Planet Earth + The number of identified impact craters on Earth is close to 190 (Planetary and +Space Science Center: PASSC.com), while, in contrast, the number of craters used for +Mars in Fig. (1) was 42,284. Therefore, in the analysis of Earth’s crater data it is +convenient to use the cumulative number of impacts of craters, 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏), defined in +Eq. (6), instead of 𝑁(𝐷, 0, 𝜏), defined in Eq. (4). Furthermore, for 𝑁(𝐷, 0, 𝜏), the +simplified expression in Eq. (19) will be used, since it reproduced the Martian impact +data very well. Thus we have + 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫ +𝑁(𝐷, 0, 𝜏)𝑑𝐷 +𝐷𝑓 +𝐷𝑖 += ∫ +𝛷𝜏𝑚(1 – 𝑒−𝑇) 𝑑𝐷 = +𝐷𝑓 +𝐷𝑖 + + ∫ +𝛷𝜏𝑚𝑑𝐷 − ∫ +𝛷𝜏𝑚𝑒−𝑇𝑑𝐷 +𝐷𝑓 +𝐷𝑖 +𝐷𝑓 +𝐷𝑖 +. (28) +In addition, let us assume that + 𝛷𝜏𝑚 = +𝐻 +𝐷𝑚` , (29) + 𝑇 = 𝐶̅𝜏 = +𝐵𝜏 +𝐷𝑝 , (30) + 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` +𝜏 +0 += 𝛷𝜏𝑚𝑇 = +𝐴𝜏 +𝐷𝑚 . (31) +where 𝐻, 𝑚`, 𝐵, 𝑝, 𝐴 𝑎𝑛𝑑 𝑚 are independent of 𝐷, and, from Eqs. (29), (30) and (31), + 𝑚 = 𝑚` + 𝑝 , (32) + 𝐴 = 𝐻𝐵 . (33) +Equations (29), (30), and (31) are a generalization for Earth of the corresponding equations, +(20), (21) and (23), describing the crater distribution for Mars. For Mars, we have 𝐻 = +1.43𝑥105, 𝐵𝜏 = 2.48𝑥104 , and 𝐴𝜏 = 3.55𝑥109. However, for our planet these values will +have to be redetermined. Also, the exponents 𝑚 and 𝑝 should come out from the fitting to +Earth data. As was discussed in previous section, a value of 𝑚 = 4.3, in the exponent of 𝐷 of +the impact flux 𝛷̅ is also consistent with the Earth observational impact rate data +(Appendix B). The value 𝑝 = 2.5 is also consistent with the Earth observations, to be +discussed in this section. + After the substitution of the expressions in Eqs. (29), (30), and (31) in Eq. (28) the first +integral in the right-hand side is elementary, hence, we will turn our attention to the second +integral: + − ∫ +𝛷𝜏𝑚𝑒−𝑇𝑑𝐷 +𝐷𝑓 +𝐷𝑖 += − ∫ +𝐻 +𝐷𝑚` {𝐸𝑥𝑝 [ +−𝐵𝜏 +𝐷𝑝 ]} 𝑑𝐷 +𝐷𝑓 +𝐷𝑖 +. (34) + +To emphasize that the variable of integration is now 𝐷, while 𝜏 is a fixed parameter, we +rename 𝑇 as 𝑈: + 𝑇 = 𝑈 = +𝐵𝜏 +𝐷𝑝 , (35) +or + 𝐷 = [ +𝐵𝜏 +𝑈 ]1/𝑝 , (36) +from which, differentiating with respect to 𝐷, holding 𝜏 fixed, + 𝑑𝐷 = −[𝐵𝜏 ] +1 +𝑝[ 𝑈 ] +−1 +𝑝 −1𝑑𝑈/𝑝 . (37) +Substituting Eqs. (35), (36), and (37) in Eq. (34) we get + − ∫ +𝛷𝜏𝑚𝑒−𝑇𝑑𝐷 +𝐷𝑓 +𝐷𝑖 += {𝐻/(𝑝[𝐵𝜏 ]𝑛)} ∫ +𝑈𝑛−1{𝐸𝑥𝑝[−𝑈]}𝑑𝑈 +𝑈𝑓 +𝑈𝑖 += {𝐻/(𝑝[𝐵𝜏 ]𝑛)}𝛤[𝑛, 𝑈𝑖,𝑈𝑓],(38) + where + 𝑛 ≡ ( 𝑚` − 1)/𝑝 , (39) + 𝑈𝑖 = +𝐵𝜏 +𝐷𝑖𝑝 , (40) + 𝑈𝑓 = +𝐵𝜏 +𝐷𝑓𝑝 , (41) +and + 𝛤[𝑛, 𝑈𝑖,𝑈𝑓] = ∫ +𝑈𝑛−1{𝐸𝑥𝑝[−𝑈]}𝑑𝑈 +𝑈𝑓 +𝑈𝑖 + (42) +is the generalized incomplete gamma function. Consequently, we can rewrite Eq. (28) +in the form + 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫ + +𝐻 +𝐷𝑚` 𝑑𝐷 + { +𝐷𝑓 +𝐷𝑖 +𝐻/(𝑝[𝐵𝜏 ]𝑛)} 𝛤[𝑛, 𝑈𝑖,𝑈𝑓] (43) + The above integral represents the number of craters with diameters in the interval +between 𝐷𝑖 and 𝐷𝑓, that are younger than 𝜏. Hence, the number of craters formed with +ages between 𝜏𝑖 and 𝜏𝑓 is + 𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) ≡ ∫ +𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷 +𝐷𝑓 +𝐷𝑖 += 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏𝑓) - 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏𝑖) = + {𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛)} 𝛤 [𝑛, +𝐵𝜏𝑓 +𝐷𝑖𝑝 , +𝐵𝜏𝑓 +𝐷𝑓𝑝] − {𝐻/(𝑝[𝐵𝜏𝑖 ]𝑛)} 𝛤 [𝑛, +𝐵𝜏𝑖 +𝐷𝑖𝑝 , +𝐵𝜏𝑖 +𝐷𝑓𝑝] . (44) + +Here, if 𝐵 is a function of time, it should be evaluated at the corresponding 𝜏𝑖 or 𝜏𝑓. + Another useful concept is the statistical mean of a function of 𝐷: 𝑓(𝐷), which is +defined using 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓), as follows + 𝑓̅ = ∫ +𝑓𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 +𝐷𝑓 +𝐷𝑖 +/{∫ +𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 +𝐷𝑓 +𝐷𝑖 +}=∫ +𝑓𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷 +𝐷𝑓 +𝐷𝑖 +/{𝑁 +̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓)}. (45) +For instance, if 𝑓 = 𝐷 we get, from definition (45), the average diameters of craters with +diameters and ages in the intervals 𝐷𝑖 ≤ 𝐷 ≤ 𝐷𝑓, and 𝜏𝑖 ≤ 𝜏 ≤ 𝜏𝑓 , respectively. In this +case it follows that the numerator of Eq. (45) is + ∫ +𝐷𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 = +𝐷𝑓 +𝐷𝑖 +{𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛`)} 𝛤 [𝑛`, +𝐵𝜏𝑓 +𝐷𝑖𝑝 , +𝐵𝜏𝑓 +𝐷𝑓𝑝] − {𝐻(𝑝[𝐵𝜏𝑖 ]𝑛`)}𝛤 [𝑛`, +𝐵𝜏𝑖 +𝐷𝑖𝑝 , +𝐵𝜏𝑖 +𝐷𝑓𝑝], (46) +where + 𝑛` ≡ + 𝑚`−2 +𝑝 += 𝑛 − 1/𝑝 . (47) +Hence + 𝐷̅ = [ +1 +𝑁̃(𝐷𝑖,𝐷𝑓,𝜏𝑖,𝜏𝑓)][{𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛`)} 𝛤 [𝑛`, +𝐵𝜏𝑓 +𝐷𝑖𝑝 , +𝐵𝜏𝑓 +𝐷𝑓𝑝] − {𝐻/(𝑝[𝐵𝜏𝑖 ]𝑛`)} 𝛤 [𝑛`, +𝐵𝜏𝑖 +𝐷𝑖𝑝 , +𝐵𝜏𝑖 +𝐷𝑓𝑝]] (48) +The above expression was adapted and applied to the Earth crater data in reference +(1). The value of 𝑝 was determined by the best fitting of the data to the model given in Eq. +(48), and yielded a value similar to that for Mars. As stated, this is interpreted to be the result of +the proportionality of 𝜏𝑚𝑒𝑎𝑛 to the initial volume of craters, and that this volume is in turn +proportional to 𝐷𝑝. From this fitting to observation also came an approximate value for +Earth’s parameter 𝐵. + The expression 𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) in Eq. (44), was also used in reference (1) to +describe the number of Earth’s craters as a function of diameter and age, as illustrated +in figures C1 and C2 in Appendix C. The values 𝑝 = 2.5, 𝑚 = 4.3, and 𝑚` = 𝑚 − 𝑝 = 1.8 +were assumed since they were observationally justified. The value of 𝐻 = 𝐴/𝐵 was also +needed, and, since 𝐵 was estimated from observations of 𝐷̅, then the value of 𝐴 remained to +be estimated, as described in Appendix C. A remarkable agreement of the model with +observations was obtained. + + + + + Appendix A + The number of impacts, during the time 𝜏, producing craters of diameter 𝐷, per bin, can be +expressed as + 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` = ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 +𝜏 +0 + +𝜏 +0 +. A1 + Using Eqs. (3) and (12), we get + 𝑑𝑇/𝑑𝜏 = 𝐶(𝐷, 𝜏) = +1 +𝜏𝑚. A2 +We can then rewrite the right hand side of Eq. (A1) in the form + ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 = +𝜏 +0 +∫ 𝛷𝜏𝑚𝑑𝑇` . +𝑇 +0 + A3 +If furthermore 𝛷𝜏𝑚 is independent of 𝑇 we have + ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 = +𝜏 +0 +∫ 𝛷𝜏𝑚𝑑𝑇` = +𝑇 +0 +𝛷𝜏𝑚 ∫ 𝑑𝑇` = 𝛷𝜏𝑚𝑇 +𝑇 +0 + . A4 +Therefore, from Eqs. (A1) and (A4), + 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` = ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 +𝜏 +0 + +𝜏 +0 += 𝛷𝜏𝑚𝑇. A5 +Note also that, since + 𝑇 = 𝐶̅𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛 , A6 +from (A5) we get + 𝛷̅𝜏𝑚𝑒𝑎𝑛 = 𝛷𝜏𝑚. A7 + + + + + + + + + Appendix B + Let us investigate the observational implications of the assumption of an average impact +flux for Earth given by + 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` +𝜏 +0 += +𝐴𝜏 +𝐷4.3, B1 +which implies the following cumulative impact flux + 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = ∫ +𝛷̅ +,∞ +𝐷𝑖 +𝑑𝐷 = +𝐴/3.3 +𝐷3.3 , B2 +where we drop the 𝑖 sub index from 𝐷, in the right-hand side of Eq. (B2). The value of 𝐴 +can be estimated for Earth from the result of Grieve and Shoemaker (1994) for 𝐷 = +20𝑘𝑚: + 𝛷̅𝐶(20𝑘𝑚, ∞, 𝜏) = +(5.5∓2.7)10−9 +(𝑚𝑦)𝑘𝑚2 4𝜋𝑅2 ≈ 2.8[ +1∓0.50 +𝑚𝑦 ], B3 +where 𝑅 is the Earth’s radius, and 𝑚𝑦 is million years. Comparing Eq. (B2), evaluated at +𝐷 = 20𝑘𝑚, with Eq. (B3) we obtain + 𝐴 = 9.24[1∓0.50] +(20)3.3 +𝑚𝑦 , B4 +and thus + 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) ≡ 𝛷̅𝐶 = 2.8[ +1∓0.50 +𝑚𝑦 ](20 𝐷 +⁄ )3.3 B5 +This equation is a generalization of the result of Grieve and Shoemaker (1994), which +gives the Earth’s impact rate for the formation of craters with diameters larger than 𝐷. It +incorporates the 3.3 exponent on 𝐷 that we deduced from the model and observations +from Mars. + The diameter of a crater corresponds to an energy, 𝐸, associated to the impact, +and hence Eq. (B5) can be re-expressed as (reference 1) + 𝛷̅𝐶(𝐸) = +[1∓0.5] +14.5𝑦 𝐸0.86, B6 +where 𝐸 is in megatons. Equation (B6) gives similar predictions to those of Poveda et +al. (1999). The predictions of Eq. (B6) are also in agreement with Silber et al. (2009), +that, for impacts with energies larger than a megaton, gives one Earth impact about +every 15 years. It is interesting to note that, according to Eq. (B6), events like the 2013 +Chelyabinsk meteorite of energy of about 0.5 megatons are predicted to happen with a + +periodicity near one every 8/(1 ∓ 0.5) years, so that this type of event is expected to be +repeated in the near future. + Observations in the last few decades of lunar meteorites, called Lunar Flashes, +provide a direct determination of the impact rate, at these low range of energies (see for +example Oberst et al. (2012), and Suggs et al. (2014)). For instance, Oberst et al. +(2012) interpreted data of lunar flashes, and concluded a rate of 10−3 impacts per 𝑘𝑚2 +per year, for energies ≥ ~8𝑥10−6 kilotons. This result, translated to the total Earth`s +surface area, becomes approximately 5.1𝑥105 impacts per year for these energies, +while from Eq. (B6) we get about 6.3[1 ∓ 0.5]𝑥105 impacts per year, which is consistent +with the above result for lunar flashes. + + + + + + + + + + + + + + + + + + +Appendix C + To reduce the uncertainties due to undercounting in the Earth crater data we +selected the following regions for the study in reference 1: +(a) Continental United States +(b) Canada up to the Arctic Circle +(c) Europe +(d) Australia +The crater data is taken from The Planetary and Space Science Centre +(www.passc.net). Then, in Eq. (44), instead of using for the total Earth’s impact flux + 𝛷̅ = (1/𝜏) ∫ 𝛷𝑑𝜏` +𝜏 +0 += +𝐴 +𝐷4.3, C1 +we used for our study the more accurate impact flux corresponding to the area under +consideration above in a,b,c,d, which is given by + 𝛷̅𝑎𝑐𝑐 = +𝐴𝑎𝑐𝑐 +𝐷4.3, C2 +where +𝐴𝑎𝑐𝑐 ≡ 𝐴 +𝐴𝑟𝑒𝑎 𝑈𝑛𝑑𝑒𝑟 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑎𝑡𝑖𝑜𝑛 +𝐸𝑎𝑟𝑡ℎ`𝑠 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 +, C3 +where 𝐴 is given, from Eq. (B4), by +𝐴 = 9.24[1 ∓ 0.50] +(20)3.3 +𝑚𝑦 += (1.82)105[1 ∓ 0.5]/𝑚𝑦. + C4 +Accordingly, 𝐻 = 𝐴/𝐵 becomes 𝐻𝑎𝑐𝑐 = +𝐴𝑎𝑐𝑐 +𝐵 , with 𝐵 estimated from the curve 𝐷̅ vs. +crater age , given by Eq. (48), fitting to the Earth’s data. Therefore, we can write the +theoretical 𝑁̃ with no free parameters, and compare it with the observations, as +described below. We do this first in table (I) and Figure (C1), for craters with 𝐷 ≥ 20𝑘𝑚 +and cumulative age starting with 𝜏 = 1𝑚𝑦 up to 𝜏 = 2,000𝑚𝑦. Furthermore, we put 𝜏𝑓 = +2,500𝑚𝑦 and 𝐷𝑓 = 300𝑘𝑚, since all craters in the field of study are within this bin size. +This theoretical curve, 𝑁̃(𝜏), is then compared with the corresponding observational +data, and the very good agreement between theory and observation is noteworthy. On +the other hand, we also compare theory and observation in Table II and Figure (C2), +where now 𝑁̃ cumulative represents the number of craters of all ages, 1𝑚𝑦 ≤ 𝜏 ≤ +2,500𝑚𝑦, with diameters greater than or equal to 𝐷. Again, the theoretical 𝑁̃(𝐷) is in +very good agreement with the observations for 𝐷 ≥ ~20𝑘𝑚, although not so good for +𝐷 ≤ ~20𝑘𝑚, which is as expected due to the undercounting of craters of these sizes. + + +Table l +𝜏(𝑚𝑦) +𝑁̃[𝜏, 𝐷 ≥ 20𝑘𝑚 ] +Observation +1 +33.14 +33 +10 +32.00 +32 +20 +30.80 +31 +40 +28.62 +29 +50 +27.62 +28 +100 +23.40 +24 +150 +20.24 +20 +200 +17.80 +17 +300 +14.20 +13 +400 +11.70 +10 +600 +8.50 +8 +800 +6.50 +5 +1000 +5.00 +5 +1200 +3.89 4 +1400 +2.99 +3 +1600 +2.25 +3 +1800 +1.62 +2 +2000 +1.08 +1 + + + +FIGURE (C1): 𝐿𝑜𝑔[𝑁] +̃ 𝑣𝑠 𝐿𝑜𝑔[𝜏 ≡ 𝐴𝑔𝑒], for all diameters 𝐷 ≥ 20𝑘𝑚. See Table l. + +Table ll +D +𝑁̃[𝐷, 1𝑚𝑦 ≤ 𝜏 ≤ 2,500𝑚𝑦 ] +Observation +1 +166.00 +121 +2 +165.00 +118 +4 +137.00 +99 +8 +82.60 +72 +16 +42.40 +37 +20 +33.14 +33 +32 +18.18 +16 +45 +10.37 +10 +64 +4.79 +5 +91 +1.82 +2 +128 +0.62 +1 + + + +Log NaccAge,D>20km +1.4 F +1.2 E +1.0 F +180 +0.6 E +0.4 E +0.2 E +0.5 +Log Age +1.0 +1.5 +2.0 +2.5 +3.0 + +FIGURE (C2): [𝐿𝑜𝑔[𝑁̃] vs. 𝐿𝑜𝑔[𝐷𝐴𝑐𝑐 ≡ 𝐷], for all ages between 1𝑚𝑦 ≤ 𝜏 ≤ 2,500𝑚𝑦 (Table II). + + + + + + + + + + + + + + + + + + + +Log NaccD +2.0 E +1.5 上 +1.0 F +0.5 +0.5 +1.0 +1.5 +2.0References +1. Bruckman, W.F., Ruiz, A., Ramos, E. (2012). +Earth and Mars Crater Size Frequency Distribution +and Impact Rates: Theoretical and Observational +Analysis; arXiv:1212.3273(astro-ph) +2. Barlow, N.G. (1988). Icarus 75, 285. +3. Robbins, S.J., and Hynex, B.M. (2012). Global +Database of Mars Impact Craters ≥ 1𝑘𝑚.; Journal +of Geophysical Research: Planets 117(E5) +4. Bruckman, W.F. (2019). Researchgate preprint. +DOI: 10.13140/R.G.2.2.33363.43047 +5. Garvin, J.B. (2002). Lunar and Planetary Science 33, 1255. +6. Boyce, J.M., Garbeil, H. (2007). Geophysical Research Letters 34(16). +7. Planetary and Space Science Centre (PASSC), Earth Impact Database +(http://www.passc.net/EarthImpactDatabase/ +8. Grieve and Shoemaker (1994). The Record of Past Impacts on Earth. In: Hazards +Due To Comets And Asteroids, T. Gehrels, ed., The University of Arizona Press. +9. Poveda, A., Herrera, M.A., Garcia, J.L., Curioca, K. (1999) Planetary and Space +Science 47, 679. +10. Silber, E.A., Revelle, D.O., Brown, P.G., Edwards, W.N. (2009). Journal of +Geophysical Research 114, E08006. +11. Oberst, J., A., Christou, A., Suggs, R., Moser, D., Daubar, I.J., McEwenf, A.S., +Burchell, M., Kawamura, T., Hiesinger, H., Wünnemann, K., Wagner, R., Robinson, +M.S. (2012); The Present day Flux of Large Meteoroids on the Lunar Surface. A +synthesis of Models and Observational Techniques. Planetary and Space Science 74, +179–193 +12. Suggs, R.M., Muser, D.E., Cooke, W.J., Suggs, R.J. (2014). The Flux of Kilogram- +Sized Meteoroids From Lunar Impact Monitoring. Icarus April 2014. + + + + diff --git a/Z9FAT4oBgHgl3EQf4R6K/content/tmp_files/load_file.txt b/Z9FAT4oBgHgl3EQf4R6K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..96f0ba6e86eb716d74bdda52f858edec84b98dcb --- /dev/null +++ b/Z9FAT4oBgHgl3EQf4R6K/content/tmp_files/load_file.txt @@ -0,0 +1,435 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf,len=434 +page_content='TITLE Using Gamma Functions in the Mathematical Formulation of the Impact Crater Size-Age Frequency Distribution on Earth and Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Author: William F Bruckman Abstract A review of a mathematical formulation that describes the number of impact craters as function of diameter and time of formation is presented, where the use of Gamma functions is emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The application of this formalism for the description of the impact crater data of Planets Earth and Mars is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Introduction When solving differential or integral equations an ideal outcome is to express the solution in terms of elementary or special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' In that case the mathematical and physical interpretation of the solutions is clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Moreover, with the use of algebraic computing, the comparison of the prediction of theoretical models with the observational data is greatly facilitated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' This paper will consider work in reference1 (Earth and Mars Crater Size Frequency Distribution and Impact Rates: Theoretical and Observational Analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' William Bruckman, Abraham Ruiz, and Elio Ramos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Arxiv: 1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3273), which presented a theoretical formulation describing impact crater data on Earth and Mars, giving the number of craters as functions of diameter, and time of formation, successfully reproducing the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The revision will emphasize the presentation of the solutions of the models in terms of Gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' General Considerations Impact craters, of a given diameter 𝐷, are formed at a certain rate 𝛷, and are also depleted, as they get older, by a variety of processes, at a rate proportional to their already existing number of craters, 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Hence, the number of craters eliminated in the time interval dt can be express as 𝐶𝑁𝑑𝑡, where 𝐶 is a parameter representing the rate of elimination per crater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' On the other hand, in this time interval we also have that the number of craters produced by impacts is 𝛷𝑑𝑡, and thus the net change in the number of crater numbers, 𝑑𝑁, is given by 𝑑𝑁 = 𝛷𝑑𝑡 − 𝐶𝑁𝑑𝑡 = ( 𝛷 𝐶 − 𝑁 )𝐶𝑑𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1) This equation is expected to represent well the observational data if the number of craters is large enough to justify the assumptions that analytical mathematical continuity is a good approximation to the discrete and probabilistic nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' We see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1) that 𝑁= constant implies that 𝑁 = 𝛷 𝐶 = 𝛷𝜏𝑚, (2) 𝜏𝑚 ≡ 1/𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (3) In this situation (saturation) the number of craters produced by impacts is equal to the number of craters eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The dimension of 𝜏𝑚 is time, and we will see later in this section that this time is related to the concept of “craters mean life.” Equation (1) was integrated in reference (1) to obtain 𝑁(𝐷, 0, 𝜏) = ∫ {𝛷(𝐷, 𝜏`) 𝜏 0 𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (4) 𝐶̅ ≡ ∫ 𝐶𝑑𝜏` 𝜏 0 𝜏 , (5) where 𝐶̅ is the time average of 𝐶, and 𝑁(𝐷, 0, 𝜏) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (4) denotes the number of craters of diameter 𝐷, per bin size, observed at the present time (𝜏` = 0), with age younger than 𝜏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Accordingly, defining the term “per bin”, we have that the integral 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫ 𝑁(𝐷, 0, 𝜏)𝑑𝐷 𝐷𝑓 𝐷𝑖 , (6) gives the total number of craters with diameters in the interval between 𝐷𝑖 and 𝐷𝑓, observed at the present time, with age of formation younger than 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Also, 𝛷(𝐷, 𝜏) is the rate of meteorite impacts, per bin, forming craters of diameter 𝐷 at time 𝜏, so that 𝛷𝐶: 𝛷𝐶(𝐷𝑖, 𝐷𝑓, 𝜏) = ∫ {𝛷(𝐷, 𝜏) 𝐷𝑓 𝐷𝑖 }𝑑𝐷, (7) Is the cumulative impact rate of formation of craters with diameters in the interval between 𝐷𝑖 and 𝐷𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' For instance, if 𝐷𝑓 → ∞, which is of common use, the above integral is the total cumulative impact rate of formation of craters with diameters larger than 𝐷𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Equations (6) and (4) can be generalized so that the lower 𝜏 limit is different from zero: 𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) ≡ ∫ 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷 𝐷𝑓 𝐷𝑖 , (8) where 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) = ∫ {𝛷(𝐷, 𝜏`) 𝜏𝑓 𝜏𝑖 𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (9) Thus, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (8) and (9) refer to craters with ages between 𝜏𝑖 and 𝜏𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Further discussion and applications of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (8) to the Earth’s crater record will be continued in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' For the planet Mars, however, we will be applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (4) in the next section, but now continue its interpretation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Since the quantity 𝛷(𝐷, 𝜏`)𝑑𝜏`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Is the number of craters formed at time 𝜏`, during the interval 𝑑𝜏`, and the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (4): 𝛷(𝐷, 𝜏`)𝑑𝜏`𝐸𝑥𝑝[−𝐶̅𝜏`], is the number of these craters, of age 𝜏` , that remain at the present time, then the expression 𝐸𝑥𝑝[−𝐶̅𝜏`] represents the fraction of these formed craters that survive after the time 𝜏`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' It is then usual to call the inverse of 𝐶̅ “the mean life”: 𝜏𝑚𝑒𝑎𝑛, 𝜏𝑚𝑒𝑎𝑛 ≡ 1 𝐶̅ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 1/𝜏𝑚𝑒𝑎𝑛 = 𝐶̅ ≡ ∫ 𝐶𝑑𝜏` 𝜏 0 𝜏 = ∫ (1 𝜏𝑚)𝑑𝜏` 𝜏 0 𝜏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (10) Thus, in this context 𝜏𝑚𝑒𝑎𝑛 can be viewed as the mean life of craters of diameter 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Also, this interpretation suggests thinking of 𝛷 as a probability of impact, rather than an impact flux, thus emphasizing the statistical nature of the impacts of asteroids and comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Conversely, if we start with the definition of 𝐸𝑥𝑝[−𝐶̅𝜏`] as the fraction of craters surviving after the interval 𝜏` from their formation, then we can construct Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (4) to represent the sum of all the contributions, to the present number, for all times 𝜏` younger than 𝜏, and then find that 𝑁 satisfies the differential equation implied in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Consider the following definition: 𝑇(𝐷, 𝜏, ) ≡ ∫ 𝐶𝑑𝜏` 𝜏 0 = 𝐶̅ 𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (11) Hence 𝑇 is a dimensionless time that measures the numbers of mean-life in an interval 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (11) it follows that 𝑑𝑇/𝑑𝜏 = 𝐶(𝐷, 𝜏) , (12) where 𝐷 is considered here as a constant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Since crater elimination is a decay process, where 𝐶 is strictly positive, we have 𝑑𝑇/𝑑𝜏 > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (13) Consequently, the function 𝑇(𝐷, 𝜏, ) can be inverted to express 𝜏 as a function of 𝑇 and 𝐷: 𝜏(𝐷, 𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Likewise, 𝐶 and 𝛷 are each expressible as functions of 𝑇 and 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' We can then rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (4), using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (11), (12) and (3), in the form 𝑁(𝐷, 0, 𝜏) = ∫ {𝛷(𝐷, 𝜏`) 𝜏 0 𝐸𝑥𝑝[−𝐶̅𝜏`]}𝑑𝜏` = ∫ {( 𝛷 𝐶) 𝑇 0 𝐸𝑥𝑝[−𝑇`]}𝑑𝑇` = ∫ {(𝛷𝜏𝑚) 𝑇 0 𝐸𝑥𝑝[−𝑇`]}𝑑𝑇`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (14) where, in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (14), 𝛷 𝐶 = 𝛷𝜏𝑚 is considered now a function of 𝑇, and the parameter 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' For instance, if 𝛷𝜏𝑚 is a sum like 𝛷𝜏𝑚 = 𝛴𝑎𝑠𝑇𝑠 , 𝑎𝑠 and 𝑠 are independent of 𝑇, (15) then we have, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (14), 𝑁(𝐷, 0, 𝑇) = 𝛴𝑎𝑠 ∫ {𝑇`𝑠 𝑇 0 𝐸𝑥𝑝[−𝑇`]}𝑑𝑇` = 𝛴𝑎𝑠 𝛾(𝑠 + 1, 𝑇) , (16) where the lower incomplete gamma function notation was used above: 𝛾(𝑠 + 1, 𝑇) = ∫ {𝑇`𝑠 𝑇 0 𝐸𝑥𝑝[−𝑇`]}𝑑𝑇`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (17) If 𝑠 is a whole number, as in a Taylor-Maclaurin series, we can also write 𝛾(𝑠 + 1, 𝑇) = 𝑠!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1 - 𝑒−𝑇 ∑ 𝑇𝑘 𝑠 𝑘=0 /𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (18) This is our first encounter with the use of gamma functions expressing the number of craters as a function of diameter and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' We will see further use of gamma functions when considering applications to Earth’s impact crater data in Section (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' We will now focus our attention on applications of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (14) to the planet Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Applications to the Crater-Size Frequency Distribution of Mars It was discussed in Section 2 that the product 𝛷 𝐶 = 𝛷𝜏𝑚 represents the value of 𝑁 when the production and the elimination of craters are equal and 𝑑𝑁 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Then in a steady state situation we will have 𝑁 = 𝛷𝜏𝑚 = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' However, in general, 𝛷𝜏𝑚 could depend on time, since both 𝛷 and 𝜏𝑚 could depend on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' On the other hand, since 𝐶 is by definition the rate of crater elimination per number of craters, we have then that 𝜏𝑚 ≡ 1/𝐶 is strongly influenced by the elimination of old craters due to impacts forming new craters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Therefore, an increase or decrease of 𝛷 would be correlated with an increase or decrease of 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Consequently, if obliterations by impacts are important, the changes in time of 𝛷 𝐶 = 𝛷𝜏𝑚 are smoothed out relative to the individual changes in time of 𝛷, 𝐶,or 𝜏𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' In such a heuristic and realistic situation, a model in which it is assumed that 𝛷 𝐶 = 𝛷𝜏𝑚 is constant should be a good representation of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' In this case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (14) becomes 𝑁 = 𝛷𝜏𝑚(1 - 𝑒−𝑇) (19) With the above simple model we were able to represent (reference 1) remarkably well the pioneering Mars crater database catalog of Barlow (1988), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Also, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 2 compares the model with the more recent Mars data catalog of Robbins and Hynek (2012), and also the model is in very good agreement with observations (Bruckman 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The values of 𝛷𝜏𝑚 and 𝑇 for Barlow’s model are 𝛷𝜏𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='43𝑥105 𝐷1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='8 , (20) 𝑇 = 𝐶̅𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='48𝑥104 𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 , (21) and then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (19) becomes 𝑁 = 𝛷𝜏𝑚(1 - 𝑒−𝑇) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='43𝑥105 𝐷1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='8 (1 − 𝐸𝑥𝑝[− 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='48𝑥104 𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 ]), (22) where the unit of 𝐷 is kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' It can also be shown (Appendix A), using the assumption that 𝛷𝜏𝑚 is independent of 𝑇, that 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏 𝜏 0 = 𝛷𝜏𝑚𝑇 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='55𝑥109 𝐷4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 , (23) where 𝛷̅ is the time average of 𝛷, and 𝛷̅𝜏 is the total number of craters, of diameter 𝐷, per bin, created over the total time of production 𝜏 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The corresponding expression for the number of craters created with diameters in the interval between 𝐷𝑖 and 𝐷𝑓 is then : 𝜏𝛷̅𝐶(𝐷𝑖, 𝐷𝑓, 𝜏) = ∫ 𝛷̅𝜏 𝐷𝑓 𝐷𝑖 𝑑𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='55/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3)109( 1 𝐷𝑖3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 - 1 𝐷𝑓3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (24) It is common to take the upper limit 𝐷𝑓 to be infinite to obtain the total number of craters produced larger than 𝐷𝑖 : 𝜏𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='076𝑥109( 1 𝐷𝑖3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3) , (25) or 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='076𝑥109/𝜏)( 1 𝐷𝑖3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3) , (26) where 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) is the time average of the cumulative impact rate for the formation of craters larger than 𝐷𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' For instance, it is interesting to note that for 𝐷𝑖 = 1 km, approximately 109 such impacts were produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Therefore, assuming that the total time of crater production 𝜏 was 3000 to 4000 million years, we get an average of one impact, making craters larger than 1 km, approximately every three to four years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Since the energy associated to impacts with a diameter of 1 km is close to one megaton, this result is of concern for explorations of Mars, assuming that the present impact flux average is comparable to that given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' It is expected that also the corresponding impact rate for Earth has, similar to Mars, a crater diameter dependency of the form 1 𝐷𝑖3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3, and indeed, we found that for our planet such a relation is consistent with the observations (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Let us continue our analysis of the implications of the above model, by looking at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (21), rewritten in the form 𝜏𝑚𝑒𝑎𝑛 𝜏 = 𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='48𝑥104 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (27) An interesting interpretation of the above equation (Reference 1) is that it represents a proportionality relation between the mean life, of a crater of diameter 𝐷, and the initial volume of this crater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' This conclusion is based on observations on Mars that established that the initial depths of pristine craters are proportional to 𝐷𝑘/2, with 𝑘 ≈ 1, and, consequently, the expected initial volumes for these craters are proportional to 𝐷2𝐷𝑘/2 ≈ 𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' For instance, Garvin (2002) gives 𝑘 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='98, while Boyce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2007) give 𝑘 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Furthermore, from the application to Earth of the above formalism, to be discussed in the next section, it was concluded that craters in our planet also have their mean-life proportional to ≈ 𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Thus, we have that the relation 𝜏𝑚𝑒𝑎𝑛 proportional to the crater initial volume is not only intuitively appealing, but also helps us understand why we have similar 𝐷 exponents in the 𝜏𝑚𝑒𝑎𝑛 for Earth and Mars, notwithstanding these planets contrasting geological evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' FIGURE (1): Log-Log plot of number of craters per bin, 𝑁(𝐷) 𝑣𝑠 𝐷 based on Barlow’s Mars catalog (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The number 𝑁(𝐷) is calculated by counting the number of craters in a bin ∆𝐷 = 𝐷𝑅 − 𝐷𝐿, and then dividing this number by the bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The point is placed at the mathematical average of 𝐷 in the bin: (𝐷𝑅 + 𝐷𝐿)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The bin size is ∆𝐷 = (√2 − 1)𝐷𝐿, so that 𝐷𝑅 𝐷𝐿 = √2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The curve is from the model implied by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' We see that the theoretical curve shown differs significantly from the observed data for 𝐷 less than about 8𝑘𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' However, according to Barlow, the empirical data undercounts the actual crater population for 𝐷 less than 8𝑘𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' However, more recent Mars crater data by Robbins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2012) was used to update the observations, yielding similar results to the model in Figure 1, but extending the range to craters with diameters down to 1 km (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' FIGURE (2): Log-Log plot of 𝑵(𝐷), 𝑣𝑠 𝐷(km), based on the Mars catalog of Robbins et al (2012), (Bruckman (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Bin size is ∆𝐷 = ( 21/6 − 1)𝐷𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Note that for 𝐷 > ~300 𝑘𝑚, the data points are above the curve of the analytic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' However, we expect that the analytical model will be less reliable when the number of craters in a given bin is so small that statistical continuous models break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Moreover, another source of discrepancy could be that these very large craters were being formed at high proportions at older times 𝜏, thus perhaps belonging to the so-called heavy bombardment era, characterized by a much higher impact flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 10 20 50 100 200 500 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='1 1 10 100 1000 N Log[N(D)] 3 E 2 E 1上 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 Log[P] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Applications to Planet Earth The number of identified impact craters on Earth is close to 190 (Planetary and Space Science Center: PASSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='com), while, in contrast, the number of craters used for Mars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1) was 42,284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Therefore, in the analysis of Earth’s crater data it is convenient to use the cumulative number of impacts of craters, 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (6), instead of 𝑁(𝐷, 0, 𝜏), defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Furthermore, for 𝑁(𝐷, 0, 𝜏), the simplified expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (19) will be used, since it reproduced the Martian impact data very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Thus we have 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫ 𝑁(𝐷, 0, 𝜏)𝑑𝐷 𝐷𝑓 𝐷𝑖 = ∫ 𝛷𝜏𝑚(1 – 𝑒−𝑇) 𝑑𝐷 = 𝐷𝑓 𝐷𝑖 ∫ 𝛷𝜏𝑚𝑑𝐷 − ∫ 𝛷𝜏𝑚𝑒−𝑇𝑑𝐷 𝐷𝑓 𝐷𝑖 𝐷𝑓 𝐷𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (28) In addition, let us assume that 𝛷𝜏𝑚 = 𝐻 𝐷𝑚` , (29) 𝑇 = 𝐶̅𝜏 = 𝐵𝜏 𝐷𝑝 , (30) 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` 𝜏 0 = 𝛷𝜏𝑚𝑇 = 𝐴𝜏 𝐷𝑚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (31) where 𝐻, 𝑚`, 𝐵, 𝑝, 𝐴 𝑎𝑛𝑑 𝑚 are independent of 𝐷, and, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (29), (30) and (31), 𝑚 = 𝑚` + 𝑝 , (32) 𝐴 = 𝐻𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (33) Equations (29), (30), and (31) are a generalization for Earth of the corresponding equations, (20), (21) and (23), describing the crater distribution for Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' For Mars, we have 𝐻 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='43𝑥105, 𝐵𝜏 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='48𝑥104 , and 𝐴𝜏 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='55𝑥109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' However, for our planet these values will have to be redetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Also, the exponents 𝑚 and 𝑝 should come out from the fitting to Earth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' As was discussed in previous section, a value of 𝑚 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3, in the exponent of 𝐷 of the impact flux 𝛷̅ is also consistent with the Earth observational impact rate data (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The value 𝑝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 is also consistent with the Earth observations, to be discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' After the substitution of the expressions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (29), (30), and (31) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (28) the first integral in the right-hand side is elementary, hence, we will turn our attention to the second integral: − ∫ 𝛷𝜏𝑚𝑒−𝑇𝑑𝐷 𝐷𝑓 𝐷𝑖 = − ∫ 𝐻 𝐷𝑚` {𝐸𝑥𝑝 [ −𝐵𝜏 𝐷𝑝 ]} 𝑑𝐷 𝐷𝑓 𝐷𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (34) To emphasize that the variable of integration is now 𝐷, while 𝜏 is a fixed parameter, we rename 𝑇 as 𝑈: 𝑇 = 𝑈 = 𝐵𝜏 𝐷𝑝 , (35) or 𝐷 = [ 𝐵𝜏 𝑈 ]1/𝑝 , (36) from which, differentiating with respect to 𝐷, holding 𝜏 fixed, 𝑑𝐷 = −[𝐵𝜏 ] 1 𝑝[ 𝑈 ] −1 𝑝 −1𝑑𝑈/𝑝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (37) Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (35), (36), and (37) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (34) we get − ∫ 𝛷𝜏𝑚𝑒−𝑇𝑑𝐷 𝐷𝑓 𝐷𝑖 = {𝐻/(𝑝[𝐵𝜏 ]𝑛)} ∫ 𝑈𝑛−1{𝐸𝑥𝑝[−𝑈]}𝑑𝑈 𝑈𝑓 𝑈𝑖 = {𝐻/(𝑝[𝐵𝜏 ]𝑛)}𝛤[𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 𝑈𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='𝑈𝑓],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='(38) where 𝑛 ≡ ( 𝑚` − 1)/𝑝 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (39) 𝑈𝑖 = 𝐵𝜏 𝐷𝑖𝑝 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (40) 𝑈𝑓 = 𝐵𝜏 𝐷𝑓𝑝 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (41) and 𝛤[𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 𝑈𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='𝑈𝑓] = ∫ 𝑈𝑛−1{𝐸𝑥𝑝[−𝑈]}𝑑𝑈 𝑈𝑓 𝑈𝑖 (42) is the generalized incomplete gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Consequently, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (28) in the form 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏) ≡ ∫ 𝐻 𝐷𝑚` 𝑑𝐷 + { 𝐷𝑓 𝐷𝑖 𝐻/(𝑝[𝐵𝜏 ]𝑛)} 𝛤[𝑛, 𝑈𝑖,𝑈𝑓] (43) The above integral represents the number of craters with diameters in the interval between 𝐷𝑖 and 𝐷𝑓, that are younger than 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Hence, the number of craters formed with ages between 𝜏𝑖 and 𝜏𝑓 is 𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) ≡ ∫ 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷 𝐷𝑓 𝐷𝑖 = 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏𝑓) - 𝑁̃(𝐷𝑖, 𝐷𝑓, 0, 𝜏𝑖) = {𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛)} 𝛤 [𝑛, 𝐵𝜏𝑓 𝐷𝑖𝑝 , 𝐵𝜏𝑓 𝐷𝑓𝑝] − {𝐻/(𝑝[𝐵𝜏𝑖 ]𝑛)} 𝛤 [𝑛, 𝐵𝜏𝑖 𝐷𝑖𝑝 , 𝐵𝜏𝑖 𝐷𝑓𝑝] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (44) Here, if 𝐵 is a function of time, it should be evaluated at the corresponding 𝜏𝑖 or 𝜏𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Another useful concept is the statistical mean of a function of 𝐷: 𝑓(𝐷), which is defined using 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓), as follows 𝑓̅ = ∫ 𝑓𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 𝐷𝑓 𝐷𝑖 /{∫ 𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 𝐷𝑓 𝐷𝑖 }=∫ 𝑓𝑁(𝐷, 𝜏𝑖, 𝜏𝑓)𝑑𝐷 𝐷𝑓 𝐷𝑖 /{𝑁 ̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (45) For instance, if 𝑓 = 𝐷 we get, from definition (45), the average diameters of craters with diameters and ages in the intervals 𝐷𝑖 ≤ 𝐷 ≤ 𝐷𝑓, and 𝜏𝑖 ≤ 𝜏 ≤ 𝜏𝑓 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' In this case it follows that the numerator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (45) is ∫ 𝐷𝑁(𝐷, 𝜏𝑖, 𝜏𝑓) 𝑑𝐷 = 𝐷𝑓 𝐷𝑖 {𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛`)} 𝛤 [𝑛`, 𝐵𝜏𝑓 𝐷𝑖𝑝 , 𝐵𝜏𝑓 𝐷𝑓𝑝] − {𝐻(𝑝[𝐵𝜏𝑖 ]𝑛`)}𝛤 [𝑛`, 𝐵𝜏𝑖 𝐷𝑖𝑝 , 𝐵𝜏𝑖 𝐷𝑓𝑝], (46) where 𝑛` ≡ 𝑚`−2 𝑝 = 𝑛 − 1/𝑝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (47) Hence 𝐷̅ = [ 1 𝑁̃(𝐷𝑖,𝐷𝑓,𝜏𝑖,𝜏𝑓)][{𝐻/(𝑝[𝐵𝜏𝑓 ]𝑛`)} 𝛤 [𝑛`, 𝐵𝜏𝑓 𝐷𝑖𝑝 , 𝐵𝜏𝑓 𝐷𝑓𝑝] − {𝐻/(𝑝[𝐵𝜏𝑖 ]𝑛`)} 𝛤 [𝑛`, 𝐵𝜏𝑖 𝐷𝑖𝑝 , 𝐵𝜏𝑖 𝐷𝑓𝑝]] (48) The above expression was adapted and applied to the Earth crater data in reference (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The value of 𝑝 was determined by the best fitting of the data to the model given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (48), and yielded a value similar to that for Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' As stated, this is interpreted to be the result of the proportionality of 𝜏𝑚𝑒𝑎𝑛 to the initial volume of craters, and that this volume is in turn proportional to 𝐷𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' From this fitting to observation also came an approximate value for Earth’s parameter 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The expression 𝑁̃(𝐷𝑖, 𝐷𝑓, 𝜏𝑖, 𝜏𝑓) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (44), was also used in reference (1) to describe the number of Earth’s craters as a function of diameter and age, as illustrated in figures C1 and C2 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The values 𝑝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5, 𝑚 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3, and 𝑚` = 𝑚 − 𝑝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='8 were assumed since they were observationally justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The value of 𝐻 = 𝐴/𝐵 was also needed, and, since 𝐵 was estimated from observations of 𝐷̅, then the value of 𝐴 remained to be estimated, as described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' A remarkable agreement of the model with observations was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Appendix A The number of impacts, during the time 𝜏, producing craters of diameter 𝐷, per bin, can be expressed as 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` = ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 𝜏 0 𝜏 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' A1 Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (3) and (12), we get 𝑑𝑇/𝑑𝜏 = 𝐶(𝐷, 𝜏) = 1 𝜏𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' A2 We can then rewrite the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (A1) in the form ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 = 𝜏 0 ∫ 𝛷𝜏𝑚𝑑𝑇` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 𝑇 0 A3 If furthermore 𝛷𝜏𝑚 is independent of 𝑇 we have ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 = 𝜏 0 ∫ 𝛷𝜏𝑚𝑑𝑇` = 𝑇 0 𝛷𝜏𝑚 ∫ 𝑑𝑇` = 𝛷𝜏𝑚𝑇 𝑇 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' A4 Therefore, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (A1) and (A4), 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` = ∫ 𝛷𝜏𝑚𝑑𝜏`/𝜏𝑚 𝜏 0 𝜏 0 = 𝛷𝜏𝑚𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' A5 Note also that, since 𝑇 = 𝐶̅𝜏 = 𝜏/𝜏𝑚𝑒𝑎𝑛 , A6 from (A5) we get 𝛷̅𝜏𝑚𝑒𝑎𝑛 = 𝛷𝜏𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' A7 Appendix B Let us investigate the observational implications of the assumption of an average impact flux for Earth given by 𝛷̅𝜏 = ∫ 𝛷𝑑𝜏` 𝜏 0 = 𝐴𝜏 𝐷4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3, B1 which implies the following cumulative impact flux 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) = ∫ 𝛷̅ ,∞ 𝐷𝑖 𝑑𝐷 = 𝐴/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 𝐷3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 , B2 where we drop the 𝑖 sub index from 𝐷, in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The value of 𝐴 can be estimated for Earth from the result of Grieve and Shoemaker (1994) for 𝐷 = 20𝑘𝑚: 𝛷̅𝐶(20𝑘𝑚, ∞, 𝜏) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5∓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='7)10−9 (𝑚𝑦)𝑘𝑚2 4𝜋𝑅2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='8[ 1∓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='50 𝑚𝑦 ], B3 where 𝑅 is the Earth’s radius, and 𝑚𝑦 is million years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B2), evaluated at 𝐷 = 20𝑘𝑚, with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B3) we obtain 𝐴 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='24[1∓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='50] (20)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 𝑚𝑦 , B4 and thus 𝛷̅𝐶(𝐷𝑖, ∞, 𝜏) ≡ 𝛷̅𝐶 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='8[ 1∓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='50 𝑚𝑦 ](20 𝐷 ⁄ )3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 B5 This equation is a generalization of the result of Grieve and Shoemaker (1994), which gives the Earth’s impact rate for the formation of craters with diameters larger than 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' It incorporates the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 exponent on 𝐷 that we deduced from the model and observations from Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The diameter of a crater corresponds to an energy, 𝐸, associated to the impact, and hence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B5) can be re-expressed as (reference 1) 𝛷̅𝐶(𝐸) = [1∓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5𝑦 𝐸0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='86, B6 where 𝐸 is in megatons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Equation (B6) gives similar predictions to those of Poveda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The predictions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B6) are also in agreement with Silber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2009), that, for impacts with energies larger than a megaton, gives one Earth impact about every 15 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' It is interesting to note that, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B6), events like the 2013 Chelyabinsk meteorite of energy of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 megatons are predicted to happen with a periodicity near one every 8/(1 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5) years, so that this type of event is expected to be repeated in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Observations in the last few decades of lunar meteorites, called Lunar Flashes, provide a direct determination of the impact rate, at these low range of energies (see for example Oberst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2012), and Suggs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' For instance, Oberst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2012) interpreted data of lunar flashes, and concluded a rate of 10−3 impacts per 𝑘𝑚2 per year, for energies ≥ ~8𝑥10−6 kilotons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' This result, translated to the total Earth`s surface area, becomes approximately 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='1𝑥105 impacts per year for these energies, while from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B6) we get about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3[1 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5]𝑥105 impacts per year, which is consistent with the above result for lunar flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Appendix C To reduce the uncertainties due to undercounting in the Earth crater data we selected the following regions for the study in reference 1: (a) Continental United States (b) Canada up to the Arctic Circle (c) Europe (d) Australia The crater data is taken from The Planetary and Space Science Centre (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='passc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Then, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (44), instead of using for the total Earth’s impact flux 𝛷̅ = (1/𝜏) ∫ 𝛷𝑑𝜏` 𝜏 0 = 𝐴 𝐷4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3, C1 we used for our study the more accurate impact flux corresponding to the area under consideration above in a,b,c,d, which is given by 𝛷̅𝑎𝑐𝑐 = 𝐴𝑎𝑐𝑐 𝐷4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3, C2 where 𝐴𝑎𝑐𝑐 ≡ 𝐴 𝐴𝑟𝑒𝑎 𝑈𝑛𝑑𝑒𝑟 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝐸𝑎𝑟𝑡ℎ`𝑠 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 , C3 where 𝐴 is given, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (B4), by 𝐴 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='24[1 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='50] (20)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3 𝑚𝑦 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='82)105[1 ∓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5]/𝑚𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' C4 Accordingly, 𝐻 = 𝐴/𝐵 becomes 𝐻𝑎𝑐𝑐 = 𝐴𝑎𝑐𝑐 𝐵 , with 𝐵 estimated from the curve 𝐷̅ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' crater age , given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (48), fitting to the Earth’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Therefore, we can write the theoretical 𝑁̃ with no free parameters, and compare it with the observations, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' We do this first in table (I) and Figure (C1), for craters with 𝐷 ≥ 20𝑘𝑚 and cumulative age starting with 𝜏 = 1𝑚𝑦 up to 𝜏 = 2,000𝑚𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Furthermore, we put 𝜏𝑓 = 2,500𝑚𝑦 and 𝐷𝑓 = 300𝑘𝑚, since all craters in the field of study are within this bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' This theoretical curve, 𝑁̃(𝜏), is then compared with the corresponding observational data, and the very good agreement between theory and observation is noteworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' On the other hand, we also compare theory and observation in Table II and Figure (C2), where now 𝑁̃ cumulative represents the number of craters of all ages, 1𝑚𝑦 ≤ 𝜏 ≤ 2,500𝑚𝑦, with diameters greater than or equal to 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Again, the theoretical 𝑁̃(𝐷) is in very good agreement with the observations for 𝐷 ≥ ~20𝑘𝑚, although not so good for 𝐷 ≤ ~20𝑘𝑚, which is as expected due to the undercounting of craters of these sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Table l 𝜏(𝑚𝑦) 𝑁̃[𝜏, 𝐷 ≥ 20𝑘𝑚 ] Observation 1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='14 33 10 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='00 32 20 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='80 31 40 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='62 29 50 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='62 28 100 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='40 24 150 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='24 20 200 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='80 17 300 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='20 13 400 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='70 10 600 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='50 8 800 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='50 5 1000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='00 5 1200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='89 4 1400 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='99 3 1600 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='25 3 1800 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='62 2 2000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='08 1 FIGURE (C1): 𝐿𝑜𝑔[𝑁] ̃ 𝑣𝑠 𝐿𝑜𝑔[𝜏 ≡ 𝐴𝑔𝑒], for all diameters 𝐷 ≥ 20𝑘𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' See Table l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Table ll D 𝑁̃[𝐷, 1𝑚𝑦 ≤ 𝜏 ≤ 2,500𝑚𝑦 ] Observation 1 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='00 121 2 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='00 118 4 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='00 99 8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='60 72 16 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='40 37 20 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='14 33 32 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='18 16 45 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='37 10 64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='79 5 91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='82 2 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='62 1 Log NaccAge,D>20km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='4 F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='2 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 F 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='6 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='4 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 Log Age 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 FIGURE (C2): [𝐿𝑜𝑔[𝑁̃] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 𝐿𝑜𝑔[𝐷𝐴𝑐𝑐 ≡ 𝐷], for all ages between 1𝑚𝑦 ≤ 𝜏 ≤ 2,500𝑚𝑦 (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Log NaccD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 上 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='0References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Bruckman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Ruiz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Ramos, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Earth and Mars Crater Size Frequency Distribution and Impact Rates: Theoretical and Observational Analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='3273(astro-ph) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Barlow, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Icarus 75, 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Robbins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', and Hynex, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Global Database of Mars Impact Craters ≥ 1𝑘𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Journal of Geophysical Research: Planets 117(E5) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Bruckman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Researchgate preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='13140/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='33363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='43047 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Garvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Lunar and Planetary Science 33, 1255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Boyce, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Garbeil, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Geophysical Research Letters 34(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Planetary and Space Science Centre (PASSC), Earth Impact Database (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='passc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='net/EarthImpactDatabase/ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Grieve and Shoemaker (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The Record of Past Impacts on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' In: Hazards Due To Comets And Asteroids, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Gehrels, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', The University of Arizona Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Poveda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Herrera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Garcia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Curioca, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (1999) Planetary and Space Science 47, 679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Silber, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Revelle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Brown, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Edwards, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Journal of Geophysical Research 114, E08006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Oberst, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Christou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Suggs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Moser, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Daubar, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', McEwenf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Burchell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Kawamura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Hiesinger, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Wünnemann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Wagner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Robinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The Present day Flux of Large Meteoroids on the Lunar Surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' A synthesis of Models and Observational Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Planetary and Space Science 74, 179–193 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Suggs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Muser, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Cooke, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=', Suggs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' The Flux of Kilogram- Sized Meteoroids From Lunar Impact Monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} +page_content=' Icarus April 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FAT4oBgHgl3EQf4R6K/content/2301.08725v1.pdf'} diff --git a/aNE1T4oBgHgl3EQfcwSy/content/tmp_files/2301.03188v1.pdf.txt b/aNE1T4oBgHgl3EQfcwSy/content/tmp_files/2301.03188v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..37d051dfbe52550bc4e1be2dca05f3694a56fa02 --- /dev/null +++ b/aNE1T4oBgHgl3EQfcwSy/content/tmp_files/2301.03188v1.pdf.txt @@ -0,0 +1,998 @@ +Linear optical quantum computation with frequency-comb qubits and passive devices +Tomohiro Yamazaki,1, 2, ∗ Tomoaki Arizono,1 Toshiki Kobayashi,1, 2 Rikizo Ikuta,1, 2 and Takashi Yamamoto1, 2 +1Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan +2Center for Quantum Information and Quantum Biology, +Osaka University, Toyonaka, Osaka 560-8531, Japan +We propose a linear optical quantum computation scheme using time–frequency degree of freedom. +In this scheme, a qubit is encoded in single-photon frequency combs, and manipulation of the +qubits is performed using time-resolving detectors, beam splitters, and optical interleavers. This +scheme does not require active devices such as high-speed switches and electro-optic modulators +and is robust against temporal and spectral errors, which are mainly caused by the detectors’ +finite resolution. We show that current technologies almost meet the requirements for fault-tolerant +quantum computation. +Introduction.— Photons and their manipulation using +linear optics play an indispensable role in quantum in- +formation processing [1, 2]. There has been considerable +interest in the choice of the degrees of freedom (DoF) of +photons as quantum information carriers [3–7]. The use +of time–frequency DoF has several advantages. +First, +qubits formed by time–frequency DoF are usually less +susceptible to errors because most optical components +do not depend on small temporal and spectral differ- +ences. Second, time–frequency DoF is suitable for real- +izing high-dimensional quantum information processing +with qudits because it is a continuous variable. +There are variations pertaining to encoding using the +time–frequency DoF. In time-bin encoding, the tempo- +ral peaks of a photon form the computational basis, and +its manipulation has been demonstrated by a series of +fast switches via spatial or polarization DoF [4, 8–10]. In +frequency-bin encoding, the spectral peaks of a photon +form the computational basis, and its manipulation has +been demonstrated by a series of electro-optic modula- +tors (EOMs) and pulse shapers [11–16]. +However, the +use of many active devices in these approaches is prone +to errors and losses and poses challenges in scaling up. +Instead, the manipulation of frequency-bin qubits using +time-resolving detectors was recently proposed [17], but +the finite resolution of these detectors causes serious er- +rors because frequency-bin qudits are susceptible to tem- +poral shift errors. +In this study, we propose a new scheme for linear opti- +cal quantum computation (LOQC) using time–frequency +DoF. We use encoding in which single-photon frequency +combs form the computational basis. +The state in +this encoding is called the time–frequency Gottesman– +Kitaev–Preskill (TFGKP) state [18, 19] derived from the +analog of GKP code [20] for quadrature amplitudes of +light [21]. The TFGKP state is robust against time- and +frequency-shift errors because it is discretized in both the +time and frequency domains. +We show that universal +quantum computation can be achieved using TFGKP- +state generators, time-resolving detectors, beam splitters +(BSs), and optical interleavers (OIs). Thus, active de- +vices such as high-speed switches and electro-optic mod- +ulators are not required. TFGKP-state generators can be +realized using a cavity-enhanced nonlinear optical pro- +cess [22–28]. +Furthermore, in contrast to the passive +scheme that uses frequency-bin encoding [17], quantum +computation can be performed robustly despite the de- +tectors’ finite resolutions and other temporal and spec- +tral errors. +We estimate the errors occurring in this +scheme and show that the experimental requirements for +fault-tolerant quantum computation are almost achiev- +able with current state-of-the-art technologies. +Time–frequency DoF of a photon.— We first summa- +rize the expressions and properties of the time–frequency +DoF of a photon. We consider that all probability density +functions (PDFs) are localized at the origin. A complex +function f(ω) is referred to as a probability amplitude +function (PAF) when |f(ω)|2 is a PDF. We represent the +Fourier transformation of a function f by ˆf and the point- +wise product and convolution of functions f and g by f ·g +and f ∗ g, respectively. We introduce the functions Tω′ +and Mτ ′ as +Tω′(f)(ω) = f(ω + ω′), Mτ ′(f)(ω) = e−iωτ ′f(ω). +(1) +The annihilation and creation operators of a photon with +frequency ω are represented as a(ω) and a†(ω), respec- +tively. The Fourier transformation of a(ω) and its adjoint +ˆa(τ) and ˆa†(τ) represent the annihilation and creation +operators of the photon that arrives at time τ at a certain +point. Propagation with distance L corresponds to the +change in creation operators as a†(ω) → a†(ω)e−ik(ω)L, +where k represents the wave number. +In practice, photons have finite temporal and spectral +widths as wave packets. +A photon wave packet with +central frequency ω0 can be described using PAF ξ as +(ξ∗a†)(ω0) |0⟩. Assuming that PAF ξ is sufficiently local- +ized, we can approximate k around ω0 to the first order as +k(ω) ≃ k(ω0)+k′(ω −ω0). Omitting the constant phase, +the propagation of the photon wave packet with distance +L corresponds to the transformation ξ → Mτ0(ξ), where +τ0 = k′L is the propagation time. The state after propa- +gation time τ0 is +(Mτ0(ξ) ∗ a†)(ω0) |0⟩ = eiω0τ0(M−ω0(ˆξ) ∗ ˆa†)(τ0) |0⟩ . (2) +arXiv:2301.03188v1 [quant-ph] 9 Jan 2023 + +2 +The right side of this equation denotes the temporal pho- +ton wave packet centered at τ0 [29]. +A qudit of time–frequency DoF is affected by unitary +and non-unitary errors. One of the major unitary errors +is caused by group velocity dispersion (GVD). Account- +ing for the effect of GVD, propagation with distance L +corresponds to the transformation of ξ → D · Mτ0(ξ), +where D(ω) = e−ik′′Lω2/2. Typically, this corresponds +to coherent temporal broadening by ∼ +√ +8 ln 2k′′L. Co- +herent spectral broadenings rarely occur except during +manipulation. Probabilistic temporal/spectral shifts can +be represented as incoherent temporal/spectral broaden- +ing by using a PDF. +Consider the frequency-bin qudit as an example. +It +is robust against incoherent spectral broadening because +each bin is spectrally isolated unless the bins overlap +owing to the broadening. +However, incoherent tempo- +ral broadening causes fluctuations in the relative phase +between bins. The error due to this fluctuation is not +small even if the broadening is relatively small compared +with the inverse of the frequency difference between bins. +Therefore, frequency-bin qudits are susceptible to tempo- +ral errors. +Time–frequency GKP qudit.— The GKP and TFGKP +qudits were introduced by assuming that the PAFs were +Gaussian [18, 20, 30]. By contrast, we introduce TFGKP +qudits without the Gaussian PAF assumption. The fre- +quency basis states of the ideal d-dimensional TFGKP +qudit are defined by the frequency combs formed by a +photon with shifted central frequencies. Using a Dirac +comb, that is, the sum of shifted Dirac delta functions +Cωr(ω) = � +n∈Z δ(ω + nωr), they are represented as +|jf⟩ = (Cωr ∗ a†) +�j +dωr +� +|0⟩ = (Tjωr/d(Cωr) ∗ a†)(0) |0⟩ +(3) +for j = 0, · · · , d − 1. +Each frequency basis state |jf⟩ +differs from |0f⟩ by the frequency offset j +dωr. The time +basis states of the TFGKP qudit are defined by the dis- +crete Fourier transformation of the frequency basis states +as +|jt⟩ ≡ +1 +√ +d +� +k +e−i2πjk/d |kf⟩ = (Cτr ∗ ˆa†) +�j +dτr +� +|0⟩ +(4) +for τr = 2πd +ωr and j = 0, · · · , d−1. |jt⟩ forms the temporal +comb with time period τr and time offset +j +dτr. +Since +the Fourier transformation of the Dirac comb is another +Dirac comb, this encoding discretizes the states in both +the frequency and time domains. +The states in Eq. (3) are not normalizable in two re- +spects. +First, each peak consists of a monochromatic +mode a†((n + j +d)ωr) |0⟩, and second, the summation of +the peaks is performed over an infinite range. To deal +with realistic situations, we introduce PAFs φf and φt +to represent the spectral broadening of each peak and +Time basis +Frequency basis + +FIG. 1: Probability distributions of a time–frequency +Gottesmann–Kitaev–Preskill (TFGKP) qudit in the +frequency and time bases for d = 2. The blue and orange +lines in the frequency/time basis represent |˜0f/t⟩ and |˜1f/t⟩, +respectively. +the envelope of the peaks, respectively. +This corre- +sponds to the replacement Tjωr/d(Cωr) in Eq. (3) using +φt · (Tjωr/d(Cωr) ∗ φf). +For central frequency ω0, the +frequency basis states after propagation for time τ0 are +|˜jf⟩ ∝ ((Mτ0(φt) · (Tjωr/d(Cωr) ∗ φf)) ∗ a†)(ω0) |0⟩ . (5) +Then, the time basis states are [29] +|˜jt⟩ ∝ +� +k +e−i2πjk |˜kf⟩ += eiω0τ0(M−ω0((Tjτr/d(Cτr) · ˆ +φf) ∗ ˆφt) ∗ ˆa†)(τ0) |0⟩ . +(6) +We call these normalizable states physical TFGKP states +in contrast with the ideal ones. Fig. 1 shows an example +of physical TFGKP states. Coherent broadenings of the +envelope on the frequency basis are equivalent to coher- +ent compressions of each peak on the time basis and vice +versa. +A comb-shaped structure in the frequency domain of +light often appears as a series of the transmission peaks +of a cavity. +For example, the heralded generation of +a TFGKP state is possible using a broadband time– +frequency entangled photon pair and a cavity. +When +one of the two photons passes through the cavity and is +detected by a time-resolving detector, the state of the +other photon corresponds to a TFGKP state. A cavity- +enhanced nonlinear optical process [22–28] can be applied +to the generation of a TFGKP state in this manner [31]. +By letting the free spectral range (FSR) of the cavity +be ∆FSR, the generated state corresponds to |˜0f⟩ for +ωr = ∆FSR. In addition, by setting the frequency pe- +riod to ωr = d∆FSR, the generated state corresponds to +|˜0t⟩ for any dimension d. A major incoherent temporal +broadening is caused by the finite temporal resolution of +the detector in this state-preparation method. +Optical components and detectors.— We introduce our +toolbox for the manipulation of TFGKP qudits, which +consists of BSs, OIs, and time- and frequency-resolving + +3 +FIG. 2: Graphical representation of 2:2 optical interleaver +(OI). Orange and blue spectral peaks represent |˜0f⟩ and +|˜1f⟩, respectively. +detectors. Herein, BSs generally refer to spatial linear +optical circuits that are independent of the other DoFs +of photons, including time–frequency DoF. As we will +see below, 50:50 BSs are sufficient for universal quantum +computation. +OI is a spectrally periodic optical filter that spatially +combines or separates frequency combs [32, 33]. Herein, +we use d:d OIs that have d input and output ports [34] +as shown in Fig. 2. The transformation of the creation +operator a† +j in each mode by d:d OI is represented as +a† +j → �d−1 +k=0 Ik · a† +j+k (mod d), where +Ij(ω) = (gI · (Tjωr/d(Cωr) ∗ fI))(ω0 − ω) +(7) +for central frequencies ω0 and j = 0, · · · , d − 1. Func- +tions gI and fI represent the transmission coefficients for +the envelope and each peak, respectively. Ideally, an OI +routes the frequency basis states of a TFGKP qudit into +spatially different d paths [35]. +Time- and frequency-resolving detectors are used for +photon detection. When we detect a photon in state |˜it⟩ +using an ideal time-resolving detector with the infinite +resolution, the temporal probability distribution of the +photon detection has periodical peaks at τ0 + n i +dτr for +integers n. +Even if each peak is blurred by the finite +resolution of the detector, we can identify the time basis +states of a TFGKP qudit from which the detection tim- +ing is closest unless the finite resolution is as large as the +time period. Similarly, we can identify the frequency ba- +sis states of a TFGKP qudit using a frequency-resolving +detector with a finite resolution. +A frequency-resolving detector can be substituted by +combining a 1:d OI with d detectors. +In that case, +a discrete result indicating which frequency basis state +was detected will be obtained. By contrast, the use of +frequency-resolving detectors, which have been actively +studied recently [36–38], would be advantageous in error +analysis [39]. +Universal quantum computation.— Let us consider the +frequency basis of TFGKP states as the computational +(a) +OI +OI +T +(b) +T +BS +T +OI +OI +OI +(c) +OI +T +(d) +BS +T +T +(e) +BS +T +FIG. 3: Concrete setups for quantum operations. All +detectors, beam splitters (BSs), and OIs are time-resolving +detectors, 50:50 BSs, and 2:2 OIs, respectively. (a) +Measurement in the cos +� θ +2 +� +X + sin +� θ +2 +� +Y basis. θ is +adjustable by changing the relative lengths between the two +arms. (b) Bell-state generation, which succeeds when +detectors detect two photons in different states. The OI +enclosed by the dotted lines denotes a feed-forward +operation required in 1/3 of the success cases. Its total +success probability is 3/16. (c) Type-I fusion gate, which +succeeds when a detector detects a photon with a +probability 1/2. (d) Type-II’ fusion gate, which succeeds +when a detector or detectors detect |0⟩ and |1⟩ with a +probability of 1/2. (e) Type-I’ fusion gate, which succeeds +when a detector detects a photon with a probability of 1/4. +basis of a qubit (d = 2) as |0⟩ ≃ |˜0f⟩ and |1⟩ ≃ |˜1f⟩. +The approximation here is due to the use of physical +TFGKP states rather than ideal ones. +Then, |+⟩ = +( |0⟩ + |1⟩)/ +√ +2 ≃ |˜0t⟩ and |−⟩ = ( |0⟩ − |1⟩)/ +√ +2 ≃ |˜1t⟩ +hold. The frequency- and time-resolving measurements +correspond to the measurements in the Z and X bases, +respectively. We can realize an arbitrary phase gate by +spatially separating each computational basis state by a +1:2 OI, adding a small relative time delay ∆τ satisfying +|∆τ| ≤ π/(2ω0) ≪ τr, and then combining them with a +2:1 OI. As shown in Fig. 3a, the phase gate followed by +the measurement in the X basis corresponds to that in +the cos +� θ +2 +� +X + sin +� θ +2 +� +Y basis. The measurements in the +Z and X bases and the phase gates are readily extendable +to qudits with arbitrary dimensions. +For cluster-state generations, we refer to the proto- +col in polarization encoding using type-I and type-II fu- +sion gates [40]; this enables the generation of an ar- +bitrary cluster state from 2-qubit cluster states, such +as ( |0+⟩ + |1−⟩)/ +√ +2. However, we need to modify it +because our toolbox does not include operations corre- +sponding to polarization rotations [41]. +While we can + +4 +realize type-I fusion gate as shown in Fig. 3c, we need to +introduce type-II’ fusion gate shown in Fig. 3d instead +of type-II fusion gate. Type-II’ fusion gate works as well +as type-II fusion gate, except for X gate on one of the +qubits [42]. We also can realize a Bell-state generation +setup as shown in Fig. 3b. +The generated Bell states +differ from the two-qubit cluster state by one Hadamard +gate. Thus, we additionally introduce type-I’ gate shown +in Fig. 3e, to generate a three-qubit cluster state from +the two Bell states with a success probability of 1/4. +Fault-tolerant measurement-based quantum computa- +tion can be performed by one-qubit measurements in the +X, Z and (X + Y )/ +√ +2 bases on a three-dimensional +cluster state [43, 44]. Therefore, fault-tolerant quantum +computation is possible by combining the quantum op- +erations shown in Fig. 3 and the time- and frequency- +resolving measurements. +Error analysis.— As the error analysis specific to this +scheme, we calculate the errors on the qubits caused by +temporal and spectral broadenings. We call the insuf- +ficient separation between states corresponding to the +different basis states “factor I.” This causes flips in the +measurement results and decreases in the success prob- +abilities of the entangling gates. +This is characterized +by the total amount of coherent and incoherent broaden- +ings. By contrast, we call the insufficient overlap between +states corresponding to the same basis state “factor II.” +This degrades the entangling gates and phase gates. This +is characterized by the amount of incoherent broadening +relative to that of coherent broadening. In our scheme, +we use frequency-resolving measurements only for one- +qubit measurements; therefore, we do not have to con- +sider factor II on the frequency basis. On the time basis, +there is an optimal amount of coherent temporal broad- +ening owing to a tradeoff between factors I and II. +Herein, we consider only the computational errors. We +make several assumptions to obtain specific error thresh- +olds. We assume that the coherent spectral broadening is +characterized by a Lorentzian because it is mainly deter- +mined by the transmission spectrum of the cavity used +in the state preparation. This assumption considerably +increases the amount of errors compared with the Gaus- +sian assumption. For simplicity, we ignore the influences +of using OIs instead of frequency-resolving detectors and +the incoherent spectral broadening [45]. Conversely, we +assume that coherent and incoherent temporal broaden- +ings are characterized by Gaussian functions [46]. The +condition subject to which the major error probabilities +are less than 0.01 corresponds to the following condi- +tions [29], +∆t,i +∆t,c +≲ 0.202, +∆t,c +τr/d ≲ 0.476, +∆f,c +ωr/d ≲ 0.016. +(8) +∆t,i, ∆t,c, and ∆f,c are the full-width-at-half-maximum +(FWHM) values of the incoherent temporal, coherent +temporal, and coherent spectral broadenings, respec- +tively. +Let us assume the use of telecom photons around +1.55 µm, which corresponds to ω0/2π ∼ 1.9 × 102 THz. +The group indices ng = ck′ and GVDs k′′ are typi- +cally 1.5 and −2.3 × 104 fs2/m for an optical fiber [47] +and 4.2 and −5.6 × 106 fs2/m for a silicon-on-insulator +waveguide [48], respectively. Using these parameters, the +lengths corresponding to 1 ps of time delay and tempo- +ral broadening due to GVD are 2.0 × 10−4 and 7.8 m +for the optical fiber and 7.1 × 10−2 and 3.2 × 10−2 m +for the silicon-on-insulator waveguide, respectively. By +contrast, the best value of the time resolution of a de- +tector is 4.3 ps for telecom wavelengths [49]. Thus, we +consider only the resolution of the detectors as the major +temporal error source for quantum computational appli- +cations [50]. For ∆t,i = 4.3 ps, we obtained the required +experimental parameters from Eq (8) as ∆t,c ≳ 21.5 ps, +ωr/2π ≲ 21 GHz, and ∆f,c/2π ≲ 0.33/d ≲ 0.17 GHz. +The first inequality corresponds to that the FWHM of +the spectral envelope of a qudit is ≲ 42 GHz. Therefore, +the number of frequency bins and the finesse of a state +|˜0t⟩ are approximately equal to 2d and 66, respectively. +OIs with 12.5 GHz comb spacings are commercially avail- +able [51]. The generation of a biphoton frequency comb +with finesse ∼ 60 [27, 28] and FSR = 12.5 GHz [52] +has been demonstrated using nonlinear optical waveguide +resonators. Thus, the current state-of-the-art technolo- +gies largely meet the experimental requirements for fault- +tolerant quantum computation based on our scheme. +Conclusion.— We proposed a LOQC scheme with +TFGKP state generators, time-resolving detectors, BSs, +and OIs, and demonstrated the possibility of fault- +tolerant quantum computation with currently achievable +technologies. +The discretization in both the time and +frequency domains owing to TFGKP qubits leads to er- +ror robustness against both temporal and spectral errors. +Furthermore, by treating the time and frequency basis +asymmetrically, we realized universal quantum computa- +tion without active devices. In addition, this asymmetric +structure enabled this scheme to yield good performance, +despite the assumption of a Lorentzian coherent spectral +broadening. +Additional optimization of TFGKP state +generators and OIs for this scheme would relax the re- +quirements of other devices or increase the dimension of +the qudit. +Although we show the universality of this +scheme for qubits, that is, d = 2, its components can +be extended to qudits. Thus, they are a good platform +for realizing the recently developed field of LOQC with +qudits [53–55]. This scheme has high error robustness +and ease of operation due to its use of time–frequency +DoF and passive devices. Therefore, this is a practical +approach, especially for quantum computation with in- +tegrated photonic circuits [56] and quantum communica- +tion requiring multiphoton entangled states [57–59]. +This work was supported by JST Moonshot R&D + +5 +JPMJMS2066, MEXT/JSPS KAKENHI JP20H01839, +JP20J20261, JP21H04445, and Asahi Glass Foundation. +∗ yamazaki-t@qi.mp.es.osaka-u.ac.jp +[1] E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, +46 (2001). +[2] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. +Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 +(2007). +[3] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature +412, 313 (2001). +[4] P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, +X.-M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. +Walmsley, Phys. Rev. Lett. 111, 150501 (2013). +[5] J. M. Lukens, A. Dezfooliyan, C. Langrock, M. M. Fejer, +D. E. Leaird, and A. M. Weiner, Phys. Rev. Lett. 112, +133602 (2014). +[6] B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. +Raymer, Phys. Rev. X 5, 041017 (2015). +[7] J. M. Lukens and P. Lougovski, Optica, OPTICA 4, 8 +(2017). +[8] H. Takesue, Phys. Rev. A 89, 062328 (2014). +[9] H.-P. Lo, T. Ikuta, N. Matsuda, T. Honjo, and H. Take- +sue, Appl. Phys. Express 11, 092801 (2018). +[10] H.-P. Lo, T. Ikuta, N. Matsuda, T. Honjo, W. J. Munro, +and H. Takesue, Phys. Rev. Appl. 13, 034013 (2020). +[11] M. Kues, C. Reimer, P. Roztocki, L. R. Cort´es, S. Sciara, +B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, +D. J. Moss, L. Caspani, J. Aza˜na, and R. Morandotti, +Nature 546, 622 (2017). +[12] H.-H. +Lu, +J. +M. +Lukens, +N. +A. +Peters, +B. +P. +Williams, A. M. Weiner, and P. Lougovski, +(2018), +arXiv:1803.10712 [quant-ph]. +[13] H.-H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. +Leaird, A. M. Weiner, and P. Lougovski, Phys. Rev. Lett. +120, 030502 (2018). +[14] H.-H. Lu, J. M. Lukens, B. P. Williams, P. Imany, N. A. +Peters, A. M. Weiner, and P. Lougovski, npj Quantum +Information 5, 1 (2019). +[15] H.-H. Lu, J. M. Lukens, N. A. Peters, B. P. Williams, +A. M. Weiner, and P. Lougovski, Optica, OPTICA 5, +1455 (2018). +[16] H.-H. Lu, E. M. Simmerman, P. Lougovski, A. M. +Weiner, and J. M. Lukens, Phys. Rev. Lett. 125, 120503 +(2020). +[17] C. Cui, K. P. Seshadreesan, S. Guha, and L. Fan, Phys. +Rev. Lett. 124, 190502 (2020). +[18] N. Fabre, G. Maltese, F. Appas, S. Felicetti, A. Ket- +terer, A. Keller, T. Coudreau, F. Baboux, M. I. Amanti, +S. Ducci, and P. Milman, Phys. Rev. A 102, 012607 +(2020). +[19] N. Fabre, A. Keller, and P. Milman, Phys. Rev. A 105, +052429 (2022). +[20] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A +64, 012310 (2001). +[21] C. Weedbrook, S. Pirandola, R. Garc´ıa-Patr´on, N. J. +Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. +Mod. Phys. 84, 621 (2012). +[22] C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, +B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, +L. Caspani, D. J. Moss, and R. Morandotti, Science 351, +1176 (2016). +[23] P. Imany, J. A. Jaramillo-Villegas, O. D. Odele, K. Han, +D. E. Leaird, J. M. Lukens, P. Lougovski, M. Qi, and +A. M. Weiner, Opt. Express 26, 1825 (2018). +[24] C. +Reimer, +S. +Sciara, +P. +Roztocki, +M. +Islam, +L. Romero Cort´es, Y. Zhang, B. Fischer, S. Loranger, +R. Kashyap, A. Cino, S. T. Chu, B. E. Little, D. J. +Moss, L. Caspani, W. J. Munro, J. Aza˜na, M. Kues, and +R. Morandotti, Nat. Phys. 15, 148 (2018). +[25] P. Imany, J. A. Jaramillo-Villegas, M. S. Alshaykh, J. M. +Lukens, O. D. Odele, A. J. Moore, D. E. Leaird, M. Qi, +and A. M. Weiner, npj Quantum Information 5, 1 (2019). +[26] M. Kues, C. Reimer, J. M. Lukens, W. J. Munro, A. M. +Weiner, D. J. Moss, and R. Morandotti, Nat. Photonics +13, 170 (2019). +[27] R. Ikuta, R. Tani, M. Ishizaki, S. Miki, M. Yabuno, +H. Terai, N. Imoto, and T. Yamamoto, Phys. Rev. Lett. +123, 193603 (2019). +[28] T. Yamazaki, R. Ikuta, T. Kobayashi, S. Miki, F. China, +H. Terai, N. Imoto, and T. Yamamoto, Sci. Rep. 12, 8964 +(2022). +[29] See the Supplemental material for its derivation and de- +tails. +[30] T. Matsuura, H. Yamasaki, and M. Koashi, Phys. Rev. +A 102, 032408 (2020). +[31] Another way to generate a TFGKP state is to use a deter- +ministic broadband single-photon generator and a cavity. +The spectrum of the photons generated inside the cavity +corresponds to the transmission spectrum of the cavity. +Therefore, we can generate the TFGKP state in a deter- +ministic manner. However, the bandwidth of the photon +generated by a quantum dot is usually not sufficiently +large [60]. +[32] S. Cao, J. Chen, J. N. Damask, C. R. Doerr, L. Guiziou, +G. Harvey, Y. Hibino, H. Li, S. Suzuki, K.-Y. Wu, and +P. Xie, J. Lightwave Technol. 22, 281 (2004). +[33] L.-W. Luo, S. Ibrahim, A. Nitkowski, Z. Ding, C. B. +Poitras, S. J. Ben Yoo, and M. Lipson, Opt. Express 18, +23079 (2010). +[34] A d:d OI can be made by connecting 2d pieces of com- +monly used 1:d OIs. +[35] A similar functionality can also be implemented by spa- +tially decomposing all the spectral peaks of an input state +and recombining a part of them into the same path. How- +ever, the method based on an OI that makes effective use +of interference would be better. +[36] O. Kahl, S. Ferrari, V. Kovalyuk, A. Vetter, G. Lewes- +Malandrakis, +C. +Nebel, +A. +Korneev, +G. +Golts- +man, and W. Pernice, Optica, OPTICA 10.1364/OP- +TICA.4.000557. +[37] R. Cheng, C.-L. Zou, X. Guo, S. Wang, X. Han, and +H. X. Tang, Nat. Commun. 10, 4104 (2019). +[38] S. M. Young, M. Sarovar, and F. L´eonard, +(2022), +arXiv:2205.05817 [quant-ph]. +[39] K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, Phys. +Rev. X 8, 021054 (2018). +[40] D. E. Browne and T. Rudolph, Phys. Rev. Lett. 95, +010501 (2005). +[41] A type of time–frequency rotation could be realized with +active operations such as chirped-pulse up-conversion [61, +62]. +[42] The effect of this extra X gate can be eliminated by the +way the measurement results are interpreted. + +6 +[43] R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, +190504 (2007). +[44] R. Raussendorf, J. Harrington, and K. Goyal, Ann. Phys. +321, 2242 (2006). +[45] This assumption would cause only quantitative differ- +ences since there is no need to consider Factor II in the +frequency basis. +[46] On the state-preparation using time–frequency entangle- +ment, the shape of coherent temporal broadening can be +designed depending on the filter used. +[47] Corning® SMF-28e+® optical fiber product infor- +mation, +https://www.corning.com/media/worldwide/ +coc/documents/Fiber/PI-1463-AEN.pdf (). +[48] E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and +Y. A. Vlasov, Opt. Express 14, 3853 (2006). +[49] B. Korzh, Q.-Y. Zhao, J. P. Allmaras, S. Frasca, T. M. +Autry, E. A. Bersin, A. D. Beyer, R. M. Briggs, B. Bum- +ble, M. Colangelo, G. M. Crouch, A. E. Dane, T. Gerrits, +A. E. Lita, F. Marsili, G. Moody, C. Pe˜na, E. Ramirez, +J. D. Rezac, N. Sinclair, M. J. Stevens, A. E. Velasco, +V. B. Verma, E. E. Wollman, S. Xie, D. Zhu, P. D. Hale, +M. Spiropulu, K. L. Silverman, R. P. Mirin, S. W. Nam, +A. G. Kozorezov, M. D. Shaw, and K. K. Berggren, Nat. +Photonics 14, 250 (2020). +[50] An implementation of a phase gate induces a time delay +< π/2ω0 ∼ 1.3 fs, which is negligible. +[51] Optoplex optical interleaver / optical De-Interleaver - +symmetric interleaver and asymmetric interleaver, http: +//www.optoplex.com/Optical_Interleaver.htm (), ac- +cessed: 2022-10-10. +[52] R. Fujimoto, T. Yamazaki, T. Kobayashi, S. Miki, +F. China, H. Terai, R. Ikuta, and T. Yamamoto, Opt. +Express 30, 36711 (2022). +[53] S. Paesani, J. F. F. Bulmer, A. E. Jones, R. Santagati, +and A. Laing, Phys. Rev. Lett. 126, 230504 (2021). +[54] C. Zhang, J. F. Chen, C. Cui, J. P. Dowling, Z. Y. Ou, +and T. Byrnes, Phys. Rev. A 100, 032330 (2019). +[55] Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.- +C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. +Lu, A. Zeilinger, and J.-W. Pan, Phys. Rev. Lett. 123, +070505 (2019). +[56] D. Dai, L. Liu, S. Gao, D.-X. Xu, and S. He, Laser Pho- +ton. Rev. 7, 303 (2013). +[57] K. Azuma, K. Tamaki, and H.-K. Lo, Nat. Commun. 6, +6787 (2015). +[58] F. Ewert, M. Bergmann, and P. van Loock, Phys. Rev. +Lett. 117, 210501 (2016). +[59] J. Borregaard, H. Pichler, T. Schr¨oder, M. D. Lukin, +P. Lodahl, and A. S. Sørensen, Phys. Rev. X 10, 021071 +(2020). +[60] A. V. Kuhlmann, J. H. Prechtel, J. Houel, A. Ludwig, +D. Reuter, A. D. Wieck, and R. J. Warburton, Nat. Com- +mun. 6, 8204 (2015). +[61] J. Lavoie, J. M. Donohue, L. G. Wright, A. Fedrizzi, and +K. J. Resch, Nat. Photonics 7, 363 (2013). +[62] J. M. Donohue, M. Agnew, J. Lavoie, and K. J. Resch, +Phys. Rev. Lett. 111, 153602 (2013). + +1 +SUPPLEMENTAL MATERIAL +A. Derivations of Equations (2) and (6) +Herein, we list useful formulas and derive Eqs. (2) and (6) in a more general form that includes the effect of group +velocity dispersion as a function D. +Note that the following relations hold. +Tω(f · g) = Tω(f) · Tω(g), +(S1) +Tω(f ∗ g) = Tω(f) ∗ g = f ∗ Tω(g), +(S2) +Mτ(f · g) = Mτ(f) · g = f · Mτ(g), +(S3) +Mτ(f ∗ g) = Mτ(f) ∗ Mτ(g), +(S4) +ˆ +(Tω(f)) = M−ω( ˆf), +ˆ +(Mτ(f)) = Tτ( ˆf) +(S5) +and +TωMτ = eiωτMτTω +(S6) +for functions f and g. +A photon wave packet can be converted between the time and frequency domains as follows, +(ξ ∗ a†)(0) = (ˆξ ∗ ˆa†)(0). +(S7) +It holds that +(Tω0(D · Mτ0(ξ)) ∗ a†)(0) = (M−ω0( ˆD ∗ Tτ0(ˆξ)) ∗ ˆa†)(0) = eiω0τ0(Tτ0(M−ω0( ˆD ∗ ˆξ)) ∗ ˆa†)(0), +(S8) +where the first equality holds from Eqs. (S7) and (S5), and the second equality holds from Eqs. (S2) and (S6). Thus, +we have +((D · Mτ0(ξ)) ∗ a†)(ω0) |0⟩ = eiω0τ0(M−ω0( ˆD ∗ ˆξ) ∗ ˆa†)(τ0) |0⟩ . +(S9) +We obtain Eq. (2) by omitting D from Eq. (S9). +Using the Poisson summation formula for a Dirac comb, +Cx(x) = +� +n +δ(x − nx) = 1 +x +� +n +ei2πn x +x , +(S10) +it is derived that +d−1 +� +k=0 +e−i2πjk/dTkx/d(Cx) = M2πj/x(Cx/d). +(S11) +The Fourier transformation of a Dirac comb is also a Dirac comb, as follows: +ˆCωr = 2π +ωr +C2π/ωr. +(S12) +The subsequent state following the propagation of a physical time–frequency Gottesmann–Kitaev–Preskill (TFGKP) +state with central frequency ω0 for time τ0 is described as +|˜if⟩ ∝ ((D · Mτ0(φt) · (Tiωr/d(Cωr) ∗ φf)) ∗ a†)(ω0) |0⟩ +(S13) +Correspondingly, it holds that +|˜jt⟩ ∝ +d−1 +� +k=0 +e−i2πjk/d |˜kf⟩ = ((D · Mτ0(φt) · (M2πj/ωr(Cωr/d) ∗ φf)) ∗ a†)(ω0) |0⟩ +(S14) +from Eq. (S11). The Fourier transformation of D′ = D · (M2πj/ωr(Cωr/d) ∗ φf) is +ˆ +D′ = ˆD ∗ (Tjτr/d(Cτr) · ˆ +φf) +(S15) +from Eqs. (S5) and (S12). Therefore, by replacing D by D′ in Eq. (S9), we obtain +|˜jt⟩ ∝ eiω0τ0(M−ω0( ˆD ∗ (Tjτr/d(Cτr) · ˆ +φf) ∗ ˆφt) ∗ ˆa†)(τ0) |0⟩ . +(S16) +Omitting D from Eq. (S16) yields Eq. (6). +arXiv:2301.03188v1 [quant-ph] 9 Jan 2023 + +2 +B. Detailed error analysis +Herein, we derive formulas for the error probabilities. We explicitly represent |˜if⟩ with central frequency ω0 as +|˜if(ω0)⟩ and |˜it⟩ with central timing τ0 as |˜it(τ0)⟩. We use PAFs φt/f for coherent temporal/spectral broadening and +PDFs Φt/f for incoherent temporal/spectral broadening. For all the calculations listed below, we assume that tem- +poral/spectral broadenings of TFGKP states are sufficiently narrow that the overlaps between the bins are negligible, +except within half of the time/frequency periods. +The error probability of qudit measurement in the temporal basis with a time-resolving detector is, +et,1 = 1 − +� +n +� +τr +2d +− τr +2d +dτ +� ∞ +−∞ +dτ ′Φt(τ ′) = +��⟨0| ˆa(τ0 + τ + nτr) |˜0t(τ0 − τ ′)⟩ +��2. +(S17) +We have +��⟨0| ˆa(τ0 + τ + nτr) |˜0t(τ0 − τ ′)⟩ +��2 = 1 +N +���((ˆφf · Cτr) ∗ ˆφt)(−(τ ′ + τ + nτr)) +��� +2 +(S18) +≃ +1 +� +m |ˆφf(mτr)| +2 |ˆφf(−nτr)ˆφt(−(τ ′ + τ))| +2, +(S19) +with a normalization factor N ≃ � +m |ˆφf(mτr)| +2. Therefore, +et,1 ≃ 1 − +� +τr +2d +− τr +2d +dτ(Φt ∗ (ˆφ∗ +t · ˆφt))(τ) +(S20) +holds. Similarly, we express the error probability of qudit measurement in the frequency basis with a frequency- +resolving detector as, +ef,1 ≃ 1 − +� +ωr +2d +− ωr +2d +dω(Φf ∗ (φ∗ +f · φf))(ω). +(S21) +When we use the 1:d optical interleavers (OIs) and d detectors instead of a frequency-resolving detector for mea- +surement in the frequency basis, the error probability becomes +ef,1 = +�d−1 +k=1 Fk +�d−1 +k=0 Fk +≃ F1 + Fd−1 +F0 +(S22) +where +Fk = +� ∞ +−∞ +dω +� ∞ +−∞ +dω′Φf(ω′) +��⟨0| (T−ω′(Ik) · a)(ω) |˜0f(ω0)⟩ +��2. +(S23) +Assuming good OI, that is, Tωr(gI) ≃ gI and fI(ω) ≃ 0 for |ω| > ωr/2, +Fx ≃ 1 +N +� ∞ +−∞ +dω +� ∞ +−∞ +dω′Φf(ω′) +��(Tω′(gI) · φt · (Cωr ∗ (Txωr/d+ω′(fI) · φf)))(−ω) +��2 +(S24) +≃ 1 +N +� +n +|(gI · φt)(nωr)|2 +� ∞ +−∞ +dω(Txωr/d(f ∗ +I · fI) · (Φf ∗ (φ∗ +f · φf)))(ω), +(S25) +where x = −1, 0, 1 and F−1 = Fd−1. Therefore, ef,1 hardly depends on the envelope of the OI gI. When fI(ω) = 1 for +|ω| ≤ ωr/2d and fI(ω) = 0 for |ω| > ωr/2d, Eq. (S22) becomes the same as Eq. (S21). We assume this for simplicity. +In the generations of entangled states and measurement on entangled bases, partial distinguishability causes a +deviation of the states from the maximally entangled states. The temporal distinguishability between the two input +photons is, +et,2 = 1 − +� ∞ +−∞ +dτΦt(τ) +�� ⟨˜0t(τ0)|˜0t(τ0 − τ)⟩ +��2 ≃ 1 − (Φt ∗ Gt)(0), +(S26) + +3 +where +Gt(δ) = +���� +� ∞ +−∞ +dτ ˆφ∗ +t (τ)ˆφt(τ + δ) +���� +2 +. +(S27) +In principle, this formula should be generalized to multiphoton interference, such as four-photon interference in +bell-state generations. Herein, we use Eq. (S26) as the error probability to characterize the effect of temporal distin- +guishability. +Assuming that the concrete shapes of coherent/incoherent temporal/spectral broadenings, we calculated the error +probabilities. We assumed that the incoherent temporal broadening was mainly caused by the finite temporal resolu- +tion of the detectors used for the generation and measurement of a qudit, which is usually characterized by a Gaussian +function Φt = fG,σt,i, where +fG,σ(x) = +1 +(2πσ2)1/2 exp +� +− x2 +2σ2 +� +. +(S28) +We assume that the coherent temporal broadening is mainly determined by the state-generation method based on +which we can design its PAF using the appropriate filter during state generation. Thus, we consider φ∗ +t · φt to be +Gaussian, that is, φt = (8πσ2 +t,c)1/4fG, +√ +2σt,c. We also assumed that the coherent spectral broadening is determined by +the state generation method, but it is reasonable to assume that φ∗ +f · φf is Lorentzian, that is, φf = fL,γf,c for +fL,γ(x) = +�γ +π +1 +γ − ix, +(S29) +because this is the case for the transmission spectrum of the cavities. For example, incoherent broadening may be +induced by the instability of OIs. However, we ignored this for simplicity because incoherent spectral broadening +plays the same role as coherent spectral broadening in our scheme. +Eqs. (S20), (S21), and (S26) correspond to the following equations, +et,1 ≃ 1 − erf +� τr +2d(σ2 +t,c + σ2 +t,i)− 1 +2 +� +(S30) +ef,1 ≃ 1 − 2 +π Tan−1�ωr/2d +γf,c +� +(S31) +et,2 ≃ 1 − +� +1 + σ2 +t,i +2σ2 +t,c +�−1/2 +. +(S32) +We can translate them into the expression with the FWHMs ∆FWHM of each PDF, where ∆FWHM = +√ +8 ln 2σ for a +Gaussian and ∆FWHM = 2γ for a Lorentzian. Setting their common error threshold e, we finally set the following +inequalities as a sufficient condition: +∆t,i +∆t,c +≲ +√ +A +(S33) +∆t,c +τr/d ≲ +� +2 ln 2 +1 + A(erf−1(1 − e))−1 +(S34) +∆f,c +ωr/d ≲ tan +�π +2 (1 − e) +�−1 +, +(S35) +where A = 2((1 − e)−2 − 1), and ∆t,i, ∆t,c, and ∆f,c are the full-width-at-half-maximum (FWHM) values of the +incoherent temporal, coherent temporal, and coherent spectral broadenings, respectively. +We obtain Eq. (8), for +e = 0.01. + diff --git a/aNE1T4oBgHgl3EQfcwSy/content/tmp_files/load_file.txt b/aNE1T4oBgHgl3EQfcwSy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..808e4857c82a1d4310e784c2badaa174e898efc1 --- /dev/null +++ b/aNE1T4oBgHgl3EQfcwSy/content/tmp_files/load_file.txt @@ -0,0 +1,909 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf,len=908 +page_content='Linear optical quantum computation with frequency-comb qubits and passive devices Tomohiro Yamazaki,1, 2, ∗ Tomoaki Arizono,1 Toshiki Kobayashi,1, 2 Rikizo Ikuta,1, 2 and Takashi Yamamoto1, 2 1Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan 2Center for Quantum Information and Quantum Biology, Osaka University, Toyonaka, Osaka 560-8531, Japan We propose a linear optical quantum computation scheme using time–frequency degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In this scheme, a qubit is encoded in single-photon frequency combs, and manipulation of the qubits is performed using time-resolving detectors, beam splitters, and optical interleavers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' This scheme does not require active devices such as high-speed switches and electro-optic modulators and is robust against temporal and spectral errors, which are mainly caused by the detectors’ finite resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We show that current technologies almost meet the requirements for fault-tolerant quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='— Photons and their manipulation using linear optics play an indispensable role in quantum in- formation processing [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' There has been considerable interest in the choice of the degrees of freedom (DoF) of photons as quantum information carriers [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The use of time–frequency DoF has several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' First, qubits formed by time–frequency DoF are usually less susceptible to errors because most optical components do not depend on small temporal and spectral differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Second, time–frequency DoF is suitable for real- izing high-dimensional quantum information processing with qudits because it is a continuous variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' There are variations pertaining to encoding using the time–frequency DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In time-bin encoding, the tempo- ral peaks of a photon form the computational basis, and its manipulation has been demonstrated by a series of fast switches via spatial or polarization DoF [4, 8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In frequency-bin encoding, the spectral peaks of a photon form the computational basis, and its manipulation has been demonstrated by a series of electro-optic modula- tors (EOMs) and pulse shapers [11–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' However, the use of many active devices in these approaches is prone to errors and losses and poses challenges in scaling up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Instead, the manipulation of frequency-bin qubits using time-resolving detectors was recently proposed [17], but the finite resolution of these detectors causes serious er- rors because frequency-bin qudits are susceptible to tem- poral shift errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In this study, we propose a new scheme for linear opti- cal quantum computation (LOQC) using time–frequency DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We use encoding in which single-photon frequency combs form the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The state in this encoding is called the time–frequency Gottesman– Kitaev–Preskill (TFGKP) state [18, 19] derived from the analog of GKP code [20] for quadrature amplitudes of light [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The TFGKP state is robust against time- and frequency-shift errors because it is discretized in both the time and frequency domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We show that universal quantum computation can be achieved using TFGKP- state generators, time-resolving detectors, beam splitters (BSs), and optical interleavers (OIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Thus, active de- vices such as high-speed switches and electro-optic mod- ulators are not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' TFGKP-state generators can be realized using a cavity-enhanced nonlinear optical pro- cess [22–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Furthermore, in contrast to the passive scheme that uses frequency-bin encoding [17], quantum computation can be performed robustly despite the de- tectors’ finite resolutions and other temporal and spec- tral errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We estimate the errors occurring in this scheme and show that the experimental requirements for fault-tolerant quantum computation are almost achiev- able with current state-of-the-art technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Time–frequency DoF of a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='— We first summa- rize the expressions and properties of the time–frequency DoF of a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We consider that all probability density functions (PDFs) are localized at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A complex function f(ω) is referred to as a probability amplitude function (PAF) when |f(ω)|2 is a PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We represent the Fourier transformation of a function f by ˆf and the point- wise product and convolution of functions f and g by f ·g and f ∗ g, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We introduce the functions Tω′ and Mτ ′ as Tω′(f)(ω) = f(ω + ω′), Mτ ′(f)(ω) = e−iωτ ′f(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (1) The annihilation and creation operators of a photon with frequency ω are represented as a(ω) and a†(ω), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The Fourier transformation of a(ω) and its adjoint ˆa(τ) and ˆa†(τ) represent the annihilation and creation operators of the photon that arrives at time τ at a certain point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Propagation with distance L corresponds to the change in creation operators as a†(ω) → a†(ω)e−ik(ω)L, where k represents the wave number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In practice, photons have finite temporal and spectral widths as wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A photon wave packet with central frequency ω0 can be described using PAF ξ as (ξ∗a†)(ω0) |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Assuming that PAF ξ is sufficiently local- ized, we can approximate k around ω0 to the first order as k(ω) ≃ k(ω0)+k′(ω −ω0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Omitting the constant phase, the propagation of the photon wave packet with distance L corresponds to the transformation ξ → Mτ0(ξ), where τ0 = k′L is the propagation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The state after propa- gation time τ0 is (Mτ0(ξ) ∗ a†)(ω0) |0⟩ = eiω0τ0(M−ω0(ˆξ) ∗ ˆa†)(τ0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (2) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='03188v1 [quant-ph] 9 Jan 2023 2 The right side of this equation denotes the temporal pho- ton wave packet centered at τ0 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A qudit of time–frequency DoF is affected by unitary and non-unitary errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' One of the major unitary errors is caused by group velocity dispersion (GVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Account- ing for the effect of GVD, propagation with distance L corresponds to the transformation of ξ → D · Mτ0(ξ), where D(ω) = e−ik′′Lω2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Typically, this corresponds to coherent temporal broadening by ∼ √ 8 ln 2k′′L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Co- herent spectral broadenings rarely occur except during manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Probabilistic temporal/spectral shifts can be represented as incoherent temporal/spectral broaden- ing by using a PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Consider the frequency-bin qudit as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' It is robust against incoherent spectral broadening because each bin is spectrally isolated unless the bins overlap owing to the broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' However, incoherent tempo- ral broadening causes fluctuations in the relative phase between bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The error due to this fluctuation is not small even if the broadening is relatively small compared with the inverse of the frequency difference between bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, frequency-bin qudits are susceptible to tempo- ral errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Time–frequency GKP qudit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='— The GKP and TFGKP qudits were introduced by assuming that the PAFs were Gaussian [18, 20, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' By contrast, we introduce TFGKP qudits without the Gaussian PAF assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The fre- quency basis states of the ideal d-dimensional TFGKP qudit are defined by the frequency combs formed by a photon with shifted central frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Using a Dirac comb, that is, the sum of shifted Dirac delta functions Cωr(ω) = � n∈Z δ(ω + nωr), they are represented as |jf⟩ = (Cωr ∗ a†) �j dωr � |0⟩ = (Tjωr/d(Cωr) ∗ a†)(0) |0⟩ (3) for j = 0, · · · , d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Each frequency basis state |jf⟩ differs from |0f⟩ by the frequency offset j dωr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The time basis states of the TFGKP qudit are defined by the dis- crete Fourier transformation of the frequency basis states as |jt⟩ ≡ 1 √ d � k e−i2πjk/d |kf⟩ = (Cτr ∗ ˆa†) �j dτr � |0⟩ (4) for τr = 2πd ωr and j = 0, · · · , d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' |jt⟩ forms the temporal comb with time period τr and time offset j dτr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Since the Fourier transformation of the Dirac comb is another Dirac comb, this encoding discretizes the states in both the frequency and time domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (3) are not normalizable in two re- spects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' First, each peak consists of a monochromatic mode a†((n + j d)ωr) |0⟩, and second, the summation of the peaks is performed over an infinite range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' To deal with realistic situations, we introduce PAFs φf and φt to represent the spectral broadening of each peak and Time basis Frequency basis FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 1: Probability distributions of a time–frequency Gottesmann–Kitaev–Preskill (TFGKP) qudit in the frequency and time bases for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The blue and orange lines in the frequency/time basis represent |˜0f/t⟩ and |˜1f/t⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' the envelope of the peaks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' This corre- sponds to the replacement Tjωr/d(Cωr) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (3) using φt · (Tjωr/d(Cωr) ∗ φf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' For central frequency ω0, the frequency basis states after propagation for time τ0 are |˜jf⟩ ∝ ((Mτ0(φt) · (Tjωr/d(Cωr) ∗ φf)) ∗ a†)(ω0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (5) Then, the time basis states are [29] |˜jt⟩ ∝ � k e−i2πjk |˜kf⟩ = eiω0τ0(M−ω0((Tjτr/d(Cτr) · ˆ φf) ∗ ˆφt) ∗ ˆa†)(τ0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (6) We call these normalizable states physical TFGKP states in contrast with the ideal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 1 shows an example of physical TFGKP states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Coherent broadenings of the envelope on the frequency basis are equivalent to coher- ent compressions of each peak on the time basis and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A comb-shaped structure in the frequency domain of light often appears as a series of the transmission peaks of a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' For example, the heralded generation of a TFGKP state is possible using a broadband time– frequency entangled photon pair and a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' When one of the two photons passes through the cavity and is detected by a time-resolving detector, the state of the other photon corresponds to a TFGKP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A cavity- enhanced nonlinear optical process [22–28] can be applied to the generation of a TFGKP state in this manner [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' By letting the free spectral range (FSR) of the cavity be ∆FSR, the generated state corresponds to |˜0f⟩ for ωr = ∆FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In addition, by setting the frequency pe- riod to ωr = d∆FSR, the generated state corresponds to |˜0t⟩ for any dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A major incoherent temporal broadening is caused by the finite temporal resolution of the detector in this state-preparation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Optical components and detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='— We introduce our toolbox for the manipulation of TFGKP qudits, which consists of BSs, OIs, and time- and frequency-resolving 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 2: Graphical representation of 2:2 optical interleaver (OI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Orange and blue spectral peaks represent |˜0f⟩ and |˜1f⟩, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Herein, BSs generally refer to spatial linear optical circuits that are independent of the other DoFs of photons, including time–frequency DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' As we will see below, 50:50 BSs are sufficient for universal quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' OI is a spectrally periodic optical filter that spatially combines or separates frequency combs [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Herein, we use d:d OIs that have d input and output ports [34] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The transformation of the creation operator a† j in each mode by d:d OI is represented as a† j → �d−1 k=0 Ik · a† j+k (mod d), where Ij(ω) = (gI · (Tjωr/d(Cωr) ∗ fI))(ω0 − ω) (7) for central frequencies ω0 and j = 0, · · · , d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Func- tions gI and fI represent the transmission coefficients for the envelope and each peak, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ideally, an OI routes the frequency basis states of a TFGKP qudit into spatially different d paths [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Time- and frequency-resolving detectors are used for photon detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' When we detect a photon in state |˜it⟩ using an ideal time-resolving detector with the infinite resolution, the temporal probability distribution of the photon detection has periodical peaks at τ0 + n i dτr for integers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Even if each peak is blurred by the finite resolution of the detector, we can identify the time basis states of a TFGKP qudit from which the detection tim- ing is closest unless the finite resolution is as large as the time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Similarly, we can identify the frequency ba- sis states of a TFGKP qudit using a frequency-resolving detector with a finite resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A frequency-resolving detector can be substituted by combining a 1:d OI with d detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In that case, a discrete result indicating which frequency basis state was detected will be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' By contrast, the use of frequency-resolving detectors, which have been actively studied recently [36–38], would be advantageous in error analysis [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Universal quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='— Let us consider the frequency basis of TFGKP states as the computational (a) OI OI T (b) T BS T OI OI OI (c) OI T (d) BS T T (e) BS T FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 3: Concrete setups for quantum operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' All detectors, beam splitters (BSs), and OIs are time-resolving detectors, 50:50 BSs, and 2:2 OIs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (a) Measurement in the cos � θ 2 � X + sin � θ 2 � Y basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' θ is adjustable by changing the relative lengths between the two arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (b) Bell-state generation, which succeeds when detectors detect two photons in different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The OI enclosed by the dotted lines denotes a feed-forward operation required in 1/3 of the success cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Its total success probability is 3/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (c) Type-I fusion gate, which succeeds when a detector detects a photon with a probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (d) Type-II’ fusion gate, which succeeds when a detector or detectors detect |0⟩ and |1⟩ with a probability of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (e) Type-I’ fusion gate, which succeeds when a detector detects a photon with a probability of 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' basis of a qubit (d = 2) as |0⟩ ≃ |˜0f⟩ and |1⟩ ≃ |˜1f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The approximation here is due to the use of physical TFGKP states rather than ideal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Then, |+⟩ = ( |0⟩ + |1⟩)/ √ 2 ≃ |˜0t⟩ and |−⟩ = ( |0⟩ − |1⟩)/ √ 2 ≃ |˜1t⟩ hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The frequency- and time-resolving measurements correspond to the measurements in the Z and X bases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We can realize an arbitrary phase gate by spatially separating each computational basis state by a 1:2 OI, adding a small relative time delay ∆τ satisfying |∆τ| ≤ π/(2ω0) ≪ τr, and then combining them with a 2:1 OI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 3a, the phase gate followed by the measurement in the X basis corresponds to that in the cos � θ 2 � X + sin � θ 2 � Y basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The measurements in the Z and X bases and the phase gates are readily extendable to qudits with arbitrary dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' For cluster-state generations, we refer to the proto- col in polarization encoding using type-I and type-II fu- sion gates [40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' this enables the generation of an ar- bitrary cluster state from 2-qubit cluster states, such as ( |0+⟩ + |1−⟩)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' However, we need to modify it because our toolbox does not include operations corre- sponding to polarization rotations [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' While we can 4 realize type-I fusion gate as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 3c, we need to introduce type-II’ fusion gate shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 3d instead of type-II fusion gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Type-II’ fusion gate works as well as type-II fusion gate, except for X gate on one of the qubits [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We also can realize a Bell-state generation setup as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The generated Bell states differ from the two-qubit cluster state by one Hadamard gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Thus, we additionally introduce type-I’ gate shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 3e, to generate a three-qubit cluster state from the two Bell states with a success probability of 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fault-tolerant measurement-based quantum computa- tion can be performed by one-qubit measurements in the X, Z and (X + Y )/ √ 2 bases on a three-dimensional cluster state [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, fault-tolerant quantum computation is possible by combining the quantum op- erations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 3 and the time- and frequency- resolving measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='— As the error analysis specific to this scheme, we calculate the errors on the qubits caused by temporal and spectral broadenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We call the insuf- ficient separation between states corresponding to the different basis states “factor I.” This causes flips in the measurement results and decreases in the success prob- abilities of the entangling gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' This is characterized by the total amount of coherent and incoherent broaden- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' By contrast, we call the insufficient overlap between states corresponding to the same basis state “factor II.” This degrades the entangling gates and phase gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' This is characterized by the amount of incoherent broadening relative to that of coherent broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In our scheme, we use frequency-resolving measurements only for one- qubit measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' therefore, we do not have to con- sider factor II on the frequency basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' On the time basis, there is an optimal amount of coherent temporal broad- ening owing to a tradeoff between factors I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Herein, we consider only the computational errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We make several assumptions to obtain specific error thresh- olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We assume that the coherent spectral broadening is characterized by a Lorentzian because it is mainly deter- mined by the transmission spectrum of the cavity used in the state preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' This assumption considerably increases the amount of errors compared with the Gaus- sian assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' For simplicity, we ignore the influences of using OIs instead of frequency-resolving detectors and the incoherent spectral broadening [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Conversely, we assume that coherent and incoherent temporal broaden- ings are characterized by Gaussian functions [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The condition subject to which the major error probabilities are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='01 corresponds to the following condi- tions [29], ∆t,i ∆t,c ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='202, ∆t,c τr/d ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='476, ∆f,c ωr/d ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (8) ∆t,i, ∆t,c, and ∆f,c are the full-width-at-half-maximum (FWHM) values of the incoherent temporal, coherent temporal, and coherent spectral broadenings, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Let us assume the use of telecom photons around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='55 µm, which corresponds to ω0/2π ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='9 × 102 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The group indices ng = ck′ and GVDs k′′ are typi- cally 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='5 and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='3 × 104 fs2/m for an optical fiber [47] and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='2 and −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='6 × 106 fs2/m for a silicon-on-insulator waveguide [48], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Using these parameters, the lengths corresponding to 1 ps of time delay and tempo- ral broadening due to GVD are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='0 × 10−4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='8 m for the optical fiber and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='1 × 10−2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='2 × 10−2 m for the silicon-on-insulator waveguide, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' By contrast, the best value of the time resolution of a de- tector is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='3 ps for telecom wavelengths [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Thus, we consider only the resolution of the detectors as the major temporal error source for quantum computational appli- cations [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' For ∆t,i = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='3 ps, we obtained the required experimental parameters from Eq (8) as ∆t,c ≳ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='5 ps, ωr/2π ≲ 21 GHz, and ∆f,c/2π ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='33/d ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='17 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The first inequality corresponds to that the FWHM of the spectral envelope of a qudit is ≲ 42 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, the number of frequency bins and the finesse of a state |˜0t⟩ are approximately equal to 2d and 66, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' OIs with 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='5 GHz comb spacings are commercially avail- able [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The generation of a biphoton frequency comb with finesse ∼ 60 [27, 28] and FSR = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='5 GHz [52] has been demonstrated using nonlinear optical waveguide resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Thus, the current state-of-the-art technolo- gies largely meet the experimental requirements for fault- tolerant quantum computation based on our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='— We proposed a LOQC scheme with TFGKP state generators, time-resolving detectors, BSs, and OIs, and demonstrated the possibility of fault- tolerant quantum computation with currently achievable technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The discretization in both the time and frequency domains owing to TFGKP qubits leads to er- ror robustness against both temporal and spectral errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Furthermore, by treating the time and frequency basis asymmetrically, we realized universal quantum computa- tion without active devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In addition, this asymmetric structure enabled this scheme to yield good performance, despite the assumption of a Lorentzian coherent spectral broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Additional optimization of TFGKP state generators and OIs for this scheme would relax the re- quirements of other devices or increase the dimension of the qudit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Although we show the universality of this scheme for qubits, that is, d = 2, its components can be extended to qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Thus, they are a good platform for realizing the recently developed field of LOQC with qudits [53–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' This scheme has high error robustness and ease of operation due to its use of time–frequency DoF and passive devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, this is a practical approach, especially for quantum computation with in- tegrated photonic circuits [56] and quantum communica- tion requiring multiphoton entangled states [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' This work was supported by JST Moonshot R&D 5 JPMJMS2066, MEXT/JSPS KAKENHI JP20H01839, JP20J20261, JP21H04445, and Asahi Glass Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' ∗ yamazaki-t@qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='jp [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Knill, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Laflamme, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Milburn, Nature 409, 46 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kok, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Munro, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Nemoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ralph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Dowling, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Milburn, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 79, 135 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Mair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Vaziri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weihs, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zeilinger, Nature 412, 313 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Humphreys, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Metcalf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Spring, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Moore, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Jin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Barbieri, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kolthammer, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Walmsley, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 111, 150501 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Dezfooliyan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Langrock, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fejer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Leaird, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 112, 133602 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Brecht, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Reddy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Silberhorn, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Raymer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' X 5, 041017 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lougovski, Optica, OPTICA 4, 8 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Takesue, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A 89, 062328 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ikuta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Matsuda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Honjo, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Take- sue, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Express 11, 092801 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [10] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ikuta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Matsuda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Honjo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Munro, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Takesue, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 13, 034013 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kues, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Reimer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Roztocki, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cort´es, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Sciara, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wetzel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Chu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Little, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Moss, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Caspani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Aza˜na, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Morandotti, Nature 546, 622 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Peters, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Williams, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lougovski, (2018), arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='10712 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Peters, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Odele, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Leaird, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lougovski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 120, 030502 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [14] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Williams, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Imany, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Peters, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lougovski, npj Quantum Information 5, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Peters, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Williams, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lougovski, Optica, OPTICA 5, 1455 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Simmerman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lougovski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 125, 120503 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cui, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Seshadreesan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Guha, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 124, 190502 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fabre, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Maltese, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Appas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Felicetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ket- terer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Keller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Coudreau, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Baboux, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Amanti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ducci, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Milman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A 102, 012607 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [19] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fabre, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Keller, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Milman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A 105, 052429 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Gottesman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kitaev, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Preskill, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A 64, 012310 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weedbrook, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Pirandola, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Garc´ıa-Patr´on, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cerf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ralph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Shapiro, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lloyd, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 84, 621 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Reimer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kues, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Roztocki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wetzel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Grazioso, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Little, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Chu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Johnston, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Bromberg, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Caspani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Moss, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Morandotti, Science 351, 1176 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Imany, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Jaramillo-Villegas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Odele, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Leaird, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lougovski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Qi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Express 26, 1825 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Reimer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Sciara, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Roztocki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Islam, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Romero Cort´es, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fischer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Loranger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kashyap, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Chu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Little, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Moss, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Caspani, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Munro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Aza˜na, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kues, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Morandotti, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 15, 148 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Imany, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Jaramillo-Villegas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Alshaykh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Odele, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Moore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Leaird, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Qi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, npj Quantum Information 5, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kues, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Reimer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukens, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Munro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Weiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Moss, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Morandotti, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Photonics 13, 170 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [27] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ikuta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Tani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ishizaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Miki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Yabuno, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Terai, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Imoto, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Yamamoto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 123, 193603 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Yamazaki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ikuta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kobayashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Miki, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' China, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Terai, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Imoto, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Yamamoto, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 12, 8964 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [29] See the Supplemental material for its derivation and de- tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Matsuura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Yamasaki, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Koashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A 102, 032408 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [31] Another way to generate a TFGKP state is to use a deter- ministic broadband single-photon generator and a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The spectrum of the photons generated inside the cavity corresponds to the transmission spectrum of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, we can generate the TFGKP state in a deter- ministic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' However, the bandwidth of the photon generated by a quantum dot is usually not sufficiently large [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Damask, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Doerr, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Guiziou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Harvey, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Hibino, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Suzuki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Xie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lightwave Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 22, 281 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [33] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Luo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ibrahim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Nitkowski, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ding, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Poitras, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ben Yoo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lipson, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Express 18, 23079 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [34] A d:d OI can be made by connecting 2d pieces of com- monly used 1:d OIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [35] A similar functionality can also be implemented by spa- tially decomposing all the spectral peaks of an input state and recombining a part of them into the same path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' How- ever, the method based on an OI that makes effective use of interference would be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [36] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kahl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ferrari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kovalyuk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Vetter, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lewes- Malandrakis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Nebel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Korneev, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Golts- man, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Pernice, Optica, OPTICA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='1364/OP- TICA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='000557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [37] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Guo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Han, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Tang, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 10, 4104 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Young, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Sarovar, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' L´eonard, (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='05817 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [39] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fukui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Tomita, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Okamoto, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fujii, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' X 8, 021054 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [40] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Browne and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rudolph, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 95, 010501 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [41] A type of time–frequency rotation could be realized with active operations such as chirped-pulse up-conversion [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [42] The effect of this extra X gate can be eliminated by the way the measurement results are interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 6 [43] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Raussendorf and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Harrington, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 98, 190504 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [44] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Raussendorf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Harrington, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Goyal, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 321, 2242 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [45] This assumption would cause only quantitative differ- ences since there is no need to consider Factor II in the frequency basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [46] On the state-preparation using time–frequency entangle- ment, the shape of coherent temporal broadening can be designed depending on the filter used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [47] Corning® SMF-28e+® optical fiber product infor- mation, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='corning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='com/media/worldwide/ coc/documents/Fiber/PI-1463-AEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='pdf ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [48] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Dulkeith, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Xia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Schares, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Green, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Vlasov, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Express 14, 3853 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [49] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Korzh, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zhao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Allmaras, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Frasca, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Autry, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Bersin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Beyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Briggs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Bum- ble, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Colangelo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Crouch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Dane, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Gerrits, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lita, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Marsili, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Moody, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Pe˜na, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ramirez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rezac, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Sinclair, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Stevens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Velasco, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Verma, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wollman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Xie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zhu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Hale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Spiropulu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Silverman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Mirin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Nam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kozorezov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Shaw, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Berggren, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Photonics 14, 250 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [50] An implementation of a phase gate induces a time delay < π/2ω0 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='3 fs, which is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [51] Optoplex optical interleaver / optical De-Interleaver - symmetric interleaver and asymmetric interleaver, http: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='optoplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='com/Optical_Interleaver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='htm (), ac- cessed: 2022-10-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [52] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fujimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Yamazaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kobayashi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Miki, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' China, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Terai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ikuta, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Yamamoto, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Express 30, 36711 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [53] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Paesani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Bulmer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Santagati, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Laing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 126, 230504 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [54] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Cui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Dowling, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ou, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Byrnes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A 100, 032330 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [55] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Luo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zhong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Erhard, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Peng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Krenn, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Zeilinger, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Pan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 123, 070505 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [56] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Dai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Gao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Xu, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' He, Laser Pho- ton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 7, 303 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [57] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Azuma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Tamaki, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lo, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 6, 6787 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [58] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ewert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Bergmann, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' van Loock, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 117, 210501 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [59] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Borregaard, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Pichler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Schr¨oder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lukin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lodahl, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Sørensen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' X 10, 021071 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [60] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Kuhlmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Prechtel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Houel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Ludwig, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Reuter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wieck, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Warburton, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Com- mun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 6, 8204 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [61] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lavoie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Donohue, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Wright, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Fedrizzi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Resch, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Photonics 7, 363 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' [62] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Donohue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Agnew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lavoie, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Resch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 111, 153602 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' 1 SUPPLEMENTAL MATERIAL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Derivations of Equations (2) and (6) Herein, we list useful formulas and derive Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (2) and (6) in a more general form that includes the effect of group velocity dispersion as a function D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Note that the following relations hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Tω(f · g) = Tω(f) · Tω(g), (S1) Tω(f ∗ g) = Tω(f) ∗ g = f ∗ Tω(g), (S2) Mτ(f · g) = Mτ(f) · g = f · Mτ(g), (S3) Mτ(f ∗ g) = Mτ(f) ∗ Mτ(g), (S4) ˆ (Tω(f)) = M−ω( ˆf), ˆ (Mτ(f)) = Tτ( ˆf) (S5) and TωMτ = eiωτMτTω (S6) for functions f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' A photon wave packet can be converted between the time and frequency domains as follows, (ξ ∗ a†)(0) = (ˆξ ∗ ˆa†)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S7) It holds that (Tω0(D · Mτ0(ξ)) ∗ a†)(0) = (M−ω0( ˆD ∗ Tτ0(ˆξ)) ∗ ˆa†)(0) = eiω0τ0(Tτ0(M−ω0( ˆD ∗ ˆξ)) ∗ ˆa†)(0), (S8) where the first equality holds from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S7) and (S5), and the second equality holds from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S2) and (S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Thus, we have ((D · Mτ0(ξ)) ∗ a†)(ω0) |0⟩ = eiω0τ0(M−ω0( ˆD ∗ ˆξ) ∗ ˆa†)(τ0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S9) We obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (2) by omitting D from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Using the Poisson summation formula for a Dirac comb, Cx(x) = � n δ(x − nx) = 1 x � n ei2πn x x , (S10) it is derived that d−1 � k=0 e−i2πjk/dTkx/d(Cx) = M2πj/x(Cx/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S11) The Fourier transformation of a Dirac comb is also a Dirac comb, as follows: ˆCωr = 2π ωr C2π/ωr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S12) The subsequent state following the propagation of a physical time–frequency Gottesmann–Kitaev–Preskill (TFGKP) state with central frequency ω0 for time τ0 is described as |˜if⟩ ∝ ((D · Mτ0(φt) · (Tiωr/d(Cωr) ∗ φf)) ∗ a†)(ω0) |0⟩ (S13) Correspondingly, it holds that |˜jt⟩ ∝ d−1 � k=0 e−i2πjk/d |˜kf⟩ = ((D · Mτ0(φt) · (M2πj/ωr(Cωr/d) ∗ φf)) ∗ a†)(ω0) |0⟩ (S14) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The Fourier transformation of D′ = D · (M2πj/ωr(Cωr/d) ∗ φf) is ˆ D′ = ˆD ∗ (Tjτr/d(Cτr) · ˆ φf) (S15) from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S5) and (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, by replacing D by D′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S9), we obtain |˜jt⟩ ∝ eiω0τ0(M−ω0( ˆD ∗ (Tjτr/d(Cτr) · ˆ φf) ∗ ˆφt) ∗ ˆa†)(τ0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S16) Omitting D from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S16) yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='03188v1 [quant-ph] 9 Jan 2023 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Detailed error analysis Herein, we derive formulas for the error probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We explicitly represent |˜if⟩ with central frequency ω0 as |˜if(ω0)⟩ and |˜it⟩ with central timing τ0 as |˜it(τ0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We use PAFs φt/f for coherent temporal/spectral broadening and PDFs Φt/f for incoherent temporal/spectral broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' For all the calculations listed below, we assume that tem- poral/spectral broadenings of TFGKP states are sufficiently narrow that the overlaps between the bins are negligible, except within half of the time/frequency periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The error probability of qudit measurement in the temporal basis with a time-resolving detector is, et,1 = 1 − � n � τr 2d − τr 2d dτ � ∞ −∞ dτ ′Φt(τ ′) = ��⟨0| ˆa(τ0 + τ + nτr) |˜0t(τ0 − τ ′)⟩ ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S17) We have ��⟨0| ˆa(τ0 + τ + nτr) |˜0t(τ0 − τ ′)⟩ ��2 = 1 N ���((ˆφf · Cτr) ∗ ˆφt)(−(τ ′ + τ + nτr)) ��� 2 (S18) ≃ 1 � m |ˆφf(mτr)| 2 |ˆφf(−nτr)ˆφt(−(τ ′ + τ))| 2, (S19) with a normalization factor N ≃ � m |ˆφf(mτr)| 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, et,1 ≃ 1 − � τr 2d − τr 2d dτ(Φt ∗ (ˆφ∗ t · ˆφt))(τ) (S20) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Similarly, we express the error probability of qudit measurement in the frequency basis with a frequency- resolving detector as, ef,1 ≃ 1 − � ωr 2d − ωr 2d dω(Φf ∗ (φ∗ f · φf))(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S21) When we use the 1:d optical interleavers (OIs) and d detectors instead of a frequency-resolving detector for mea- surement in the frequency basis, the error probability becomes ef,1 = �d−1 k=1 Fk �d−1 k=0 Fk ≃ F1 + Fd−1 F0 (S22) where Fk = � ∞ −∞ dω � ∞ −∞ dω′Φf(ω′) ��⟨0| (T−ω′(Ik) · a)(ω) |˜0f(ω0)⟩ ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S23) Assuming good OI, that is, Tωr(gI) ≃ gI and fI(ω) ≃ 0 for |ω| > ωr/2, Fx ≃ 1 N � ∞ −∞ dω � ∞ −∞ dω′Φf(ω′) ��(Tω′(gI) · φt · (Cωr ∗ (Txωr/d+ω′(fI) · φf)))(−ω) ��2 (S24) ≃ 1 N � n |(gI · φt)(nωr)|2 � ∞ −∞ dω(Txωr/d(f ∗ I · fI) · (Φf ∗ (φ∗ f · φf)))(ω), (S25) where x = −1, 0, 1 and F−1 = Fd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Therefore, ef,1 hardly depends on the envelope of the OI gI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' When fI(ω) = 1 for |ω| ≤ ωr/2d and fI(ω) = 0 for |ω| > ωr/2d, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S22) becomes the same as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We assume this for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' In the generations of entangled states and measurement on entangled bases, partial distinguishability causes a deviation of the states from the maximally entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' The temporal distinguishability between the two input photons is, et,2 = 1 − � ∞ −∞ dτΦt(τ) �� ⟨˜0t(τ0)|˜0t(τ0 − τ)⟩ ��2 ≃ 1 − (Φt ∗ Gt)(0), (S26) 3 where Gt(δ) = ���� � ∞ −∞ dτ ˆφ∗ t (τ)ˆφt(τ + δ) ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S27) In principle, this formula should be generalized to multiphoton interference, such as four-photon interference in bell-state generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Herein, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S26) as the error probability to characterize the effect of temporal distin- guishability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Assuming that the concrete shapes of coherent/incoherent temporal/spectral broadenings, we calculated the error probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We assumed that the incoherent temporal broadening was mainly caused by the finite temporal resolu- tion of the detectors used for the generation and measurement of a qudit, which is usually characterized by a Gaussian function Φt = fG,σt,i, where fG,σ(x) = 1 (2πσ2)1/2 exp � − x2 2σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S28) We assume that the coherent temporal broadening is mainly determined by the state-generation method based on which we can design its PAF using the appropriate filter during state generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Thus, we consider φ∗ t · φt to be Gaussian, that is, φt = (8πσ2 t,c)1/4fG, √ 2σt,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We also assumed that the coherent spectral broadening is determined by the state generation method, but it is reasonable to assume that φ∗ f · φf is Lorentzian, that is, φf = fL,γf,c for fL,γ(x) = �γ π 1 γ − ix, (S29) because this is the case for the transmission spectrum of the cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' For example, incoherent broadening may be induced by the instability of OIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' However, we ignored this for simplicity because incoherent spectral broadening plays the same role as coherent spectral broadening in our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S20), (S21), and (S26) correspond to the following equations, et,1 ≃ 1 − erf � τr 2d(σ2 t,c + σ2 t,i)− 1 2 � (S30) ef,1 ≃ 1 − 2 π Tan−1�ωr/2d γf,c � (S31) et,2 ≃ 1 − � 1 + σ2 t,i 2σ2 t,c �−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (S32) We can translate them into the expression with the FWHMs ∆FWHM of each PDF, where ∆FWHM = √ 8 ln 2σ for a Gaussian and ∆FWHM = 2γ for a Lorentzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' Setting their common error threshold e, we finally set the following inequalities as a sufficient condition: ∆t,i ∆t,c ≲ √ A (S33) ∆t,c τr/d ≲ � 2 ln 2 1 + A(erf−1(1 − e))−1 (S34) ∆f,c ωr/d ≲ tan �π 2 (1 − e) �−1 , (S35) where A = 2((1 − e)−2 − 1), and ∆t,i, ∆t,c, and ∆f,c are the full-width-at-half-maximum (FWHM) values of the incoherent temporal, coherent temporal, and coherent spectral broadenings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' We obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content=' (8), for e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfcwSy/content/2301.03188v1.pdf'} diff --git a/b9E0T4oBgHgl3EQfWQBc/content/tmp_files/2301.02275v1.pdf.txt b/b9E0T4oBgHgl3EQfWQBc/content/tmp_files/2301.02275v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..15e30f07814d627aa2bc92f60d8469f06a0d5a44 --- /dev/null +++ b/b9E0T4oBgHgl3EQfWQBc/content/tmp_files/2301.02275v1.pdf.txt @@ -0,0 +1,1242 @@ +DEEP LATENT VARIABLE MODELS FOR SEMI-SUPERVISED +PARAPHRASE GENERATION ∗ +Jialin Yu, Alexandra I. Cristea, Anoushka Harit, Zhongtian Sun, +Olanrewaju Tahir Aduragba, Lei Shi, Noura Al Moubayed +Department of Computer Science +Durham University +Durham, UK +ABSTRACT +This paper explores deep latent variable models for semi-supervised paraphrase generation, where +the missing target pair is modelled as a latent paraphrase sequence. We present a novel unsupervised +model named variational sequence auto-encoding reconstruction (VSAR), which performs latent +sequence inference given an observed text. To leverage information from text pairs, we introduce a su- +pervised model named dual directional learning (DDL). Combining VSAR with DDL (DDL+VSAR) +enables us to conduct semi-supervised learning; however, the combined model suffers from a cold- +start problem. To combat this issue, we propose to deal with better weight initialisation, leading to a +two-stage training scheme named knowledge reinforced training. Our empirical evaluations suggest +that the combined model yields competitive performance against the state-of-the-art supervised +baselines on complete data. Furthermore, in scenarios where only a fraction of the labelled pairs are +available, our combined model consistently outperforms the strong supervised model baseline (DDL) +by a significant margin. +Keywords Paraphrase Generation · Semi-supervised Learning · Deep Latent Variable Model +1 +Introduction +Paraphrase generation is an important Natural Language Processing (NLP) problem, useful in many NLP applications, +such as question answering [1], information retrieval [2], information extraction [3] and summarisation [4]. Natural +language itself is complicated and may be expressed in various alternative surface forms of the same underlying +semantic content [5, 6]. Hence is critically important to integrate the paraphrase generation model as a component in +real-world NLP systems to offer robust responses to end users’ inputs. Traditional solutions to paraphrase generation are +generally rule-based [7, 8], utilising lexical resources, such as WordNet [9], to find word replacements. The recent trend +brings to fore neural network models [10, 11, 12, 13], which are typically based on a sequence-to-sequence learning +paradigm [14]. +These models have achieved remarkable success for paraphrase generation due to complex architectures and sophis- +ticated conditioning mechanisms, e.g. soft, hard and self-attention. However, the advancement of such models is +primarily based on the availability of large-scale labelled data pairs. Instead, this paper explores semi-supervised +learning scenarios where only a fraction of the labels are available. This semi-supervised learning setting is favourable +and extremely useful for industry scenarios due to the effort of time and money to obtain good quality human anno- +tations. A semi-supervised learning model often consists of two components: an unsupervised learning model and a +supervised learning model. For unsupervised learning, we propose a novel deep generative model, motivated by the +classic variational autoencoder (VAE) [15, 16, 17]. Furthermore, we propose a novel supervised model that can be +integrated with our proposed VAE model. Combining both unsupervised and supervised models enable semi-supervised +learning by exploiting the VAEs’ ability to marginalise latent variables for unlabelled data. +∗Manuscript under review +arXiv:2301.02275v1 [cs.CL] 5 Jan 2023 + +Running Title for Header +Traditional VAEs typically embed data representations in a fixed latent space, with the general purpose of dimensionality +reduction [18]. This paper alternatively considers a latent variable in the form of a discrete language sequence +with various lengths. This additionally can enhance the model interpretability, as language is naturally preserved as +discrete variables [19]. Following the recent prior works successfully incorporating discrete latent variables to improve +paraphrasing [20, 6], we propose a novel model with more expressive discrete latent variable: variational sequence +auto-encoding reconstruction (VSAR) model. +In order to further boost the model performance on paraphrase generation, motivated by dual learning [21, 22, 23, 24], +we employ a novel Dual Directional Learning (DDL) model trained on labelled data. The DDL model shares model +parameters with the VSAR model and can be jointly optimised for semi-supervised scenarios, and we discuss this later +in this paper. Our contributions in this paper include: +• presenting the first study on semi-supervised learning for paraphrasing with deep latent variable models; +• introducing two novel models: VSAR (unsupervised) and DDL (supervised), which can be combined for +semi-supervised learning; +• proposing a novel training scheme (Knowledge Reinforced Learning) to deal with cold start problem in the +combined model (DDL+VSAR) when used in semi-supervised learning; +• studying semi-supervised learning scenarios with the combined model on the full fraction of data and +empirically showing that our model achieves competitive state-of-the-art results; +• presenting a study of semi-supervised scenarios with a fraction of the labelled data and our models show +significantly better results than very strong supervised baselines. +bos +bos +eos +Source +Reconstruction +bos +eos +eos +Transformer Target Decoder +bos +Transformer Target Encoder +bos +eos +bos +eos +eos +bos +eos +eos +Transformer Source Encoder +bos +Transformer Source Decoder +Latent +Inference +Weak Supervision +Figure 1: Variational Sequence Auto-Encoding Reconstruction Model. +2 +Variational Sequence Auto-Encoding Reconstruction (VSAR) +In this section, we present the VSAR model (Figure 1)2. The model consists of four separate neural network models +- a source encoder, a target decoder, a target encoder, and a source decoder. Under the unsupervised learning setup, +we only observe source text s and no target text t. We reformulate the problem of modelling fully observed source +text s as modelling partially observed parallel source text s and its associated latent target pair ¯t. We adopt Bayesian +inference to marginalise the latent target string ¯t from the joint probability distribution pθ(s,¯t) as shown in Figure 1 +based on Equation 7 and hence only the observed source text s is required for VSAR model. +2The language model prior and weak supervision decoding is omitted for clarity. +2 + +Running Title for Header +In the VSAR model, the latent inference network, parameterised as qφ(¯t|s), takes source text s and generates a latent +target sample ¯t. The source reconstruction network, parameterised as pθ(s|¯t), reconstructs the observed source text s +back, based on the latent target sample ¯t. As the prior distribution, a language model is pre-trained on unlabelled source +text corpus to approximate the prior distribution p(¯t)3. The prior is introduced for regularisation purposes [25, 26], +which enforces samples are more likely to be reasonable natural language text. +Motivated by the benefits of parameters sharing in multi-task learning for natural language generation [27, 28, 29, 30], +we share model parameters for the source encoder and the target encoder, denoted as fencode; similarly, we share +model parameters for the source decoder and the target decoder, denoted as fdecode. In the following sections, we use +fencode and fdecode to represent all encoders and decoders in the VSAR model, respectively. +2.1 +Weak Supervision +In the VSAR model, we empirically found that the quality of latent sequence ¯t is very unstable, especially at the +beginning of the training. To combat this issue, motivated by the idea of weak supervision [31, 32], we propose to use +pseudo-labels to guide VSAR throughout training. Before each model forward pass, we first assign pseudo-labels to +each token in unobserved latent target sample ¯t with the current model parameter. The pseudo-labels are detached from +the computational graph; hence no gradient is updated during the weak supervision process. The pseudo-labels can be +considered as a weak supervision signal for ‘teacher forcing training’ [33]. +The encoder model takes the source string s = (s1, ..., sn) as input and produces its corresponding contextual vector +hs = (hs +1, ..., hs +n): +hs = fencode(s) +(1) +We adopt a greedy decoding scheme to assign pseudo-target labels t∗ and assume that a good paraphrase ought to have +a similar length as the original sentence [34, 35]; such that t∗ = (t∗ +1, ..., t∗ +n). Let t∗ +i be the ith word in the pseudo target +sequence; we construct this sequence in an auto-regressive manner: +t∗ +i = fdecode(hs; t∗ +1:i−1) +(2) +2.2 +Target Inference +Once the pseudo-target labels t∗ are assigned, we perform latent variable inference with the latent inference network. +Since the source string s remains the same, we reuse the value of the contextual vector hs in the weak supervision +section. Let ¯tj be the jth words in the latent sample and ej be the corresponding output of the target decoder model. +We construct the latent sample ¯t using contextual vector hs and all t∗ +1:j−1 words in the pseudo-labels: +ej = fdecode(hs; t∗ +1:j−1) +¯tj ∼ Gumbel-TOPk(ej, τ) +(3) +The ¯ti is drawn via the Gumbel-Trick [36, 37] and TOP-k subset sampling technique [38] based on temperature τ, +which controls the probability distribution of the samples. At a high temperature τ, we equivalently sample from a +uniform distribution; at a low temperature τ, we equivalently sample from a categorical distribution. +We explore two different schemes commonly used in the literature: (1) we use a fixed temperature τ of 0.1, as in +[39]; and (2) we gradually anneal the temperature τ from a high temperature of 10 to a low temperature of 0.01, as in +[40]. Our empirical results suggest that annealing the temperature τ during training yields significantly better results +(p < .05; Wilcoxon test) and are thus used to report the final results. We use a k-value of 10 as suggested in [19]. +2.3 +Source Reconstruction +For the source reconstruction network, the encoder model takes the latent target sequence string ¯t = (¯t1, ..., ¯tn) as input +and produces its corresponding contextual vector h¯t = (h¯t +1, ..., h¯t +n): +h¯t = fencode(¯t) +(4) +3We leverage linguistic knowledge of paraphrase generation task, in which a paraphrase text string can be considered as its own +paraphrase. +3 + +Running Title for Header +Let ˆsk be the kth word in the reconstructed source string, during the training; we retrieve the reconstructed source string +ˆs via: +ˆsk = fdecode(h¯t; s1:k−1) +(5) +2.4 +Learning and Inference for VSAR +In the SVAR model, there are two sets of parameters, φ and θ, which are required to be updated. Let S be the observed +random variable for the source text, ¯T be the latent random variable for the target text, and N be the total number of the +unlabelled source text. We have the following joint likelihood for the SVAR model, parameterised by θ: +p(S, ¯T ; θ) = +N +� +i=1 +p(s(i)|¯t(i); θ)p(¯t(i)) +(6) +The log marginal likelihood L1 of the observed data that we will be approximated during training is log p(S; θ). We +adopt amortised variational inference [15, 16, 17] and build a surrogate function approximated with a neural network +q( ¯T |S; φ), parameterised by φ, to derive the evidence lower bound (ELBO) for the joint likelihood: +L1 = log +� +¯ +T +p(S, ¯T ; θ) ≥ LELBO(S, ¯T ; θ, φ) += +N +� +i=1 +{Eq(¯t|s(i);φ)[log p(s(i)|¯t; θ)] − DKL[q(¯t|s(i); φ)||p(¯t)]} +(7) +The most common variational family in the VAE framework relies on the reparameterisation trick [15], which is not +applicable for the non-differentiable discrete latent variable. An approach for optimising learning with such latent +variables uses the REINFORCE algorithm [17, 41]; however, this algorithm generally suffers from high variance. +In this paper, we instead use Gumbel-Softmax [36, 37] with differentiable subset sampling [38] to retrieve top-k +samples without replacement. Nevertheless, since sampling a one-hot form vector induces high variance, we apply the +straight-through technique [42] as a biased estimator of the gradient, to combat this variance. +During training, while optimising the log-likelihood, we perform learning (θ) and inference (φ) at the same time. The +parameters are jointly optimised with the same optimiser. Since we are sharing parameters in our model, in practice, we +are updating the same set of parameters (shared by θ and φ) with source data only. +3 +Dual Directional Learning (DDL) +In this section, we introduce the Dual Directional Learning (DDL) model, which we use for supervised paraphrase +generation. The DDL model consists of two sets of standard Transformer models [43], each with its own separate +neural networks - an encoder and a decoder. We perform standard sequence-to-sequence learning, with fully observed +parallel source text s and its associated target pair t, in dual directions. The target generation network pθt|s(t|s) takes +source text s as input and generates target text t and the source generation network pθs|t(s|t) takes target text t as +input and generates source text s. +3.1 +Parameter Learning +In the DDL model, there are two sets of parameters, θs|t and θt|s, which are required to be updated. Let S be the +observed random variable for source text, T be the observed random variable for target text, and M be the number of +labelled pairs; we then have the following conditional likelihood for our DDL model: +p(S|T ; θs|t) = +M +� +i=1 +p(s(i)|t(i); θs|t) +p(T |S; θt|s) = +M +� +i=1 +p(t(i)|s(i); θt|s) +(8) +4 + +Running Title for Header +The log conditional likelihood L2 of the observed data pairs can be jointly learnt during training as: +L2 = +M +� +i=1 +(log p(s(i)|t(i); θs|t) + log p(t(i)|s(i); θt|s)) +(9) +During training, we perform dual learning (θs|t and θt|s) at the same time and the parameters are jointly optimised +with the same optimiser. +3.2 +Parameter Sharing +Once again, motivated by the benefits of multi-task learning for natural language generation [27, 28, 29, 30], we share +model parameters for the target generation and the source generation network. Although sharing parameters is a +very simple technique, as shown in Table 1 and Table 2, the DDL model significantly improves the performance of +paraphrase generation with respect to the Transformer baseline (p < .05; Wilcoxon test), which only handles sequence +to sequence learning in a single direction. +Source +Target Generation ( + ) +Target +Target +Source Generation ( + ) +Source +Dual Directional Learning (DDL) +Source +Latent Inference +Latent Target +Source Reconstruction +Source +Variational Sequence Auto-Encoding Reconstruction (VSAR) +Source +Target Generation ( + ) +Target +Dual Directional Learning (DDL) +Target +Source Generation ( + ) +Source +Knowledge Reinforced Fine-Tuning +Knowledge Reinforced Pre-Training +Figure 2: Knowledge Reinforced Learning. +5 + +Running Title for Header +4 +Combining VSAR and DDL for Semi-supervised Learning +In this section, we introduce our semi-supervised learning model (VSAR+DDL), which combines models presented in +previous sections. For semi-supervised learning, the log-likelihood of the data can be expressed as follow: +L = L1 + L2 += +N +� +i=1 +{Eq(¯t|s(i);φ)[log p(s(i)|¯t; θ)] − DKL[q(¯t|s(i); φ)||p(¯t)]} ++ +M +� +i=1 +(log p(s(i)|t(i); θs|t) + log p(t(i)|s(i); θt|s)) +(10) +As suggested in equation 10, for unsupervised learning and supervised learning, the likelihood function involves +the same set of conditional probability between s and t. We hypothesise that sharing parameters between these +two models is beneficial and we share two sets of neural network parameters from the VSAR and DDL models (i.e. +qφ(¯t|s) ≡ pθt|s(t|s) and pθ(s|¯t) ≡ pθs|t(s|t)). This allows the strong supervision signal from the DDL model to +directly contribute to the VSAR model. At the same time, the unsupervised signal from the VSAR model can benefit +the generalisation of the DDL model. +4.1 +Knowledge Reinforced Learning +Our empirical experiments suggest that our combined model (DDL+VSAR) suffers from a cold-start problem for +parameter optimisation when conducting semi-supervised learning from scratch. We found that a key to the success +of our model is to have better initialisation of the model weight. Hence, we present a novel training scheme called +knowledge reinforced learning (Figure 2), which includes two-stage training. In stage one (pre-training), we conduct +supervised learning with our DDL model on paired training sets, as demonstrated in Algorithm 1. In stage two +(fine-tuning), we initialise the VSAR model parameter with the best performance DDL model from stage one; and we +conduct semi-supervised learning with labelled and unlabelled data, as demonstrated in Algorithm 2. The intuition is to +inject better preliminary information into training the SVAR model. +Algorithm 1 Knowledge Reinforced Pre-Training +Input: +Supervised Training Data (DS +T = {(s1, t1), ..., (sN, tN)}), Supervised Validation Data (DS +V ) +Parameter: +DDL Model: θs|t and θt|s +Parameter Sharing: +Set θs|t equals to θt|s through out knowledge reinforced pre-training +Output: θs|t +∗ and θt|s +∗ +1: Initialise θs|t and θt|s with a random seed; set maximum training epochs as T ; set L2 +∗ = 0 +2: while Maximum epochs not reached do +3: +Update θs|t and θt|s with mini-batch data from DS +T based on Equation 9 +4: +if L2 in Equation 9 calculated based on DS +V bigger than L2 +∗ then +5: +Set L2 +∗ ← L2 +6: +Set θs|t +∗ ← θs|t +7: +Set θt|s +∗ ← θt|s +8: +end if +9: end while +Return: θs|t +∗ and θt|s +∗ +4.2 +Effect of Language Model Prior +In literature [44, 25, 45, 26], a language model prior is introduced for regularisation purposes, which enforces samples +to more likely contain a ‘reasonable’ natural language, especially at the beginning of the training. Hence, we adopt +the same approach and use a prior in our model. We empirically found the prior useful when the labelled dataset is +relatively small. However, surprisingly, we found that training without a prior in the VSAR model yields better results +6 + +Running Title for Header +Algorithm 2 Knowledge Reinforced Fine-Training +Input: +Unsupervised Data (DU = {s1, ..., sM}) +Supervised Training Data (DS +T = {(s1, t1), ..., (sN, tN)}), Supervised Validation Data (DS +V ) +Parameter: +VSAR Model: φ and θ; DDL Model: θs|t and θt|s +Parameter Sharing: +Set φ equals to θt|s; θ equals to θs|t; and θs|t equals to θt|s through out knowledge reinforced fine-tuning +Output: θs|t +∗∗, θt|s +∗∗; φ∗∗ and θ∗∗ +1: Initialise φ and θt|s with θt|s +∗; and initialise θ and θs|t with θs|t +∗; set maximum training epochs as T ; set +L2 +∗ = 0. +2: while Maximum epochs not reached do +3: +Update θs|t and θt|s with mini-batch data from DS +T based on Equation 9 +4: +Update φ and θ with mini-batch data from DU based on Equation 7 +5: +if L2 in Equation 9 calculated based on DS +V bigger than L2 +∗ then +6: +Set L2 +∗ ← L2 +7: +Set θs|t +∗∗ ← θs|t +8: +Set θt|s +∗∗ ← θt|s +9: +Set φ∗∗ ← φ +10: +Set θ∗∗ ← θ +11: +end if +12: end while +Return: θs|t +∗∗, θt|s +∗∗; φ∗∗ and θ∗∗ +when the dataset is large with our parameter initialisation method. The improvement is significant (p < .05; Wilcoxon +test), as shown in Table 3 and Table 4. We report the results without language model prior as DDL +VSAR∗, and the +log-likelihood becomes: +L∗ = +N +� +i=1 +{Eq(¯t|s(i);φ)[log p(s(i)|¯t; θ)]} ++ +M +� +i=1 +(log p(s(i)|t(i); θs|t) + log p(t(i)|s(i); θt|s)) +(11) +To further investigate this issue, we conducted experiments to compare the performance of semi-supervised learning +when training with Equation 10 (with prior) and 11 (without prior) under different data portion setting. We empirically +found that with a low portion of labelled data, the combined model (DDL+VSAR) with a prior grant significantly +(p < .05; Wilcoxon test) better performances and is more stable. This aligns with the observations in [44, 25, 45, 26]. +However, with a large portion of labelled data, the combined model (DDL+VSAR) without the prior is significantly +(p < .05; Wilcoxon test) better. +We argue that this phenomenon relates to our choice of the prior as it is pre-trained on unlabelled source text corpus +instead of on the target text corpus. This approximation leads to a distribution shift from the true prior distribution +p(¯t). Thus, when a low portion of labelled data is used in Algorithm 1, the final DDL parameters θs|t +∗ and θt|s +∗ for +initialisation VSAR model in Algorithm 2 is not good enough. The prior in this case can still benefit the combined +model in the semi-supervised learning setting. However, with a large portion of labelled data, the initialisation is good +enough, and the distribution shift can harm the combined model in this case. +4.3 +Semi-supervised Learning Setup +Under the semi-supervised learning setting, we limit the size of the supervised source and target pairs to be less than or +equal to the unsupervised source text (M ≤ N), as we could otherwise just conduct supervised learning to take full +advantage of observed data pairs. This paper presents a thorough study on different sizes of M and N. Experimental +results under this setting are presented in Table 1 and Table 2. +7 + +Running Title for Header +5 +Related Work +5.1 +Paraphrase Generation +Paraphrases express the surface forms of the underlying semantic content [6] and capture the essence of language +diversity [46]. Early work on automatic generation of paraphrase are generally rule-based [7, 8], but the recent trend +brings to fore neural network solutions [47, 19, 10, 12, 13, 20, 6]. Current research for paraphrasing mainly focuses +on supervised methods, which require the availability of a large number of source and target pairs. In this work, we +alternatively explore a semi-supervised paraphrasing method, where only a fraction of source and target pairs are +observed, and where a large number of unlabelled source text exists. We made an assumption that each missing target +text can be considered as a latent variable in deep generative models. In this paper, we present two models and combine +them for paraphrasing: one for unsupervised learning and one for supervised learning. Our combined model extends +[19] and models jointly the distribution of source and target, instead of the conditional probability of a target, given the +source. Furthermore, our combined model is associated with prior works that introduce a discrete latent variable [20, 6], +and it uses an arguably more expressive latent variable, in the form of language. +5.2 +Deep Latent Variable Models for Text +Deep latent variable models have been studied for text modelling [48, 49]. The most common and widely adopted latent +variable model is the standard VAE model with a Gaussian prior [50], which suffers from posterior collapse [51, 52]. +Multiple studies have been conducted to combat this issue [44, 53, 54]. In particular, β-VAE [44] introduces a penalty +term to balance VAE reconstruction and prior regularisation intuitively and is adopted as one of our baselines. +While much of the research focuses on continuous latent variable models, the text is naturally presented in discrete +form and may not be well represented with continuous latent variables. Early work on discrete deep latent variable +models [25, 55] adopted the REINFORCE algorithm [17, 41]; however, it suffers from very high variance. With the +recent advancement in statistical relaxation techniques, Gumbel-Trick [36, 37] was utilised, to model discrete structures +in the latent variable model of the text [56, 19, 26, 57]. Our work adopts Gumbel-Trick with subset sampling for +natural language generation tasks and, for the first time, studies latent variables as a discrete language sequence for +the paraphrasing task. Our proposed model is strongly associated with [25, 26]; however, we study the problem under +the semi-supervised setup for the paraphrase generation tasks. Furthermore, we present a novel inference algorithm +(our knowledge reinforced learning scheme) to help aid learning in deep generative models and achieve competitive +performance for both full data and data fraction settings. +6 +Experiments +Here, we describe the datasets, experimental setup, evaluation metrics and experimental results. +6.1 +Datasets +MSCOCO [58]: This dataset has been widely adopted to evaluate paraphrase generation methods and contains human- +annotated captions of images. Each image is associated with five captions from different annotators, who describe the +most prominent object or action in an image. We use the 2017 version for our experiments; from the five captions +accompanying each image, we randomly choose one as the source string and one as the target string for training. We +randomly choose one as the source string for testing and use the rest four as the references. +Quora4: This dataset consists of 150K lines of question duplicate pairs, and it has been used as a benchmark dataset for +paraphrase generation since 2017. However, since this dataset does not contain a specific split for training and testing, +prior models are evaluated based on different subset sizes of data. +For both datasets (MSCOCO and Quora), in order to improve re-producibility of our results, we use a pre-trained +tokenizer (’bert-base-uncased’ version) from [59]5 and set the maximum token length as 20 (by removing the tokens +beyond the first 20). Following [60, 19, 13], we use training, validation and test sets as 100K, 4K and 20K for Quora +dataset; and 93K, 4K and 20K for MSCOCO. For the complementary study in Table 5 and Table 6, we use training, +validation and test sets as 100K, 24K and 24K for Quora dataset; and 100K, 5K and 5K for MSCOCO, in order to have +a fair comparison with the results reported in [20, 6]. +4https://quoradata.quora.com/First-Quora-Dataset-Release-Question-Pairs +5https://github.com/huggingface/transformers +8 + +Running Title for Header +6.2 +Baselines +We consider several state-of-the-art baselines, presented in Table 3, Table 4, Table 5, and Table 6. Note that these +experimental results are directly taken from [13]6 and [6]. For evaluation, we start with our implementation of +the Transformer model as the absolute baseline, which achieves competitive performance as reported in [13]. The +Transformer model [43] is consider as the SOTA model which is very ‘hard to beat’. We report our model performance +based on a similar setup as in [13] and [6]. +6.3 +Experimental Setup +In this section, we introduce our primary experimental setup. We do not use any external word embedding such as +Glove [61], word2vec[62] or BERT [59] for initialisation; rather, we obtain word embedding with end-to-end training, +in order not to use any prior knowledge and better understand the impact of our model. We use the ‘base’ version +of the Transformer model [43], which is a 6-layer model with 512 hidden units and 8 heads for each encoder and +decoder network. In each encoder and decoder, we have a separate learnable position embedding and its associated +word embedding component. +We use a greedy decoding scheme for paraphrase generation, which is fast and cheap to compute. For model optimisation, +we use Adam [63] as our optimiser with default hyper-parameters (β1 = 0.9, β2 = 0.999, ϵ = 1e − 8). We conduct +all the experiments with a batch size of 512 for the Quora and MSCOCO datasets. We set the learning rate as 1e − 4 +for MSCOCO and 2e − 4 for Quora based on empirical experiments. All experiments are run for a maximum of 30 +epochs on NVidia GPU Cluster with A100 GPU. Experiments are repeated three times with different random seeds +(1000, 2000 and 3000) and the average result is reported in Tables 1-6. +6.4 +Evaluation +In this paper, we evaluate our models based on quantitative metrics: BLEU [64]7, ROUGE [65]8, and i-BLEU [66]. +BLEU (Bilingual Evaluation Understudy) and ROUGE (Recall-Oriented Understudy for Gisting Evaluation) scores are +based on ‘n-gram’ coverage between system-generated paraphrase(s) and reference sentences. They have been used +widely to automatically evaluate the quality and accuracy of natural language generation tasks. +Previous work has shown that automatic evaluation metrics can perform well for paraphrase identification tasks [67] +and correlate well with human judgements in evaluating generated paraphrases [68]. Recent papers introduce additional +i-BLEU [66] metrics to balance the fidelity of generated outputs to reference paraphrases (BLEU) as well as the level of +diversity introduced (self-B). For all metrics apart from self-B, the higher the value, the better the model performs. +6.5 +Results and Discussion +6.5.1 +Learning with a Fraction of Data +In this section, we present results which are based on a fraction of labelled data in Table 1 and Table 2. In both tables, +we present the results of two models - the supervised learning model, DDL and the semi-supervised learning model, +DDL + VSAR. In a semi-supervised learning setting, VSAR is trained on unlabelled data, and DDL is trained on +labelled data. The DDL+VSAR1 model employs equivalent sized labelled and unlabelled datasets, which come from +the same source and target pairs, so there is no additional information applied in this case. The DDL+VSAR2 model +employs the full unlabelled dataset in addition to the existing labelled dataset, which is the true semi-supervised setting. +Results suggest that the DDL+VSAR1 model achieves competitive or better performance on most metrics’ scores +compared to the supervised DDL model only trained on labelled data; especially with a lower fraction of the data (for +example, the significant improvement for 20K is more noticeable than for 50K). Furthermore, fixing the labelled data +size, the DDL+VSAR2 model achieves significantly better performance by using additional unlabelled data than all +other models reported in both tables (p < .05; Wilcoxon test), which means the semi-supervised learning does work in +this scenario. +6.5.2 +Learning with Complete Data +In this section, we present results based on all labelled data in Table 3 and Table 4. Each table comes with three sections. +In the first section, we present an upper bound (copying the source as a paraphrase) and a lower bound (randomly +6The authors do not make their code publicly available. +7https://www.nltk.org/ +8https://github.com/huggingface/datasets/tree/master/metrics/rouge +9 + +Running Title for Header +Table 1: Semi-Supervised Learning Experiment Results for Quora. +Model +Labelled +Unlabelled +B-1 +B-2 +B-3 +B-4 +i-B +R-1 +R-2 +R-L +DDL +20K +− +46.68 +33.44 +25.46 +20.18 +11.08 +47.57 +25.42 +45.50 +DDL+VSAR1 +20K +20K +47.80 ↑ +34.33 ↑ +26.17 ↑ +20.76 ↑ +11.25 ↑ +48.03 ↑ +25.82 ↑ +45.84 ↑ +DDL+VSAR2 +20K +100K +50.26 ↑ +36.87 ↑ +28.50 ↑ +22.82 ↑ +11.60 ↑ +51.51 ↑ +28.45 ↑ +49.07 ↑ +DDL +50K +− +53.31 +40.22 +31.70 +25.80 +13.80 +55.63 +32.15 +53.13 +DDL+VSAR1 +50K +50K +53.33 ↑ +39.93 ↓ +31.39 ↓ +25.49 ↓ +13.45 ↓ +55.51 ↓ +31.90 ↓ +52.95 ↓ +DDL+VSAR2 +50K +100K +53.79 ↑ +40.47 ↑ +31.86 ↑ +25.93 ↑ +13.67 ↓ +55.58 ↓ +31.89 ↓ +52.93 ↓ +Table 2: Semi-Supervised Learning Experiment Results for MSCOCO. +Model +Labelled +Unlabelled +B-1 +B-2 +B-3 +B-4 +i-B +R-1 +R-2 +R-L +DDL +20K +− +66.82 +47.25 +33.14 +23.75 +16.66 +40.53 +14.95 +36.94 +DDL+VSAR1 +20K +20K +66.98 ↑ +47.28 ↑ +33.10 ↓ +23.72 ↓ +16.54 ↓ +40.60 ↑ +14.95 ↑ +36.94 ↑ +DDL+VSAR2 +20K +93K +67.64 ↑ +48.00 ↑ +33.96 ↑ +24.55 ↑ +16.68 ↑ +40.87 ↑ +15.12 ↑ +37.01 ↑ +DDL +50K +− +69.39 +50.17 +36.06 +26.49 +18.43 +42.08 +16.31 +38.27 +DDL+VSAR1 +50K +50K +69.43 ↑ +50.21 ↑ +36.08 ↑ +26.45 ↓ +18.31 ↓ +42.20 ↑ +16.33 ↑ +38.31 ↑ +DDL+VSAR2 +50K +93K +69.91 ↑ +50.65 ↑ +36.52 ↑ +26.93 ↑ +18.51 ↑ +42.39 ↑ +16.46 ↑ +38.40 ↑ +Table 3: Experiment Results for Quora. +Model +B-1 +B-2 +B-3 +B-4 +i-B +R-1 +R-2 +R-L +Upper Bound (Copy Source) +63.36 +49.99 +40.47 +33.54 +- +63.04 +38.15 +59.64 +Lower Bound (Random Select) +16.10 +4.50 +1.94 +0.79 +- +9.13 +1.54 +8.79 +Residual-LSTM [69] +53.59 +39.49 +30.25 +23.69 +15.93 +55.10 +33.86 +53.61 +β-VAE [44] +47.86 +33.21 +24.96 +19.73 +10.28 +47.62 +25.49 +45.46 +Transformer [43] +53.56 +40.47 +32.11 +25.01 +17.98 +57.82 +32.58 +56.26 +LBOW-TOPk [19] +55.79 +42.03 +32.71 +26.17 +19.03 +58.79 +34.57 +56.43 +IANet+X [13] +56.06 +42.69 +33.38 +26.52 +19.62 +59.33 +35.01 +57.13 +Transformer (our implementation) +54.73 +41.59 +32.96 +26.94 +14.50 +56.90 +33.28 +54.29 +DDL (our model) +55.97 ↑ +43.02 ↑ +34.32 ↑ +28.19 ↑ +14.83 ↑ +58.80 ↑ +35.00 ↑ +56.11 ↑ +DDL + SVAR (our model) +55.79 ↑ +42.79 ↑ +34.11 ↑ +28.01 ↑ +14.92 ↑ +58.61 ↑ +34.75 ↑ +55.91 ↑ +DDL + SVAR∗ (our model) +55.99 ↑ +43.05 ↑ +34.37 ↑ +28.23 ↑ +14.81 ↑ +58.79 ↑ +35.02 ↑ +56.14 ↑ +Table 4: Experiment Results for MSCOCO. +Model +B-1 +B-2 +B-3 +B-4 +i-B +R-1 +R-2 +R-L +Upper Bound (Copy Source) +64.97 +44.90 +30.69 +21.30 +- +39.18 +12.96 +34.61 +Lower Bound (Random Select) +32.34 +10.99 +3.81 +1.68 +- +17.58 +1.51 +16.27 +Residual-LSTM [69] +70.24 +48.65 +34.04 +23.66 +18.72 +41.07 +15.26 +37.35 +β-VAE [44] +70.04 +47.59 +32.29 +22.54 +18.34 +40.72 +14.75 +36.75 +Transformer [43] +71.31 +49.86 +35.55 +24.68 +19.81 +41.49 +15.84 +37.09 +LBOW-TOPk [19] +72.60 +51.14 +35.66 +25.27 +21.07 +42.08 +16.13 +38.16 +IANet+X [13] +72.10 +52.22 +37.39 +26.06 +21.28 +43.81 +16.35 +39.65 +Transformer (our implementation) +68.72 +49.64 +35.87 +26.63 +18.59 +42.09 +16.53 +38.35 +DDL (our model) +70.75 ↑ +51.72 ↑ +37.62 ↑ +27.95 ↑ +19.37 ↑ +43.00 ↑ +17.01 ↑ +39.06 ↑ +DDL + SVAR (our model) +70.84 ↑ +51.84 ↑ +37.75 ↑ +28.04 ↑ +19.39 ↑ +43.05 ↑ +17.04 ↑ +39.07 ↑ +DDL + SVAR∗ (our model) +70.99 ↑ +51.91 ↑ +37.82 ↑ +28.12 ↑ +19.39 ↑ +43.00 ↑ +17.03 ↑ +39.02 ↑ +Table 5: Complement Results for Quora. +Model +B-4 +self-B +i-B +Separator [20] +23.68 +24.20 +14.10 +HRQ-VAE [6] +33.11 +40.35 +18.42 +Transformer (our implementation) +26.92 +35.33 +14.47 +DDL + SVAR (our model) +28.15 ↑ +38.92 ↓ +14.73 ↑ +DDL + SVAR∗ (our model) +28.16 ↑ +39.07 ↓ +14.71 ↑ +selecting ground truth as a paraphrase) calculated based on the test split (as in [70]). This is used as an indication of +how well the model performs. In the second section, we present major state-of-the-art models published in recent years. +In the third section, we present our own implementation of the Transformer model, which we consider as our absolute +baseline, and present results for our models. Our implementation is competitive with the ones reported in recent papers. +10 + +Running Title for Header +Table 6: Complement Results for MSCOCO. +Model +B-4 +self-B +i-B +Separator [20] +20.59 +12.76 +13.92 +HRQ-VAE [6] +27.90 +16.58 +19.04 +Transformer (our implementation) +26.87 +13.50 +18.79 +DDL + SVAR (our model) +27.87 ↑ +15.42 ↓ +19.21 ↑ +DDL + SVAR∗ (our model) +27.92 ↑ +15.21 ↓ +19.29 ↑ +For our models, DDL is our supervised model, DDL+VSAR is our semi-supervised model, and DDL+VSAR∗ is our +model with no prior used. Compared with state-of-the-art supervised models, our models achieve better BLEU scores +and competitive Rouge scores for both datasets. Our complementary experimental results are presented in Table 5 and +Table 6, which we compare with two more recent state-of-the-art models. Our models once again achieve better or +competitive performance than the reported, which means our semi-supervised model is competitive with state-of-the-art +supervised baselines. +7 +Conclusions +In this paper, we have introduced a semi-supervised deep generative model for paraphrase generation. The unsupervised +model is based on the variational auto-encoding framework and provides an effective method to handle missing labels. +The supervised model conducts dual learning and injects supervised information into the unsupervised model. With our +novel knowledge reinforced training scheme, we empirically demonstrate that semi-supervised learning benefits our +combined model, given unlabelled data and a fraction of the paired data. The evaluation results show that our combined +model improves upon a very strong baseline model in a semi-supervised setting. We also observe that, even for the full +dataset, our combined model achieves competitive performance with the state-of-the-art models for two paraphrase +generation benchmark datasets. Additionally, we are able to model language as a discrete latent variable sequence for +paraphrase generation tasks. Importantly, the resultant generative model is able to effectively exploit both supervised +and unsupervised data in sequence-to-sequence tasks. +References +[1] Li Dong, Jonathan Mallinson, Siva Reddy, and Mirella Lapata. Learning to paraphrase for question answering. +In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 875–886, +Copenhagen, Denmark, 2017. Association for Computational Linguistics. +[2] Minsuk Lee, James Cimino, Hai Ran Zhu, Carl Sable, Vijay Shanker, John Ely, and Hong Yu. Beyond information +retrieval—medical question answering. In AMIA annual symposium proceedings, volume 2006, page 469. +American Medical Informatics Association, 2006. +[3] Xuchen Yao and Benjamin Van Durme. Information extraction over structured data: Question answering with +Freebase. In Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: +Long Papers), pages 956–966, Baltimore, Maryland, 2014. Association for Computational Linguistics. +[4] Yuanjie Liu, Shasha Li, Yunbo Cao, Chin-Yew Lin, Dingyi Han, and Yong Yu. Understanding and summarizing +answers in community-based question answering services. In Proceedings of the 22nd International Conference +on Computational Linguistics (Coling 2008), pages 497–504, Manchester, UK, 2008. Coling 2008 Organizing +Committee. +[5] Tim Miller. Explanation in artificial intelligence: Insights from the social sciences. Artificial intelligence, +267:1–38, 2019. +[6] Tom Hosking, Hao Tang, and Mirella Lapata. Hierarchical sketch induction for paraphrase generation. arXiv +preprint arXiv:2203.03463, 2022. +[7] David Kauchak and Regina Barzilay. Paraphrasing for automatic evaluation. In Proceedings of the Human +Language Technology Conference of the NAACL, Main Conference, pages 455–462, New York City, USA, 2006. +Association for Computational Linguistics. +[8] Shashi Narayan, Siva Reddy, and Shay B. Cohen. Paraphrase generation from latent-variable PCFGs for semantic +parsing. In Proceedings of the 9th International Natural Language Generation conference, pages 153–162, +Edinburgh, UK, 2016. Association for Computational Linguistics. +11 + +Running Title for Header +[9] George A. Miller. WordNet: A lexical database for English. In Speech and Natural Language: Proceedings of a +Workshop Held at Harriman, New York, February 23-26, 1992, 1992. +[10] Ashutosh Kumar, Kabir Ahuja, Raghuram Vadapalli, and Partha Talukdar. Syntax-guided controlled generation of +paraphrases. Transactions of the Association for Computational Linguistics, 8:329–345, 2020. +[11] Jianing Zhou and Suma Bhat. Paraphrase generation: A survey of the state of the art. In Proceedings of the 2021 +Conference on Empirical Methods in Natural Language Processing, pages 5075–5086, 2021. +[12] Yuxian Meng, Xiang Ao, Qing He, Xiaofei Sun, Qinghong Han, Fei Wu, Jiwei Li, et al. Conrpg: Paraphrase +generation using contexts as regularizer. arXiv preprint arXiv:2109.00363, 2021. +[13] Yixuan Su, David Vandyke, Simon Baker, Yan Wang, and Nigel Collier. Keep the primary, rewrite the secondary: +A two-stage approach for paraphrase generation. In Findings of the Association for Computational Linguistics: +ACL-IJCNLP 2021, pages 560–569, Online, 2021. Association for Computational Linguistics. +[14] Ilya Sutskever, Oriol Vinyals, and Quoc V. Le. Sequence to sequence learning with neural networks. In Zoubin +Ghahramani, Max Welling, Corinna Cortes, Neil D. Lawrence, and Kilian Q. Weinberger, editors, Advances in +Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, +December 8-13 2014, Montreal, Quebec, Canada, pages 3104–3112, 2014. +[15] Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In Yoshua Bengio and Yann LeCun, +editors, 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, +2014, Conference Track Proceedings, 2014. +[16] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate +inference in deep generative models. In Proceedings of the 31th International Conference on Machine Learning, +ICML 2014, Beijing, China, 21-26 June 2014, volume 32 of JMLR Workshop and Conference Proceedings, pages +1278–1286. JMLR.org, 2014. +[17] Andriy Mnih and Karol Gregor. Neural variational inference and learning in belief networks. In Proceedings of +the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21-26 June 2014, volume 32 +of JMLR Workshop and Conference Proceedings, pages 1791–1799. JMLR.org, 2014. +[18] Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. +science, 313(5786):504–507, 2006. +[19] Yao Fu, Yansong Feng, and John P. Cunningham. Paraphrase generation with latent bag of words. In Hanna M. +Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B. Fox, and Roman Garnett, editors, +Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing +Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 13623–13634, 2019. +[20] Tom Hosking and Mirella Lapata. Factorising meaning and form for intent-preserving paraphrasing. In Proceed- +ings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International +Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 1405–1418, Online, 2021. +Association for Computational Linguistics. +[21] Di He, Yingce Xia, Tao Qin, Liwei Wang, Nenghai Yu, Tie-Yan Liu, and Wei-Ying Ma. Dual learning for machine +translation. In Daniel D. Lee, Masashi Sugiyama, Ulrike von Luxburg, Isabelle Guyon, and Roman Garnett, +editors, Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information +Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pages 820–828, 2016. +[22] Shang-Yu Su, Chao-Wei Huang, and Yun-Nung Chen. Dual supervised learning for natural language understanding +and generation. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, +pages 5472–5477, Florence, Italy, 2019. Association for Computational Linguistics. +[23] Shang-Yu Su, Yung-Sung Chuang, and Yun-Nung Chen. Dual inference for improving language understanding +and generation. In Findings of the Association for Computational Linguistics: EMNLP 2020, pages 4930–4936, +Online, 2020. Association for Computational Linguistics. +[24] Shang-Yu Su, Chao-Wei Huang, and Yun-Nung Chen. Towards unsupervised language understanding and +generation by joint dual learning. In Proceedings of the 58th Annual Meeting of the Association for Computational +Linguistics, pages 671–680, Online, 2020. Association for Computational Linguistics. +[25] Yishu Miao and Phil Blunsom. Language as a latent variable: Discrete generative models for sentence compression. +In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pages 319–328, +Austin, Texas, 2016. Association for Computational Linguistics. +[26] Junxian He, Xinyi Wang, Graham Neubig, and Taylor Berg-Kirkpatrick. A probabilistic formulation of unsuper- +vised text style transfer. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, +Ethiopia, April 26-30, 2020. OpenReview.net, 2020. +12 + +Running Title for Header +[27] Minh-Thang Luong, Quoc V. Le, Ilya Sutskever, Oriol Vinyals, and Lukasz Kaiser. Multi-task sequence to +sequence learning. In Yoshua Bengio and Yann LeCun, editors, 4th International Conference on Learning +Representations, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings, 2016. +[28] Han Guo, Ramakanth Pasunuru, and Mohit Bansal. Dynamic multi-level multi-task learning for sentence +simplification. In Proceedings of the 27th International Conference on Computational Linguistics, pages 462–476, +Santa Fe, New Mexico, USA, 2018. Association for Computational Linguistics. +[29] Han Guo, Ramakanth Pasunuru, and Mohit Bansal. Soft layer-specific multi-task summarization with entailment +and question generation. In Proceedings of the 56th Annual Meeting of the Association for Computational +Linguistics (Volume 1: Long Papers), pages 687–697, Melbourne, Australia, 2018. Association for Computational +Linguistics. +[30] Yiren Wang, ChengXiang Zhai, and Hany Hassan. Multi-task learning for multilingual neural machine translation. +In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages +1022–1034, Online, 2020. Association for Computational Linguistics. +[31] Jingfei Du, Edouard Grave, Beliz Gunel, Vishrav Chaudhary, Onur Celebi, Michael Auli, Veselin Stoyanov, +and Alexis Conneau. Self-training improves pre-training for natural language understanding. In Proceedings of +the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human +Language Technologies, pages 5408–5418, Online, 2021. Association for Computational Linguistics. +[32] Ernie Chang, Vera Demberg, and Alex Marin. Jointly improving language understanding and generation with +quality-weighted weak supervision of automatic labeling. In Proceedings of the 16th Conference of the Euro- +pean Chapter of the Association for Computational Linguistics: Main Volume, pages 818–829, Online, 2021. +Association for Computational Linguistics. +[33] Ronald J Williams and David Zipser. A learning algorithm for continually running fully recurrent neural networks. +Neural computation, 1(2):270–280, 1989. +[34] Steven Burrows, Martin Potthast, and Benno Stein. Paraphrase acquisition via crowdsourcing and machine +learning. ACM Transactions on Intelligent Systems and Technology (TIST), 4(3):1–21, 2013. +[35] Ziqiang Cao, Chuwei Luo, Wenjie Li, and Sujian Li. Joint copying and restricted generation for paraphrase. In +Satinder P. Singh and Shaul Markovitch, editors, Proceedings of the Thirty-First AAAI Conference on Artificial +Intelligence, February 4-9, 2017, San Francisco, California, USA, pages 3152–3158. AAAI Press, 2017. +[36] Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. In 5th International +Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track +Proceedings. OpenReview.net, 2017. +[37] Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. The concrete distribution: A continuous relaxation of +discrete random variables. In 5th International Conference on Learning Representations, ICLR 2017, Toulon, +France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017. +[38] Sang Michael Xie and Stefano Ermon. Reparameterizable subset sampling via continuous relaxations. In Sarit +Kraus, editor, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI +2019, Macao, China, August 10-16, 2019, pages 3919–3925. ijcai.org, 2019. +[39] Jianbo Chen, Le Song, Martin J. Wainwright, and Michael I. Jordan. Learning to explain: An information-theoretic +perspective on model interpretation. In Jennifer G. Dy and Andreas Krause, editors, Proceedings of the 35th +International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, +2018, volume 80 of Proceedings of Machine Learning Research, pages 882–891. PMLR, 2018. +[40] Muhammed Fatih Balin, Abubakar Abid, and James Y. Zou. Concrete autoencoders: Differentiable feature +selection and reconstruction. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th +International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, +volume 97 of Proceedings of Machine Learning Research, pages 444–453. PMLR, 2019. +[41] Volodymyr Mnih, Nicolas Heess, Alex Graves, and Koray Kavukcuoglu. Recurrent models of visual attention. +In Zoubin Ghahramani, Max Welling, Corinna Cortes, Neil D. Lawrence, and Kilian Q. Weinberger, editors, +Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing +Systems 2014, December 8-13 2014, Montreal, Quebec, Canada, pages 2204–2212, 2014. +[42] Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic +neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013. +[43] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, +and Illia Polosukhin. Attention is all you need. In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. +Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett, editors, Advances in Neural Information +13 + +Running Title for Header +Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, +Long Beach, CA, USA, pages 5998–6008, 2017. +[44] Irina Higgins, Loïc Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, +and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. +In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, +Conference Track Proceedings. OpenReview.net, 2017. +[45] Zichao Yang, Zhiting Hu, Chris Dyer, Eric P. Xing, and Taylor Berg-Kirkpatrick. Unsupervised text style transfer +using language models as discriminators. In Samy Bengio, Hanna M. Wallach, Hugo Larochelle, Kristen Grauman, +Nicolò Cesa-Bianchi, and Roman Garnett, editors, Advances in Neural Information Processing Systems 31: Annual +Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, +Canada, pages 7298–7309, 2018. +[46] Ellie Pavlick, Pushpendre Rastogi, Juri Ganitkevitch, Benjamin Van Durme, and Chris Callison-Burch. PPDB +2.0: Better paraphrase ranking, fine-grained entailment relations, word embeddings, and style classification. In +Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International +Joint Conference on Natural Language Processing (Volume 2: Short Papers), pages 425–430, Beijing, China, +2015. Association for Computational Linguistics. +[47] Ankush Gupta, Arvind Agarwal, Prawaan Singh, and Piyush Rai. A deep generative framework for paraphrase +generation. In Sheila A. McIlraith and Kilian Q. Weinberger, editors, Proceedings of the Thirty-Second AAAI +Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI- +18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, +Louisiana, USA, February 2-7, 2018, pages 5149–5156. AAAI Press, 2018. +[48] Yishu Miao, Lei Yu, and Phil Blunsom. Neural variational inference for text processing. In Maria-Florina Balcan +and Kilian Q. Weinberger, editors, Proceedings of the 33nd International Conference on Machine Learning, ICML +2016, New York City, NY, USA, June 19-24, 2016, volume 48 of JMLR Workshop and Conference Proceedings, +pages 1727–1736. JMLR.org, 2016. +[49] Yoon Kim, Sam Wiseman, and Alexander M Rush. A tutorial on deep latent variable models of natural language. +arXiv preprint arXiv:1812.06834, 2018. +[50] Samuel R. Bowman, Luke Vilnis, Oriol Vinyals, Andrew Dai, Rafal Jozefowicz, and Samy Bengio. Generating +sentences from a continuous space. In Proceedings of The 20th SIGNLL Conference on Computational Natural +Language Learning, pages 10–21, Berlin, Germany, 2016. Association for Computational Linguistics. +[51] Adji B. Dieng, Yoon Kim, Alexander M. Rush, and David M. Blei. Avoiding latent variable collapse with generative +skip models. In Kamalika Chaudhuri and Masashi Sugiyama, editors, The 22nd International Conference on +Artificial Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, volume 89 of +Proceedings of Machine Learning Research, pages 2397–2405. PMLR, 2019. +[52] Junxian He, Daniel Spokoyny, Graham Neubig, and Taylor Berg-Kirkpatrick. Lagging inference networks and +posterior collapse in variational autoencoders. In 7th International Conference on Learning Representations, +ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019. +[53] Ali Razavi, Aäron van den Oord, Ben Poole, and Oriol Vinyals. Preventing posterior collapse with delta-vaes. In +7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. +OpenReview.net, 2019. +[54] Yixin Wang, David Blei, and John P Cunningham. Posterior collapse and latent variable non-identifiability. +Advances in Neural Information Processing Systems, 34, 2021. +[55] Tsung-Hsien Wen, Yishu Miao, Phil Blunsom, and Steve J. Young. Latent intention dialogue models. In Doina +Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, ICML +2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings of Machine Learning Research, pages +3732–3741. PMLR, 2017. +[56] Jihun Choi, Kang Min Yoo, and Sang-goo Lee. Learning to compose task-specific tree structures. In Sheila A. +McIlraith and Kilian Q. Weinberger, editors, Proceedings of the Thirty-Second AAAI Conference on Artificial +Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI +Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February +2-7, 2018, pages 5094–5101. AAAI Press, 2018. +[57] Giangiacomo Mercatali and André Freitas. Disentangling generative factors in natural language with discrete +variational autoencoders. arXiv preprint arXiv:2109.07169, 2021. +14 + +Running Title for Header +[58] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and +C Lawrence Zitnick. Microsoft coco: Common objects in context. In European conference on computer vision, +pages 740–755. Springer, 2014. +[59] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional +transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter +of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short +Papers), pages 4171–4186, Minneapolis, Minnesota, 2019. Association for Computational Linguistics. +[60] Zichao Li, Xin Jiang, Lifeng Shang, and Qun Liu. Decomposable neural paraphrase generation. In Proceedings +of the 57th Annual Meeting of the Association for Computational Linguistics, pages 3403–3414, Florence, Italy, +2019. Association for Computational Linguistics. +[61] Jeffrey Pennington, Richard Socher, and Christopher Manning. GloVe: Global vectors for word representation. In +Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages +1532–1543, Doha, Qatar, 2014. Association for Computational Linguistics. +[62] Tomás Mikolov, Ilya Sutskever, Kai Chen, Gregory S. Corrado, and Jeffrey Dean. Distributed representations of +words and phrases and their compositionality. In Christopher J. C. Burges, Léon Bottou, Zoubin Ghahramani, and +Kilian Q. Weinberger, editors, Advances in Neural Information Processing Systems 26: 27th Annual Conference +on Neural Information Processing Systems 2013. Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, +Nevada, United States, pages 3111–3119, 2013. +[63] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann +LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, +May 7-9, 2015, Conference Track Proceedings, 2015. +[64] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of +machine translation. In Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics, +pages 311–318, Philadelphia, Pennsylvania, USA, 2002. Association for Computational Linguistics. +[65] Chin-Yew Lin. ROUGE: A package for automatic evaluation of summaries. In Text Summarization Branches Out, +pages 74–81, Barcelona, Spain, 2004. Association for Computational Linguistics. +[66] Hong Sun and Ming Zhou. Joint learning of a dual SMT system for paraphrase generation. In Proceedings of the +50th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pages 38–42, +Jeju Island, Korea, 2012. Association for Computational Linguistics. +[67] Nitin Madnani, Joel Tetreault, and Martin Chodorow. Re-examining machine translation metrics for paraphrase +identification. In Proceedings of the 2012 Conference of the North American Chapter of the Association for +Computational Linguistics: Human Language Technologies, pages 182–190, Montréal, Canada, 2012. Association +for Computational Linguistics. +[68] Sander Wubben, Antal van den Bosch, and Emiel Krahmer. Paraphrase generation as monolingual translation: +Data and evaluation. In Proceedings of the 6th International Natural Language Generation Conference, 2010. +[69] Aaditya Prakash, Sadid A. Hasan, Kathy Lee, Vivek Datla, Ashequl Qadir, Joey Liu, and Oladimeji Farri. +Neural paraphrase generation with stacked residual LSTM networks. In Proceedings of COLING 2016, the 26th +International Conference on Computational Linguistics: Technical Papers, pages 2923–2934, Osaka, Japan, 2016. +The COLING 2016 Organizing Committee. +[70] Wenqing Chen, Jidong Tian, Liqiang Xiao, Hao He, and Yaohui Jin. A semantically consistent and syntactically +variational encoder-decoder framework for paraphrase generation. In Proceedings of the 28th International +Conference on Computational Linguistics, pages 1186–1198, Barcelona, Spain (Online), 2020. International +Committee on Computational Linguistics. +15 + diff --git a/b9E0T4oBgHgl3EQfWQBc/content/tmp_files/load_file.txt b/b9E0T4oBgHgl3EQfWQBc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3678d5e4ea6cf7c0f9bedb5036f36a3f2b0e9ea9 --- /dev/null +++ b/b9E0T4oBgHgl3EQfWQBc/content/tmp_files/load_file.txt @@ -0,0 +1,924 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf,len=923 +page_content='DEEP LATENT VARIABLE MODELS FOR SEMI-SUPERVISED PARAPHRASE GENERATION ∗ Jialin Yu, Alexandra I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Cristea, Anoushka Harit, Zhongtian Sun, Olanrewaju Tahir Aduragba, Lei Shi, Noura Al Moubayed Department of Computer Science Durham University Durham, UK ABSTRACT This paper explores deep latent variable models for semi-supervised paraphrase generation, where the missing target pair is modelled as a latent paraphrase sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We present a novel unsupervised model named variational sequence auto-encoding reconstruction (VSAR), which performs latent sequence inference given an observed text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' To leverage information from text pairs, we introduce a su- pervised model named dual directional learning (DDL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Combining VSAR with DDL (DDL+VSAR) enables us to conduct semi-supervised learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' however, the combined model suffers from a cold- start problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' To combat this issue, we propose to deal with better weight initialisation, leading to a two-stage training scheme named knowledge reinforced training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our empirical evaluations suggest that the combined model yields competitive performance against the state-of-the-art supervised baselines on complete data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Furthermore, in scenarios where only a fraction of the labelled pairs are available, our combined model consistently outperforms the strong supervised model baseline (DDL) by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Keywords Paraphrase Generation · Semi-supervised Learning · Deep Latent Variable Model 1 Introduction Paraphrase generation is an important Natural Language Processing (NLP) problem, useful in many NLP applications, such as question answering [1], information retrieval [2], information extraction [3] and summarisation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Natural language itself is complicated and may be expressed in various alternative surface forms of the same underlying semantic content [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Hence is critically important to integrate the paraphrase generation model as a component in real-world NLP systems to offer robust responses to end users’ inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Traditional solutions to paraphrase generation are generally rule-based [7, 8], utilising lexical resources, such as WordNet [9], to find word replacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The recent trend brings to fore neural network models [10, 11, 12, 13], which are typically based on a sequence-to-sequence learning paradigm [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' These models have achieved remarkable success for paraphrase generation due to complex architectures and sophis- ticated conditioning mechanisms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' soft, hard and self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' However, the advancement of such models is primarily based on the availability of large-scale labelled data pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Instead, this paper explores semi-supervised learning scenarios where only a fraction of the labels are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This semi-supervised learning setting is favourable and extremely useful for industry scenarios due to the effort of time and money to obtain good quality human anno- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' A semi-supervised learning model often consists of two components: an unsupervised learning model and a supervised learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' For unsupervised learning, we propose a novel deep generative model, motivated by the classic variational autoencoder (VAE) [15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Furthermore, we propose a novel supervised model that can be integrated with our proposed VAE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Combining both unsupervised and supervised models enable semi-supervised learning by exploiting the VAEs’ ability to marginalise latent variables for unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' ∗Manuscript under review arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='02275v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='CL] 5 Jan 2023 Running Title for Header Traditional VAEs typically embed data representations in a fixed latent space, with the general purpose of dimensionality reduction [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This paper alternatively considers a latent variable in the form of a discrete language sequence with various lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This additionally can enhance the model interpretability, as language is naturally preserved as discrete variables [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Following the recent prior works successfully incorporating discrete latent variables to improve paraphrasing [20, 6], we propose a novel model with more expressive discrete latent variable: variational sequence auto-encoding reconstruction (VSAR) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In order to further boost the model performance on paraphrase generation, motivated by dual learning [21, 22, 23, 24], we employ a novel Dual Directional Learning (DDL) model trained on labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The DDL model shares model parameters with the VSAR model and can be jointly optimised for semi-supervised scenarios, and we discuss this later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our contributions in this paper include: presenting the first study on semi-supervised learning for paraphrasing with deep latent variable models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' introducing two novel models: VSAR (unsupervised) and DDL (supervised), which can be combined for semi-supervised learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' proposing a novel training scheme (Knowledge Reinforced Learning) to deal with cold start problem in the combined model (DDL+VSAR) when used in semi-supervised learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' studying semi-supervised learning scenarios with the combined model on the full fraction of data and empirically showing that our model achieves competitive state-of-the-art results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' presenting a study of semi-supervised scenarios with a fraction of the labelled data and our models show significantly better results than very strong supervised baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' bos bos eos Source Reconstruction bos eos eos Transformer Target Decoder bos Transformer Target Encoder bos eos bos eos eos bos eos eos Transformer Source Encoder bos Transformer Source Decoder Latent Inference Weak Supervision Figure 1: Variational Sequence Auto-Encoding Reconstruction Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 2 Variational Sequence Auto-Encoding Reconstruction (VSAR) In this section, we present the VSAR model (Figure 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The model consists of four separate neural network models a source encoder, a target decoder, a target encoder, and a source decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Under the unsupervised learning setup, we only observe source text s and no target text t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We reformulate the problem of modelling fully observed source text s as modelling partially observed parallel source text s and its associated latent target pair ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We adopt Bayesian inference to marginalise the latent target string ¯t from the joint probability distribution pθ(s,¯t) as shown in Figure 1 based on Equation 7 and hence only the observed source text s is required for VSAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 2The language model prior and weak supervision decoding is omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 2 Running Title for Header In the VSAR model, the latent inference network, parameterised as qφ(¯t|s), takes source text s and generates a latent target sample ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The source reconstruction network, parameterised as pθ(s|¯t), reconstructs the observed source text s back, based on the latent target sample ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' As the prior distribution, a language model is pre-trained on unlabelled source text corpus to approximate the prior distribution p(¯t)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The prior is introduced for regularisation purposes [25, 26], which enforces samples are more likely to be reasonable natural language text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Motivated by the benefits of parameters sharing in multi-task learning for natural language generation [27, 28, 29, 30], we share model parameters for the source encoder and the target encoder, denoted as fencode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' similarly, we share model parameters for the source decoder and the target decoder, denoted as fdecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In the following sections, we use fencode and fdecode to represent all encoders and decoders in the VSAR model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='1 Weak Supervision In the VSAR model, we empirically found that the quality of latent sequence ¯t is very unstable, especially at the beginning of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' To combat this issue, motivated by the idea of weak supervision [31, 32], we propose to use pseudo-labels to guide VSAR throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Before each model forward pass, we first assign pseudo-labels to each token in unobserved latent target sample ¯t with the current model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The pseudo-labels are detached from the computational graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' hence no gradient is updated during the weak supervision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The pseudo-labels can be considered as a weak supervision signal for ‘teacher forcing training’ [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The encoder model takes the source string s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', sn) as input and produces its corresponding contextual vector hs = (hs 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', hs n): hs = fencode(s) (1) We adopt a greedy decoding scheme to assign pseudo-target labels t∗ and assume that a good paraphrase ought to have a similar length as the original sentence [34, 35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' such that t∗ = (t∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', t∗ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Let t∗ i be the ith word in the pseudo target sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' we construct this sequence in an auto-regressive manner: t∗ i = fdecode(hs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' t∗ 1:i−1) (2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='2 Target Inference Once the pseudo-target labels t∗ are assigned, we perform latent variable inference with the latent inference network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Since the source string s remains the same, we reuse the value of the contextual vector hs in the weak supervision section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Let ¯tj be the jth words in the latent sample and ej be the corresponding output of the target decoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We construct the latent sample ¯t using contextual vector hs and all t∗ 1:j−1 words in the pseudo-labels: ej = fdecode(hs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' t∗ 1:j−1) ¯tj ∼ Gumbel-TOPk(ej, τ) (3) The ¯ti is drawn via the Gumbel-Trick [36, 37] and TOP-k subset sampling technique [38] based on temperature τ, which controls the probability distribution of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' At a high temperature τ, we equivalently sample from a uniform distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' at a low temperature τ, we equivalently sample from a categorical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We explore two different schemes commonly used in the literature: (1) we use a fixed temperature τ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='1, as in [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' and (2) we gradually anneal the temperature τ from a high temperature of 10 to a low temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='01, as in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our empirical results suggest that annealing the temperature τ during training yields significantly better results (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wilcoxon test) and are thus used to report the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We use a k-value of 10 as suggested in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='3 Source Reconstruction For the source reconstruction network, the encoder model takes the latent target sequence string ¯t = (¯t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', ¯tn) as input and produces its corresponding contextual vector h¯t = (h¯t 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', h¯t n): h¯t = fencode(¯t) (4) 3We leverage linguistic knowledge of paraphrase generation task, in which a paraphrase text string can be considered as its own paraphrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 3 Running Title for Header Let ˆsk be the kth word in the reconstructed source string, during the training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' we retrieve the reconstructed source string ˆs via: ˆsk = fdecode(h¯t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' s1:k−1) (5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='4 Learning and Inference for VSAR In the SVAR model, there are two sets of parameters, φ and θ, which are required to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Let S be the observed random variable for the source text, ¯T be the latent random variable for the target text, and N be the total number of the unlabelled source text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We have the following joint likelihood for the SVAR model, parameterised by θ: p(S, ¯T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ) = N � i=1 p(s(i)|¯t(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ)p(¯t(i)) (6) The log marginal likelihood L1 of the observed data that we will be approximated during training is log p(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We adopt amortised variational inference [15, 16, 17] and build a surrogate function approximated with a neural network q( ¯T |S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' φ), parameterised by φ, to derive the evidence lower bound (ELBO) for the joint likelihood: L1 = log � ¯ T p(S, ¯T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ) ≥ LELBO(S, ¯T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ, φ) = N � i=1 {Eq(¯t|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='φ)[log p(s(i)|¯t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ)] − DKL[q(¯t|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' φ)||p(¯t)]} (7) The most common variational family in the VAE framework relies on the reparameterisation trick [15], which is not applicable for the non-differentiable discrete latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' An approach for optimising learning with such latent variables uses the REINFORCE algorithm [17, 41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' however, this algorithm generally suffers from high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In this paper, we instead use Gumbel-Softmax [36, 37] with differentiable subset sampling [38] to retrieve top-k samples without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Nevertheless, since sampling a one-hot form vector induces high variance, we apply the straight-through technique [42] as a biased estimator of the gradient, to combat this variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' During training, while optimising the log-likelihood, we perform learning (θ) and inference (φ) at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The parameters are jointly optimised with the same optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Since we are sharing parameters in our model, in practice, we are updating the same set of parameters (shared by θ and φ) with source data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 3 Dual Directional Learning (DDL) In this section, we introduce the Dual Directional Learning (DDL) model, which we use for supervised paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The DDL model consists of two sets of standard Transformer models [43], each with its own separate neural networks - an encoder and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We perform standard sequence-to-sequence learning, with fully observed parallel source text s and its associated target pair t, in dual directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The target generation network pθt|s(t|s) takes source text s as input and generates target text t and the source generation network pθs|t(s|t) takes target text t as input and generates source text s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='1 Parameter Learning In the DDL model, there are two sets of parameters, θs|t and θt|s, which are required to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Let S be the observed random variable for source text, T be the observed random variable for target text, and M be the number of labelled pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' we then have the following conditional likelihood for our DDL model: p(S|T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θs|t) = M � i=1 p(s(i)|t(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θs|t) p(T |S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θt|s) = M � i=1 p(t(i)|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θt|s) (8) 4 Running Title for Header The log conditional likelihood L2 of the observed data pairs can be jointly learnt during training as: L2 = M � i=1 (log p(s(i)|t(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θs|t) + log p(t(i)|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θt|s)) (9) During training, we perform dual learning (θs|t and θt|s) at the same time and the parameters are jointly optimised with the same optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='2 Parameter Sharing Once again, motivated by the benefits of multi-task learning for natural language generation [27, 28, 29, 30], we share model parameters for the target generation and the source generation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Although sharing parameters is a very simple technique, as shown in Table 1 and Table 2, the DDL model significantly improves the performance of paraphrase generation with respect to the Transformer baseline (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wilcoxon test), which only handles sequence to sequence learning in a single direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Source Target Generation ( ) Target Target Source Generation ( ) Source Dual Directional Learning (DDL) Source Latent Inference Latent Target Source Reconstruction Source Variational Sequence Auto-Encoding Reconstruction (VSAR) Source Target Generation ( ) Target Dual Directional Learning (DDL) Target Source Generation ( ) Source Knowledge Reinforced Fine-Tuning Knowledge Reinforced Pre-Training Figure 2: Knowledge Reinforced Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 5 Running Title for Header 4 Combining VSAR and DDL for Semi-supervised Learning In this section, we introduce our semi-supervised learning model (VSAR+DDL), which combines models presented in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' For semi-supervised learning, the log-likelihood of the data can be expressed as follow: L = L1 + L2 = N � i=1 {Eq(¯t|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='φ)[log p(s(i)|¯t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ)] − DKL[q(¯t|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' φ)||p(¯t)]} + M � i=1 (log p(s(i)|t(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θs|t) + log p(t(i)|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θt|s)) (10) As suggested in equation 10, for unsupervised learning and supervised learning, the likelihood function involves the same set of conditional probability between s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We hypothesise that sharing parameters between these two models is beneficial and we share two sets of neural network parameters from the VSAR and DDL models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' qφ(¯t|s) ≡ pθt|s(t|s) and pθ(s|¯t) ≡ pθs|t(s|t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This allows the strong supervision signal from the DDL model to directly contribute to the VSAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' At the same time, the unsupervised signal from the VSAR model can benefit the generalisation of the DDL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='1 Knowledge Reinforced Learning Our empirical experiments suggest that our combined model (DDL+VSAR) suffers from a cold-start problem for parameter optimisation when conducting semi-supervised learning from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We found that a key to the success of our model is to have better initialisation of the model weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Hence, we present a novel training scheme called knowledge reinforced learning (Figure 2), which includes two-stage training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In stage one (pre-training), we conduct supervised learning with our DDL model on paired training sets, as demonstrated in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In stage two (fine-tuning), we initialise the VSAR model parameter with the best performance DDL model from stage one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' and we conduct semi-supervised learning with labelled and unlabelled data, as demonstrated in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The intuition is to inject better preliminary information into training the SVAR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Algorithm 1 Knowledge Reinforced Pre-Training Input: Supervised Training Data (DS T = {(s1, t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', (sN, tN)}), Supervised Validation Data (DS V ) Parameter: DDL Model: θs|t and θt|s Parameter Sharing: Set θs|t equals to θt|s through out knowledge reinforced pre-training Output: θs|t ∗ and θt|s ∗ 1: Initialise θs|t and θt|s with a random seed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' set maximum training epochs as T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' set L2 ∗ = 0 2: while Maximum epochs not reached do 3: Update θs|t and θt|s with mini-batch data from DS T based on Equation 9 4: if L2 in Equation 9 calculated based on DS V bigger than L2 ∗ then 5: Set L2 ∗ ← L2 6: Set θs|t ∗ ← θs|t 7: Set θt|s ∗ ← θt|s 8: end if 9: end while Return: θs|t ∗ and θt|s ∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='2 Effect of Language Model Prior In literature [44, 25, 45, 26], a language model prior is introduced for regularisation purposes, which enforces samples to more likely contain a ‘reasonable’ natural language, especially at the beginning of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Hence, we adopt the same approach and use a prior in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We empirically found the prior useful when the labelled dataset is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' However, surprisingly, we found that training without a prior in the VSAR model yields better results 6 Running Title for Header Algorithm 2 Knowledge Reinforced Fine-Training Input: Unsupervised Data (DU = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', sM}) Supervised Training Data (DS T = {(s1, t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=', (sN, tN)}), Supervised Validation Data (DS V ) Parameter: VSAR Model: φ and θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' DDL Model: θs|t and θt|s Parameter Sharing: Set φ equals to θt|s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ equals to θs|t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' and θs|t equals to θt|s through out knowledge reinforced fine-tuning Output: θs|t ∗∗, θt|s ∗∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' φ∗∗ and θ∗∗ 1: Initialise φ and θt|s with θt|s ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' and initialise θ and θs|t with θs|t ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' set maximum training epochs as T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' set L2 ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 2: while Maximum epochs not reached do 3: Update θs|t and θt|s with mini-batch data from DS T based on Equation 9 4: Update φ and θ with mini-batch data from DU based on Equation 7 5: if L2 in Equation 9 calculated based on DS V bigger than L2 ∗ then 6: Set L2 ∗ ← L2 7: Set θs|t ∗∗ ← θs|t 8: Set θt|s ∗∗ ← θt|s 9: Set φ∗∗ ← φ 10: Set θ∗∗ ← θ 11: end if 12: end while Return: θs|t ∗∗, θt|s ∗∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' φ∗∗ and θ∗∗ when the dataset is large with our parameter initialisation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The improvement is significant (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wilcoxon test), as shown in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We report the results without language model prior as DDL +VSAR∗, and the log-likelihood becomes: L∗ = N � i=1 {Eq(¯t|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='φ)[log p(s(i)|¯t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θ)]} + M � i=1 (log p(s(i)|t(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θs|t) + log p(t(i)|s(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' θt|s)) (11) To further investigate this issue, we conducted experiments to compare the performance of semi-supervised learning when training with Equation 10 (with prior) and 11 (without prior) under different data portion setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We empirically found that with a low portion of labelled data, the combined model (DDL+VSAR) with a prior grant significantly (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wilcoxon test) better performances and is more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This aligns with the observations in [44, 25, 45, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' However, with a large portion of labelled data, the combined model (DDL+VSAR) without the prior is significantly (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wilcoxon test) better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We argue that this phenomenon relates to our choice of the prior as it is pre-trained on unlabelled source text corpus instead of on the target text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This approximation leads to a distribution shift from the true prior distribution p(¯t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Thus, when a low portion of labelled data is used in Algorithm 1, the final DDL parameters θs|t ∗ and θt|s ∗ for initialisation VSAR model in Algorithm 2 is not good enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The prior in this case can still benefit the combined model in the semi-supervised learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' However, with a large portion of labelled data, the initialisation is good enough, and the distribution shift can harm the combined model in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='3 Semi-supervised Learning Setup Under the semi-supervised learning setting, we limit the size of the supervised source and target pairs to be less than or equal to the unsupervised source text (M ≤ N), as we could otherwise just conduct supervised learning to take full advantage of observed data pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This paper presents a thorough study on different sizes of M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Experimental results under this setting are presented in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 7 Running Title for Header 5 Related Work 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='1 Paraphrase Generation Paraphrases express the surface forms of the underlying semantic content [6] and capture the essence of language diversity [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Early work on automatic generation of paraphrase are generally rule-based [7, 8], but the recent trend brings to fore neural network solutions [47, 19, 10, 12, 13, 20, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Current research for paraphrasing mainly focuses on supervised methods, which require the availability of a large number of source and target pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In this work, we alternatively explore a semi-supervised paraphrasing method, where only a fraction of source and target pairs are observed, and where a large number of unlabelled source text exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We made an assumption that each missing target text can be considered as a latent variable in deep generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In this paper, we present two models and combine them for paraphrasing: one for unsupervised learning and one for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our combined model extends [19] and models jointly the distribution of source and target, instead of the conditional probability of a target, given the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Furthermore, our combined model is associated with prior works that introduce a discrete latent variable [20, 6], and it uses an arguably more expressive latent variable, in the form of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='2 Deep Latent Variable Models for Text Deep latent variable models have been studied for text modelling [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The most common and widely adopted latent variable model is the standard VAE model with a Gaussian prior [50], which suffers from posterior collapse [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Multiple studies have been conducted to combat this issue [44, 53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In particular, β-VAE [44] introduces a penalty term to balance VAE reconstruction and prior regularisation intuitively and is adopted as one of our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' While much of the research focuses on continuous latent variable models, the text is naturally presented in discrete form and may not be well represented with continuous latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Early work on discrete deep latent variable models [25, 55] adopted the REINFORCE algorithm [17, 41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' however, it suffers from very high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' With the recent advancement in statistical relaxation techniques, Gumbel-Trick [36, 37] was utilised, to model discrete structures in the latent variable model of the text [56, 19, 26, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our work adopts Gumbel-Trick with subset sampling for natural language generation tasks and, for the first time, studies latent variables as a discrete language sequence for the paraphrasing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our proposed model is strongly associated with [25, 26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' however, we study the problem under the semi-supervised setup for the paraphrase generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Furthermore, we present a novel inference algorithm (our knowledge reinforced learning scheme) to help aid learning in deep generative models and achieve competitive performance for both full data and data fraction settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 6 Experiments Here, we describe the datasets, experimental setup, evaluation metrics and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='1 Datasets MSCOCO [58]: This dataset has been widely adopted to evaluate paraphrase generation methods and contains human- annotated captions of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Each image is associated with five captions from different annotators, who describe the most prominent object or action in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We use the 2017 version for our experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' from the five captions accompanying each image, we randomly choose one as the source string and one as the target string for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We randomly choose one as the source string for testing and use the rest four as the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Quora4: This dataset consists of 150K lines of question duplicate pairs, and it has been used as a benchmark dataset for paraphrase generation since 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' However, since this dataset does not contain a specific split for training and testing, prior models are evaluated based on different subset sizes of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' For both datasets (MSCOCO and Quora), in order to improve re-producibility of our results, we use a pre-trained tokenizer (’bert-base-uncased’ version) from [59]5 and set the maximum token length as 20 (by removing the tokens beyond the first 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Following [60, 19, 13], we use training, validation and test sets as 100K, 4K and 20K for Quora dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' and 93K, 4K and 20K for MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' For the complementary study in Table 5 and Table 6, we use training, validation and test sets as 100K, 24K and 24K for Quora dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' and 100K, 5K and 5K for MSCOCO, in order to have a fair comparison with the results reported in [20, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 4https://quoradata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='quora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='com/First-Quora-Dataset-Release-Question-Pairs 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='com/huggingface/transformers 8 Running Title for Header 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='2 Baselines We consider several state-of-the-art baselines, presented in Table 3, Table 4, Table 5, and Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Note that these experimental results are directly taken from [13]6 and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' For evaluation, we start with our implementation of the Transformer model as the absolute baseline, which achieves competitive performance as reported in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The Transformer model [43] is consider as the SOTA model which is very ‘hard to beat’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We report our model performance based on a similar setup as in [13] and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='3 Experimental Setup In this section, we introduce our primary experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We do not use any external word embedding such as Glove [61], word2vec[62] or BERT [59] for initialisation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' rather, we obtain word embedding with end-to-end training, in order not to use any prior knowledge and better understand the impact of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We use the ‘base’ version of the Transformer model [43], which is a 6-layer model with 512 hidden units and 8 heads for each encoder and decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In each encoder and decoder, we have a separate learnable position embedding and its associated word embedding component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We use a greedy decoding scheme for paraphrase generation, which is fast and cheap to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' For model optimisation, we use Adam [63] as our optimiser with default hyper-parameters (β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='999, ϵ = 1e − 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We conduct all the experiments with a batch size of 512 for the Quora and MSCOCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We set the learning rate as 1e − 4 for MSCOCO and 2e − 4 for Quora based on empirical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' All experiments are run for a maximum of 30 epochs on NVidia GPU Cluster with A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Experiments are repeated three times with different random seeds (1000, 2000 and 3000) and the average result is reported in Tables 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='4 Evaluation In this paper, we evaluate our models based on quantitative metrics: BLEU [64]7, ROUGE [65]8, and i-BLEU [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' BLEU (Bilingual Evaluation Understudy) and ROUGE (Recall-Oriented Understudy for Gisting Evaluation) scores are based on ‘n-gram’ coverage between system-generated paraphrase(s) and reference sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' They have been used widely to automatically evaluate the quality and accuracy of natural language generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Previous work has shown that automatic evaluation metrics can perform well for paraphrase identification tasks [67] and correlate well with human judgements in evaluating generated paraphrases [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Recent papers introduce additional i-BLEU [66] metrics to balance the fidelity of generated outputs to reference paraphrases (BLEU) as well as the level of diversity introduced (self-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' For all metrics apart from self-B, the higher the value, the better the model performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='5 Results and Discussion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='1 Learning with a Fraction of Data In this section, we present results which are based on a fraction of labelled data in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In both tables, we present the results of two models - the supervised learning model, DDL and the semi-supervised learning model, DDL + VSAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In a semi-supervised learning setting, VSAR is trained on unlabelled data, and DDL is trained on labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The DDL+VSAR1 model employs equivalent sized labelled and unlabelled datasets, which come from the same source and target pairs, so there is no additional information applied in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The DDL+VSAR2 model employs the full unlabelled dataset in addition to the existing labelled dataset, which is the true semi-supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Results suggest that the DDL+VSAR1 model achieves competitive or better performance on most metrics’ scores compared to the supervised DDL model only trained on labelled data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' especially with a lower fraction of the data (for example, the significant improvement for 20K is more noticeable than for 50K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Furthermore, fixing the labelled data size, the DDL+VSAR2 model achieves significantly better performance by using additional unlabelled data than all other models reported in both tables (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wilcoxon test), which means the semi-supervised learning does work in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='2 Learning with Complete Data In this section, we present results based on all labelled data in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Each table comes with three sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In the first section, we present an upper bound (copying the source as a paraphrase) and a lower bound (randomly 6The authors do not make their code publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='nltk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='org/ 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='com/huggingface/datasets/tree/master/metrics/rouge 9 Running Title for Header Table 1: Semi-Supervised Learning Experiment Results for Quora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Model Labelled Unlabelled B-1 B-2 B-3 B-4 i-B R-1 R-2 R-L DDL 20K − 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='68 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='44 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='46 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='18 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='08 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='57 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='42 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='50 DDL+VSAR1 20K 20K 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='80 ↑ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='33 ↑ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='17 ↑ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='76 ↑ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='25 ↑ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='03 ↑ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='82 ↑ 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='84 ↑ DDL+VSAR2 20K 100K 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='26 ↑ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='87 ↑ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='50 ↑ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='82 ↑ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='60 ↑ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='51 ↑ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='45 ↑ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='07 ↑ DDL 50K − 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='31 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='22 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='70 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='80 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='80 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='63 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='15 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='13 DDL+VSAR1 50K 50K 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='33 ↑ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='93 ↓ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='39 ↓ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='49 ↓ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='45 ↓ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='51 ↓ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='90 ↓ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='95 ↓ DDL+VSAR2 50K 100K 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 ↑ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='47 ↑ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='86 ↑ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='93 ↑ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='67 ↓ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='58 ↓ 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='89 ↓ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='93 ↓ Table 2: Semi-Supervised Learning Experiment Results for MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Model Labelled Unlabelled B-1 B-2 B-3 B-4 i-B R-1 R-2 R-L DDL 20K − 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='82 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='25 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='14 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='75 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='66 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='53 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='95 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='94 DDL+VSAR1 20K 20K 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='98 ↑ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='28 ↑ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='10 ↓ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='72 ↓ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='54 ↓ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='60 ↑ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='95 ↑ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='94 ↑ DDL+VSAR2 20K 93K 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='64 ↑ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='00 ↑ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='96 ↑ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='55 ↑ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='68 ↑ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='87 ↑ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='12 ↑ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='01 ↑ DDL 50K − 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='39 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='17 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='06 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='49 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='43 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='08 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='31 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='27 DDL+VSAR1 50K 50K 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='43 ↑ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='21 ↑ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='08 ↑ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='45 ↓ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='31 ↓ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='20 ↑ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='33 ↑ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='31 ↑ DDL+VSAR2 50K 93K 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='91 ↑ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='65 ↑ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='52 ↑ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='93 ↑ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='51 ↑ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='39 ↑ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='46 ↑ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='40 ↑ Table 3: Experiment Results for Quora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Model B-1 B-2 B-3 B-4 i-B R-1 R-2 R-L Upper Bound (Copy Source) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='36 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='99 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='47 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='54 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='04 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='15 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='64 Lower Bound (Random Select) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='54 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 Residual-LSTM [69] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='59 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='49 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='25 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='69 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='93 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='10 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='86 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='61 β-VAE [44] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='86 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='21 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='96 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='73 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='28 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='62 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='49 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='46 Transformer [43] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='56 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='47 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='11 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='01 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='98 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='82 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='58 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='26 LBOW-TOPk [19] 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='03 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='71 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='03 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='57 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='43 IANet+X [13] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='06 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='69 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='38 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='52 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='62 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='33 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='01 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='13 Transformer (our implementation) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='73 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='59 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='96 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='94 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='50 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='90 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='28 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='29 DDL (our model) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='97 ↑ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='02 ↑ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='32 ↑ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='19 ↑ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='83 ↑ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='80 ↑ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='00 ↑ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='11 ↑ DDL + SVAR (our model) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 ↑ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 ↑ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='11 ↑ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='01 ↑ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='92 ↑ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='61 ↑ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='75 ↑ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='91 ↑ DDL + SVAR∗ (our model) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='99 ↑ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05 ↑ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='37 ↑ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='23 ↑ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='81 ↑ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 ↑ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='02 ↑ 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='14 ↑ Table 4: Experiment Results for MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Model B-1 B-2 B-3 B-4 i-B R-1 R-2 R-L Upper Bound (Copy Source) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='97 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='90 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='69 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='30 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='18 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='96 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='61 Lower Bound (Random Select) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='34 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='99 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='68 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='51 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='27 Residual-LSTM [69] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='24 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='65 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='04 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='66 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='72 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='07 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='26 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='35 β-VAE [44] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='04 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='59 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='29 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='54 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='34 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='72 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='75 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='75 Transformer [43] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='31 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='86 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='55 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='68 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='81 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='49 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='84 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='09 LBOW-TOPk [19] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='60 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='14 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='66 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='27 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='07 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='08 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='13 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='16 IANet+X [13] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='10 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='22 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='39 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='06 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='28 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='81 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='35 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='65 Transformer (our implementation) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='72 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='64 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='87 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='63 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='59 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='09 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='53 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='35 DDL (our model) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='75 ↑ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='72 ↑ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='62 ↑ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='95 ↑ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='37 ↑ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='00 ↑ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='01 ↑ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='06 ↑ DDL + SVAR (our model) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='84 ↑ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='84 ↑ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='75 ↑ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='04 ↑ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='39 ↑ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='05 ↑ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='04 ↑ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='07 ↑ DDL + SVAR∗ (our model) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='99 ↑ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='91 ↑ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='82 ↑ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='12 ↑ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='39 ↑ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='00 ↑ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='03 ↑ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='02 ↑ Table 5: Complement Results for Quora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Model B-4 self-B i-B Separator [20] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='68 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='20 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='10 HRQ-VAE [6] 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='11 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='35 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='42 Transformer (our implementation) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='92 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='33 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='47 DDL + SVAR (our model) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='15 ↑ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='92 ↓ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='73 ↑ DDL + SVAR∗ (our model) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='16 ↑ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='07 ↓ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='71 ↑ selecting ground truth as a paraphrase) calculated based on the test split (as in [70]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' This is used as an indication of how well the model performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In the second section, we present major state-of-the-art models published in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In the third section, we present our own implementation of the Transformer model, which we consider as our absolute baseline, and present results for our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our implementation is competitive with the ones reported in recent papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 10 Running Title for Header Table 6: Complement Results for MSCOCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Model B-4 self-B i-B Separator [20] 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='59 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='76 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='92 HRQ-VAE [6] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='90 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='58 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='04 Transformer (our implementation) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='87 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='50 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='79 DDL + SVAR (our model) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='87 ↑ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='42 ↓ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='21 ↑ DDL + SVAR∗ (our model) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='92 ↑ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='21 ↓ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='29 ↑ For our models, DDL is our supervised model, DDL+VSAR is our semi-supervised model, and DDL+VSAR∗ is our model with no prior used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Compared with state-of-the-art supervised models, our models achieve better BLEU scores and competitive Rouge scores for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our complementary experimental results are presented in Table 5 and Table 6, which we compare with two more recent state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Our models once again achieve better or competitive performance than the reported, which means our semi-supervised model is competitive with state-of-the-art supervised baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 7 Conclusions In this paper, we have introduced a semi-supervised deep generative model for paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The unsupervised model is based on the variational auto-encoding framework and provides an effective method to handle missing labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The supervised model conducts dual learning and injects supervised information into the unsupervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' With our novel knowledge reinforced training scheme, we empirically demonstrate that semi-supervised learning benefits our combined model, given unlabelled data and a fraction of the paired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The evaluation results show that our combined model improves upon a very strong baseline model in a semi-supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' We also observe that, even for the full dataset, our combined model achieves competitive performance with the state-of-the-art models for two paraphrase generation benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Additionally, we are able to model language as a discrete latent variable sequence for paraphrase generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Importantly, the resultant generative model is able to effectively exploit both supervised and unsupervised data in sequence-to-sequence tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' References [1] Li Dong, Jonathan Mallinson, Siva Reddy, and Mirella Lapata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Learning to paraphrase for question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 875–886, Copenhagen, Denmark, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [2] Minsuk Lee, James Cimino, Hai Ran Zhu, Carl Sable, Vijay Shanker, John Ely, and Hong Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Beyond information retrieval—medical question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In AMIA annual symposium proceedings, volume 2006, page 469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' American Medical Informatics Association, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [3] Xuchen Yao and Benjamin Van Durme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Information extraction over structured data: Question answering with Freebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 956–966, Baltimore, Maryland, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [4] Yuanjie Liu, Shasha Li, Yunbo Cao, Chin-Yew Lin, Dingyi Han, and Yong Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Understanding and summarizing answers in community-based question answering services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 22nd International Conference on Computational Linguistics (Coling 2008), pages 497–504, Manchester, UK, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Coling 2008 Organizing Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [5] Tim Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Explanation in artificial intelligence: Insights from the social sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Artificial intelligence, 267:1–38, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [6] Tom Hosking, Hao Tang, and Mirella Lapata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Hierarchical sketch induction for paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='03463, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [7] David Kauchak and Regina Barzilay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Paraphrasing for automatic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the Human Language Technology Conference of the NAACL, Main Conference, pages 455–462, New York City, USA, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [8] Shashi Narayan, Siva Reddy, and Shay B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Paraphrase generation from latent-variable PCFGs for semantic parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 9th International Natural Language Generation conference, pages 153–162, Edinburgh, UK, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 11 Running Title for Header [9] George A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' WordNet: A lexical database for English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Speech and Natural Language: Proceedings of a Workshop Held at Harriman, New York, February 23-26, 1992, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [10] Ashutosh Kumar, Kabir Ahuja, Raghuram Vadapalli, and Partha Talukdar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Syntax-guided controlled generation of paraphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Transactions of the Association for Computational Linguistics, 8:329–345, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [11] Jianing Zhou and Suma Bhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Paraphrase generation: A survey of the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, pages 5075–5086, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [12] Yuxian Meng, Xiang Ao, Qing He, Xiaofei Sun, Qinghong Han, Fei Wu, Jiwei Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Conrpg: Paraphrase generation using contexts as regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='00363, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [13] Yixuan Su, David Vandyke, Simon Baker, Yan Wang, and Nigel Collier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Keep the primary, rewrite the secondary: A two-stage approach for paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pages 560–569, Online, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [14] Ilya Sutskever, Oriol Vinyals, and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Sequence to sequence learning with neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Zoubin Ghahramani, Max Welling, Corinna Cortes, Neil D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Lawrence, and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Weinberger, editors, Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, December 8-13 2014, Montreal, Quebec, Canada, pages 3104–3112, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [15] Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Auto-encoding variational bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Yoshua Bengio and Yann LeCun, editors, 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [16] Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Stochastic backpropagation and approximate inference in deep generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21-26 June 2014, volume 32 of JMLR Workshop and Conference Proceedings, pages 1278–1286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='org, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [17] Andriy Mnih and Karol Gregor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Neural variational inference and learning in belief networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 31th International Conference on Machine Learning, ICML 2014, Beijing, China, 21-26 June 2014, volume 32 of JMLR Workshop and Conference Proceedings, pages 1791–1799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='org, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [18] Geoffrey E Hinton and Ruslan R Salakhutdinov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Reducing the dimensionality of data with neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' science, 313(5786):504–507, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [19] Yao Fu, Yansong Feng, and John P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Cunningham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Paraphrase generation with latent bag of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Hanna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Fox, and Roman Garnett, editors, Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pages 13623–13634, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [20] Tom Hosking and Mirella Lapata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Factorising meaning and form for intent-preserving paraphrasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceed- ings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 1405–1418, Online, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [21] Di He, Yingce Xia, Tao Qin, Liwei Wang, Nenghai Yu, Tie-Yan Liu, and Wei-Ying Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Dual learning for machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Daniel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Lee, Masashi Sugiyama, Ulrike von Luxburg, Isabelle Guyon, and Roman Garnett, editors, Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5-10, 2016, Barcelona, Spain, pages 820–828, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [22] Shang-Yu Su, Chao-Wei Huang, and Yun-Nung Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Dual supervised learning for natural language understanding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 5472–5477, Florence, Italy, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [23] Shang-Yu Su, Yung-Sung Chuang, and Yun-Nung Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Dual inference for improving language understanding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: EMNLP 2020, pages 4930–4936, Online, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [24] Shang-Yu Su, Chao-Wei Huang, and Yun-Nung Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Towards unsupervised language understanding and generation by joint dual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 671–680, Online, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [25] Yishu Miao and Phil Blunsom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Language as a latent variable: Discrete generative models for sentence compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pages 319–328, Austin, Texas, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [26] Junxian He, Xinyi Wang, Graham Neubig, and Taylor Berg-Kirkpatrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' A probabilistic formulation of unsuper- vised text style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='net, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 12 Running Title for Header [27] Minh-Thang Luong, Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Le, Ilya Sutskever, Oriol Vinyals, and Lukasz Kaiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Multi-task sequence to sequence learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Yoshua Bengio and Yann LeCun, editors, 4th International Conference on Learning Representations, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [28] Han Guo, Ramakanth Pasunuru, and Mohit Bansal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Dynamic multi-level multi-task learning for sentence simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 27th International Conference on Computational Linguistics, pages 462–476, Santa Fe, New Mexico, USA, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [29] Han Guo, Ramakanth Pasunuru, and Mohit Bansal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Soft layer-specific multi-task summarization with entailment and question generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 687–697, Melbourne, Australia, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [30] Yiren Wang, ChengXiang Zhai, and Hany Hassan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Multi-task learning for multilingual neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1022–1034, Online, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [31] Jingfei Du, Edouard Grave, Beliz Gunel, Vishrav Chaudhary, Onur Celebi, Michael Auli, Veselin Stoyanov, and Alexis Conneau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Self-training improves pre-training for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 5408–5418, Online, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [32] Ernie Chang, Vera Demberg, and Alex Marin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Jointly improving language understanding and generation with quality-weighted weak supervision of automatic labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 16th Conference of the Euro- pean Chapter of the Association for Computational Linguistics: Main Volume, pages 818–829, Online, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [33] Ronald J Williams and David Zipser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' A learning algorithm for continually running fully recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Neural computation, 1(2):270–280, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [34] Steven Burrows, Martin Potthast, and Benno Stein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Paraphrase acquisition via crowdsourcing and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' ACM Transactions on Intelligent Systems and Technology (TIST), 4(3):1–21, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [35] Ziqiang Cao, Chuwei Luo, Wenjie Li, and Sujian Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Joint copying and restricted generation for paraphrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Satinder P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Singh and Shaul Markovitch, editors, Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, February 4-9, 2017, San Francisco, California, USA, pages 3152–3158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' AAAI Press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [36] Eric Jang, Shixiang Gu, and Ben Poole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Categorical reparameterization with gumbel-softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='net, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [37] Chris J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Maddison, Andriy Mnih, and Yee Whye Teh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The concrete distribution: A continuous relaxation of discrete random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='net, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [38] Sang Michael Xie and Stefano Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Reparameterizable subset sampling via continuous relaxations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Sarit Kraus, editor, Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, Macao, China, August 10-16, 2019, pages 3919–3925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' ijcai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='org, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [39] Jianbo Chen, Le Song, Martin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wainwright, and Michael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Learning to explain: An information-theoretic perspective on model interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Jennifer G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, volume 80 of Proceedings of Machine Learning Research, pages 882–891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [40] Muhammed Fatih Balin, Abubakar Abid, and James Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Concrete autoencoders: Differentiable feature selection and reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pages 444–453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [41] Volodymyr Mnih, Nicolas Heess, Alex Graves, and Koray Kavukcuoglu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Recurrent models of visual attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Zoubin Ghahramani, Max Welling, Corinna Cortes, Neil D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Lawrence, and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Weinberger, editors, Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems 2014, December 8-13 2014, Montreal, Quebec, Canada, pages 2204–2212, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [42] Yoshua Bengio, Nicholas Léonard, and Aaron Courville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Estimating or propagating gradients through stochastic neurons for conditional computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' arXiv preprint arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='3432, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [43] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Gomez, Lukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wallach, Rob Fergus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Vishwanathan, and Roman Garnett, editors, Advances in Neural Information 13 Running Title for Header Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA, pages 5998–6008, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [44] Irina Higgins, Loïc Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' beta-vae: Learning basic visual concepts with a constrained variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In 5th International Conference on Learning Representations, ICLR 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='net, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [45] Zichao Yang, Zhiting Hu, Chris Dyer, Eric P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Xing, and Taylor Berg-Kirkpatrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Unsupervised text style transfer using language models as discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Samy Bengio, Hanna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Wallach, Hugo Larochelle, Kristen Grauman, Nicolò Cesa-Bianchi, and Roman Garnett, editors, Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada, pages 7298–7309, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [46] Ellie Pavlick, Pushpendre Rastogi, Juri Ganitkevitch, Benjamin Van Durme, and Chris Callison-Burch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' PPDB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='0: Better paraphrase ranking, fine-grained entailment relations, word embeddings, and style classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 2: Short Papers), pages 425–430, Beijing, China, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [47] Ankush Gupta, Arvind Agarwal, Prawaan Singh, and Piyush Rai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' A deep generative framework for paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Sheila A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' McIlraith and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Weinberger, editors, Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI- 18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018, pages 5149–5156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' AAAI Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [48] Yishu Miao, Lei Yu, and Phil Blunsom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Neural variational inference for text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Maria-Florina Balcan and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Weinberger, editors, Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, volume 48 of JMLR Workshop and Conference Proceedings, pages 1727–1736.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='org, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [49] Yoon Kim, Sam Wiseman, and Alexander M Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' A tutorial on deep latent variable models of natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' arXiv preprint arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='06834, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [50] Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Bowman, Luke Vilnis, Oriol Vinyals, Andrew Dai, Rafal Jozefowicz, and Samy Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Generating sentences from a continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of The 20th SIGNLL Conference on Computational Natural Language Learning, pages 10–21, Berlin, Germany, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [51] Adji B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Dieng, Yoon Kim, Alexander M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Rush, and David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Blei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Avoiding latent variable collapse with generative skip models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Kamalika Chaudhuri and Masashi Sugiyama, editors, The 22nd International Conference on Artificial Intelligence and Statistics, AISTATS 2019, 16-18 April 2019, Naha, Okinawa, Japan, volume 89 of Proceedings of Machine Learning Research, pages 2397–2405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [52] Junxian He, Daniel Spokoyny, Graham Neubig, and Taylor Berg-Kirkpatrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Lagging inference networks and posterior collapse in variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='net, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [53] Ali Razavi, Aäron van den Oord, Ben Poole, and Oriol Vinyals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Preventing posterior collapse with delta-vaes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='net, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [54] Yixin Wang, David Blei, and John P Cunningham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Posterior collapse and latent variable non-identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [55] Tsung-Hsien Wen, Yishu Miao, Phil Blunsom, and Steve J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Young.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Latent intention dialogue models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings of Machine Learning Research, pages 3732–3741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [56] Jihun Choi, Kang Min Yoo, and Sang-goo Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Learning to compose task-specific tree structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Sheila A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' McIlraith and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Weinberger, editors, Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018, pages 5094–5101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' AAAI Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [57] Giangiacomo Mercatali and André Freitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Disentangling generative factors in natural language with discrete variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content='07169, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 14 Running Title for Header [58] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Microsoft coco: Common objects in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In European conference on computer vision, pages 740–755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [59] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' BERT: Pre-training of deep bidirectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171–4186, Minneapolis, Minnesota, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [60] Zichao Li, Xin Jiang, Lifeng Shang, and Qun Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Decomposable neural paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 3403–3414, Florence, Italy, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [61] Jeffrey Pennington, Richard Socher, and Christopher Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' GloVe: Global vectors for word representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1532–1543, Doha, Qatar, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [62] Tomás Mikolov, Ilya Sutskever, Kai Chen, Gregory S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Corrado, and Jeffrey Dean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Distributed representations of words and phrases and their compositionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Christopher J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Burges, Léon Bottou, Zoubin Ghahramani, and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Weinberger, editors, Advances in Neural Information Processing Systems 26: 27th Annual Conference on Neural Information Processing Systems 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States, pages 3111–3119, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [63] Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [64] Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Bleu: a method for automatic evaluation of machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics, pages 311–318, Philadelphia, Pennsylvania, USA, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [65] Chin-Yew Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' ROUGE: A package for automatic evaluation of summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Text Summarization Branches Out, pages 74–81, Barcelona, Spain, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [66] Hong Sun and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Joint learning of a dual SMT system for paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pages 38–42, Jeju Island, Korea, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [67] Nitin Madnani, Joel Tetreault, and Martin Chodorow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Re-examining machine translation metrics for paraphrase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 2012 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 182–190, Montréal, Canada, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [68] Sander Wubben, Antal van den Bosch, and Emiel Krahmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Paraphrase generation as monolingual translation: Data and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 6th International Natural Language Generation Conference, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [69] Aaditya Prakash, Sadid A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Hasan, Kathy Lee, Vivek Datla, Ashequl Qadir, Joey Liu, and Oladimeji Farri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' Neural paraphrase generation with stacked residual LSTM networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of COLING 2016, the 26th International Conference on Computational Linguistics: Technical Papers, pages 2923–2934, Osaka, Japan, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' The COLING 2016 Organizing Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' [70] Wenqing Chen, Jidong Tian, Liqiang Xiao, Hao He, and Yaohui Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' A semantically consistent and syntactically variational encoder-decoder framework for paraphrase generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' In Proceedings of the 28th International Conference on Computational Linguistics, pages 1186–1198, Barcelona, Spain (Online), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' International Committee on Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfWQBc/content/2301.02275v1.pdf'} diff --git a/bNAyT4oBgHgl3EQfwPk2/content/tmp_files/2301.00644v1.pdf.txt b/bNAyT4oBgHgl3EQfwPk2/content/tmp_files/2301.00644v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc119294c5f6d2d15762de22f9634864f9fa6cb5 --- /dev/null +++ b/bNAyT4oBgHgl3EQfwPk2/content/tmp_files/2301.00644v1.pdf.txt @@ -0,0 +1,789 @@ +Equivalent conditions for the nth element of +the Beatty sequence B√ +2 being even +Sela Fried∗ +friedsela@gmail.com +Abstract +We provide equivalent conditions for the nth element of the Beatty +sequence B√ +2 being even. +In particular, we show that the integer +sequences A090892 and A120752 in the OEIS are essentially identical. +1 +Introduction +A Beatty sequence is the sequence of integers obtained by taking the floor +of the positive multiples of a positive irrational number, i.e., if r > 0 is +irrational, then the corresponding Beatty sequence Br is given by +Br = (⌊nr⌋)n∈N . +Beatty sequences are named after Samuel Beatty who brought them to the +attention of the mathematical community by posing a problem in the Ameri- +can Mathematical Monthly [1], in which the readers of the journal were asked +to prove that, if r > 1 is irrational and s = r/(r − 1), then Br and Bs are +complementary sequences, i.e., every natural number belongs to exactly one +of the two sequences. +To the best of our knowledge, little attention has been given to the parities +of the elements of Beatty sequences. That these might prove interesting is +perhaps suggested by the image below, that visualizes the first 105 elements +of the parity sequence of B√ +2 (cf. A083035 in the On-Line Encyclopedia +of Integer Sequences (OEIS) [2]). The visualization relies on a simple but +ingenious method capable of visualizing any binary sequence (an)n∈N: Start +∗The author is a teaching fellow in the Department of Computer Science at the Israel +Academic College in Ramat Gan. +1 +arXiv:2301.00644v1 [math.HO] 28 Dec 2022 + +at (0, 0), facing in the direction of the vector (1, 0). Now, for every n ∈ N, +execute the following two actions: (a) Turn right if an = 0 and left otherwise. +(b) Proceed forward one step of unit length. +The result of this process is a walk in the plane that we refer to as the +Cloitre walk of (an)n∈N, since Benoit Cloitre seems to be the first to use this +method. +Figure 1: The Cloitre walk of A083035 that corresponds to the sequence of +parities of B√ +2. +In this short note we provide several equivalent conditions for the nth +element of the Beatty sequence B√ +2 being even. Along the way we prove +that A090892 and A120752 are essentially identical since deleting the first +two elements of the former sequence results in the latter (cf. [3]). +2 +Main results +We denote by N the set of natural numbers and by {x} the fractional part of +a real number x. The purpose of this note is to prove the following theorem. +2 + +Theorem 2.1. Let 2 ≤ n ∈ N. The following conditions are equivalent: +(a) +�√ +2n +� +is even, i.e., the nth element of the Beatty sequence B√ +2 is even. +(b) +� +n +√ +2 +� +≤ 1 +2. +(c) +� +n +√ +2 +� +< 1 +2. +(d) +�√ +2n +� +n +√ +2 +�� += +� +n +√ +2 +�√ +2n +�� +. +(e) +� +n +√ +2 +� += +�� +n2 − +� +n +√ +2 +�2 +� +. +(f) +� +n +√ +2 +� +< +�� +n +√ +2 +�2 ++ +� +n +√ +2 +� ++ 1 +2 − +� +n +√ +2 +� +. +Remark 2.2. Conditions (b) and (d) correspond to the definitions of A120752 +and A090892, respectively. Cloitre made a comment to A090892 stating that +conditions (a) and (d) are equivalent. +We shall make use of the following inequality. +Lemma 2.3. Let x be a nonnegative real number. Then +� +x2 + x + 1 +2 − x > 1 +2. +Proof. We have +� +x2 + x + 1 +2 − x = +x + 1 +2 +� +x2 + x + 1 +2 + x +> +x + 1 +2 +� +x2 + +√ +2x + 1 +2 + x += +x + 1 +2 +2x + +1 +√ +2 += +x + 1 +2 +2 +� +x + +1 +√ +2 +1 +2 +� +> +x + 1 +2 +2 +� +x + 1 +2 +� = 1 +2. +3 + +Proof of Theorem 2.1. “(e) ⇐⇒ (f)”: We have +� n +√ +2 +� += +���� +� +n2 − +� n +√ +2 +�2 +���� ⇐⇒ +� n +√ +2 +� +≤ +� +n2 − +� n +√ +2 +�2 +< +� n +√ +2 +� ++ 1 +⇐⇒ 2 +� n +√ +2 +�2 +≤ n2 < 2 +� n +√ +2 +�2 ++ 2 +� n +√ +2 +� ++ 1. +(1) +Now, since +2 +� n +√ +2 +�2 +≤ 2 +� n +√ +2 +�2 += n2, +we have +(1) ⇐⇒ n2 < 2 +� n +√ +2 +�2 ++ 2 +� n +√ +2 +� ++ 1. +(2) +Let us denote +� +n +√ +2 +� +by σ. Thus, σ ∈ [0, 1) and +� +n +√ +2 +� += +n +√ +2 − σ. Then +(2) ⇐⇒ n2 < 2 +� n +√ +2 − σ +�2 ++ 2 +� n +√ +2 − σ +� ++ 1 +⇐⇒ +√ +2n(2σ − 1) < +>0 for every σ +� +�� +� +2σ2 − 2σ + 1 +⇐⇒ +� +σ ≤ 1 +2 +� +or +� +σ > 1 +2 and n < 2σ2 − 2σ + 1 +√ +2(2σ − 1) +� +. +(3) +Now, +σ > 1 +2 and n < 2σ2 − 2σ + 1 +√ +2(2σ − 1) +⇐⇒ σ > 1 +2 and +� n +√ +2 +� +< +1 − 2σ2 +2(2σ − 1) +⇐⇒ σ > 1 +2 and 2σ2 + 4 +� n +√ +2 +� +σ − 1 − 2 +� n +√ +2 +� +< 0 +⇐⇒ 1 +2 < σ < +> 1 +2 , by Lemma 2.3 +� +�� +� +�� n +√ +2 +�2 ++ +� n +√ +2 +� ++ 1 +2 − +� n +√ +2 +� +. +Thus, +(3) ⇐⇒ σ < +�� n +√ +2 +�2 ++ +� n +√ +2 +� ++ 1 +2 − +� n +√ +2 +� +. +4 + +“(b) ⇐⇒ (c)”: This is clear. +“(c) ⇐⇒ (d)”: Let us denote +�√ +2n +� +n +√ +2 +�� +by σ. Thus, +�√ +2n +� n +√ +2 +�� += +√ +2n +� n +√ +2 +� +− σ. +Then, +�√ +2n +� n +√ +2 +�� += +� n +√ +2 +�√ +2n +�� +⇐⇒ +�√ +2n +� n +√ +2 +�� +≤ n +√ +2 +�√ +2n +� +< +�√ +2n +� n +√ +2 +�� ++ 1 ⇐⇒ +n +� +2 +� n +√ +2 +� +− +�√ +2n +�� +≤ +√ +2σ and +n +√ +2 +��√ +2n +� +− 2 +� n +√ +2 +�� +< 1 − σ. +(4) +It is easy to see that, for every nonnegative real number x and m ∈ N, we +have +⌊x⌋ − m +� x +m +� +∈ {0, 1, . . . , m − 1} . +In particular, +�√ +2n +� +− 2 +� n +√ +2 +� +∈ {0, 1}. +(5) +It follows that the first inequality in (4) holds trivially. Furthermore, since +n ≥ 2, if +�√ +2n +� +− 2 +� +n +√ +2 +� += 1, then the second inequality in (4) cannot hold. +We conclude that +(4) ⇐⇒ +�√ +2n +� += 2 +� n +√ +2 +� +⇐⇒ 2 +� n +√ +2 +� +≤ +√ +2n < 2 +� n +√ +2 +� ++ 1. +(6) +The inequality 2 +� +n +√ +2 +� +≤ +√ +2n holds trivially. Thus, +(6) ⇐⇒ +√ +2n < 2 +� n +√ +2 +� ++ 1. +⇐⇒ +n +√ +2 < +� n +√ +2 +� ++ 1 +2 +⇐⇒ +� n +√ +2 +� +< 1 +2. +5 + +“(c) ⇐⇒ (f)”: By Lemma 2.3, +�� n +√ +2 +�2 ++ +� n +√ +2 +� ++ 1 +2 − +� n +√ +2 +� +> 1 +2 +and that settles the “=⇒” implication. For the other implication, we need +to show that there exists no 2 ≤ n ∈ N such that +1 +2 ≤ +� n +√ +2 +� +< +�� n +√ +2 +�2 ++ +� n +√ +2 +� ++ 1 +2 − +� n +√ +2 +� +. +(7) +Setting σ = +� +n +√ +2 +� +, we have +(7) ⇐⇒ 1 +2 ≤ σ < +�� n +√ +2 − σ +�2 ++ n +√ +2 − σ + 1 +2 − +� n +√ +2 − σ +� +⇐⇒ 1 +2 + +� n +√ +2 − σ +� +≤ σ + +� n +√ +2 − σ +� +< +�� n +√ +2 − σ +�2 ++ n +√ +2 − σ + 1 +2 +⇐⇒ +� n +√ +2 + 1 +2 − σ +�2 +≤ n2 +2 < +� n +√ +2 − σ +�2 ++ n +√ +2 − σ + 1 +2 +⇐⇒ 0 < σ2 − +√ +2nσ + n +√ +2 − σ + 1 +2 ≤ 1 +4 +⇐⇒ 0 < − +� n +√ +2 − +� n +√ +2 +�� � n +√ +2 + +� n +√ +2 +�� ++ +� n +√ +2 +� ++ 1 +2 ≤ 1 +4 +⇐⇒ 0 < +� n +√ +2 +�2 ++ +� n +√ +2 +� ++ 1 +2(1 − n2) ≤ 1 +4. +(8) +Now, both +� +n +√ +2 +�2 ++ +� +n +√ +2 +� +and 1 − n2 are integers. Thus, the two inequalities +in (8) cannot hold simultaneously. +“(a) ⇐⇒ (c)”: We have +� n +√ +2 +� +< 1 +2 ⇐⇒ +n +√ +2 − +� n +√ +2 +� +< 1 +2 ⇐⇒ +√ +2n − 2 +� n +√ +2 +� +< 1. +Since +�√ +2n +� +− 2 +� n +√ +2 +� +≤ +√ +2n − 2 +� n +√ +2 +� +, +using (5), we conclude that +√ +2n − 2 +� n +√ +2 +� +< 1 ⇐⇒ +�√ +2n +� +− 2 +� n +√ +2 +� += 0 ⇐⇒ +�√ +2n +� +is even. +6 + +References +[1] S. Beatty, Problem 3173, Amer. Math. Monthly, 3 (1926), 159. +[2] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, OEIS +Foundation Inc., https://oeis.org. +[3] N. J. A. Sloane, Families of essentially identical sequences, https:// +oeis.org/A115004/a115004.txt, 2021. +7 + diff --git a/bNAyT4oBgHgl3EQfwPk2/content/tmp_files/load_file.txt b/bNAyT4oBgHgl3EQfwPk2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ace7516e85304e002557f3254ff2655210732e62 --- /dev/null +++ b/bNAyT4oBgHgl3EQfwPk2/content/tmp_files/load_file.txt @@ -0,0 +1,99 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf,len=98 +page_content='Equivalent conditions for the nth element of the Beatty sequence B√ 2 being even Sela Fried∗ friedsela@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='com Abstract We provide equivalent conditions for the nth element of the Beatty sequence B√ 2 being even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' In particular, we show that the integer sequences A090892 and A120752 in the OEIS are essentially identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 1 Introduction A Beatty sequence is the sequence of integers obtained by taking the floor of the positive multiples of a positive irrational number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=', if r > 0 is irrational, then the corresponding Beatty sequence Br is given by Br = (⌊nr⌋)n∈N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Beatty sequences are named after Samuel Beatty who brought them to the attention of the mathematical community by posing a problem in the Ameri- can Mathematical Monthly [1], in which the readers of the journal were asked to prove that, if r > 1 is irrational and s = r/(r − 1), then Br and Bs are complementary sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=', every natural number belongs to exactly one of the two sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' To the best of our knowledge, little attention has been given to the parities of the elements of Beatty sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' That these might prove interesting is perhaps suggested by the image below, that visualizes the first 105 elements of the parity sequence of B√ 2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' A083035 in the On-Line Encyclopedia of Integer Sequences (OEIS) [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' The visualization relies on a simple but ingenious method capable of visualizing any binary sequence (an)n∈N: Start ∗The author is a teaching fellow in the Department of Computer Science at the Israel Academic College in Ramat Gan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='00644v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='HO] 28 Dec 2022 at (0, 0), facing in the direction of the vector (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Now, for every n ∈ N, execute the following two actions: (a) Turn right if an = 0 and left otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (b) Proceed forward one step of unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' The result of this process is a walk in the plane that we refer to as the Cloitre walk of (an)n∈N, since Benoit Cloitre seems to be the first to use this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Figure 1: The Cloitre walk of A083035 that corresponds to the sequence of parities of B√ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' In this short note we provide several equivalent conditions for the nth element of the Beatty sequence B√ 2 being even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Along the way we prove that A090892 and A120752 are essentially identical since deleting the first two elements of the former sequence results in the latter (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 2 Main results We denote by N the set of natural numbers and by {x} the fractional part of a real number x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' The purpose of this note is to prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 2 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Let 2 ≤ n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' The following conditions are equivalent: (a) �√ 2n � is even, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=', the nth element of the Beatty sequence B√ 2 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (b) � n √ 2 � ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (c) � n √ 2 � < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (d) �√ 2n � n √ 2 �� = � n √ 2 �√ 2n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (e) � n √ 2 � = �� n2 − � n √ 2 �2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (f) � n √ 2 � < �� n √ 2 �2 + � n √ 2 � + 1 2 − � n √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Conditions (b) and (d) correspond to the definitions of A120752 and A090892, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Cloitre made a comment to A090892 stating that conditions (a) and (d) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' We shall make use of the following inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Let x be a nonnegative real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Then � x2 + x + 1 2 − x > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' We have � x2 + x + 1 2 − x = x + 1 2 � x2 + x + 1 2 + x > x + 1 2 � x2 + √ 2x + 1 2 + x = x + 1 2 2x + 1 √ 2 = x + 1 2 2 � x + 1 √ 2 1 2 � > x + 1 2 2 � x + 1 2 � = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 3 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' “(e) ⇐⇒ (f)”: We have � n √ 2 � = ���� � n2 − � n √ 2 �2 ���� ⇐⇒ � n √ 2 � ≤ � n2 − � n √ 2 �2 < � n √ 2 � + 1 ⇐⇒ 2 � n √ 2 �2 ≤ n2 < 2 � n √ 2 �2 + 2 � n √ 2 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (1) Now, since 2 � n √ 2 �2 ≤ 2 � n √ 2 �2 = n2, we have (1) ⇐⇒ n2 < 2 � n √ 2 �2 + 2 � n √ 2 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (2) Let us denote � n √ 2 � by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Thus, σ ∈ [0, 1) and � n √ 2 � = n √ 2 − σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Then (2) ⇐⇒ n2 < 2 � n √ 2 − σ �2 + 2 � n √ 2 − σ � + 1 ⇐⇒ √ 2n(2σ − 1) < >0 for every σ � �� � 2σ2 − 2σ + 1 ⇐⇒ � σ ≤ 1 2 � or � σ > 1 2 and n < 2σ2 − 2σ + 1 √ 2(2σ − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (3) Now, σ > 1 2 and n < 2σ2 − 2σ + 1 √ 2(2σ − 1) ⇐⇒ σ > 1 2 and � n √ 2 � < 1 − 2σ2 2(2σ − 1) ⇐⇒ σ > 1 2 and 2σ2 + 4 � n √ 2 � σ − 1 − 2 � n √ 2 � < 0 ⇐⇒ 1 2 < σ < > 1 2 , by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='3 � �� � �� n √ 2 �2 + � n √ 2 � + 1 2 − � n √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Thus, (3) ⇐⇒ σ < �� n √ 2 �2 + � n √ 2 � + 1 2 − � n √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 4 “(b) ⇐⇒ (c)”: This is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' “(c) ⇐⇒ (d)”: Let us denote �√ 2n � n √ 2 �� by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Thus, �√ 2n � n √ 2 �� = √ 2n � n √ 2 � − σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Then, �√ 2n � n √ 2 �� = � n √ 2 �√ 2n �� ⇐⇒ �√ 2n � n √ 2 �� ≤ n √ 2 �√ 2n � < �√ 2n � n √ 2 �� + 1 ⇐⇒ n � 2 � n √ 2 � − �√ 2n �� ≤ √ 2σ and n √ 2 ��√ 2n � − 2 � n √ 2 �� < 1 − σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (4) It is easy to see that, for every nonnegative real number x and m ∈ N, we have ⌊x⌋ − m � x m � ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' , m − 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' In particular, �√ 2n � − 2 � n √ 2 � ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (5) It follows that the first inequality in (4) holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Furthermore, since n ≥ 2, if �√ 2n � − 2 � n √ 2 � = 1, then the second inequality in (4) cannot hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' We conclude that (4) ⇐⇒ �√ 2n � = 2 � n √ 2 � ⇐⇒ 2 � n √ 2 � ≤ √ 2n < 2 � n √ 2 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (6) The inequality 2 � n √ 2 � ≤ √ 2n holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Thus, (6) ⇐⇒ √ 2n < 2 � n √ 2 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' ⇐⇒ n √ 2 < � n √ 2 � + 1 2 ⇐⇒ � n √ 2 � < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 5 “(c) ⇐⇒ (f)”: By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='3, �� n √ 2 �2 + � n √ 2 � + 1 2 − � n √ 2 � > 1 2 and that settles the “=⇒” implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' For the other implication, we need to show that there exists no 2 ≤ n ∈ N such that 1 2 ≤ � n √ 2 � < �� n √ 2 �2 + � n √ 2 � + 1 2 − � n √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (7) Setting σ = � n √ 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' we have (7) ⇐⇒ 1 2 ≤ σ < �� n √ 2 − σ �2 + n √ 2 − σ + 1 2 − � n √ 2 − σ � ⇐⇒ 1 2 + � n √ 2 − σ � ≤ σ + � n √ 2 − σ � < �� n √ 2 − σ �2 + n √ 2 − σ + 1 2 ⇐⇒ � n √ 2 + 1 2 − σ �2 ≤ n2 2 < � n √ 2 − σ �2 + n √ 2 − σ + 1 2 ⇐⇒ 0 < σ2 − √ 2nσ + n √ 2 − σ + 1 2 ≤ 1 4 ⇐⇒ 0 < − � n √ 2 − � n √ 2 �� � n √ 2 + � n √ 2 �� + � n √ 2 � + 1 2 ≤ 1 4 ⇐⇒ 0 < � n √ 2 �2 + � n √ 2 � + 1 2(1 − n2) ≤ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' (8) Now, both � n √ 2 �2 + � n √ 2 � and 1 − n2 are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Thus, the two inequalities in (8) cannot hold simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' “(a) ⇐⇒ (c)”: We have � n √ 2 � < 1 2 ⇐⇒ n √ 2 − � n √ 2 � < 1 2 ⇐⇒ √ 2n − 2 � n √ 2 � < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Since �√ 2n � − 2 � n √ 2 � ≤ √ 2n − 2 � n √ 2 � , using (5), we conclude that √ 2n − 2 � n √ 2 � < 1 ⇐⇒ �√ 2n � − 2 � n √ 2 � = 0 ⇐⇒ �√ 2n � is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 6 References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Beatty, Problem 3173, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Monthly, 3 (1926), 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Sloane, The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=', https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' Sloane, Families of essentially identical sequences, https:// oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='org/A115004/a115004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content='txt, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfwPk2/content/2301.00644v1.pdf'} diff --git a/cNAzT4oBgHgl3EQfLfum/content/tmp_files/2301.01116v1.pdf.txt b/cNAzT4oBgHgl3EQfLfum/content/tmp_files/2301.01116v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e315bf0589d17140ea7f390514f033bee6a3ef6d --- /dev/null +++ b/cNAzT4oBgHgl3EQfLfum/content/tmp_files/2301.01116v1.pdf.txt @@ -0,0 +1,860 @@ +On a probabilistic extension of the +Oldenburger-Kolakoski sequence +Chloé Boisson∗, Damien Jamet†and Irène Marcovici‡ +chloe.boisson@ens-lyon.fr, damien.jamet@loria.fr, irene.marcovici@univ-lorraine.fr +January 4, 2023 +Abstract +The Oldenburger-Kolakoski sequence is the only infinite sequence over the alpha- +bet {1, 2} that starts with 1 and is its own run-length encoding. In the present work, +we take a step back from this largely known and studied sequence by introducing +some randomness in the choice of the letters written. This enables us to provide some +results on the convergence of the density of 1’s in the resulting sequence. When the +choice of the letters is given by an infinite sequence of i.i.d. random variables or by a +Markov chain, the average densities of letters converge. Moreover, in the case of i.i.d. +random variables, we are able to prove that the densities even almost surely converge. +1 +Introduction +The Oldenburger-Kolakoski sequence 122112122122112 . . . introduced by R. Oldenburger +[12] and lately mentioned by W. Kolakoski [10] is the unique sequence x1x2x3 . . . over the +alphabet {1, 2} with x1 = 1 and whose k-th block has length xk for k ∈ N⋆. +In [9] M.S. Keane asked whether the density of 1’s in this sequence is 1/2. In [7], +V. Chvátal showed that the upper density of 1’s (resp. 2’s) is less than 0.50084. This +bound has been slightly improved by M. Rao but Keane’s question still stands: « Is the +density of 1’s in Oldenburger-Kolakoski sequence defined and equal to 0.5? » +By definition, the Oldenburger-Kolakoski sequence O = (xn)n∈N⋆ is a fixed point of the +run-length encoding operator denoted ∆: +∗École Normale Supérieure de Lyon, 15 parvis René Descartes, F-69342, Lyon, France. +†Université de Lorraine, Loria, UMR 7503, Vandœuvre-lès-Nancy, F-54506, France. +‡Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France. +1 +arXiv:2301.01116v1 [cs.DM] 3 Jan 2023 + +O +∆(O) += += +1�� +1 +22 +�� +2 +11 +�� +2 +2�� +1 +1�� +1 +22 +�� +2 +1�� +1 +22 +�� +2 +11 +�� +2 +· · · = 112212211122112212 . . . +(1) +O += +1x12x21x32x41x52x61x72x81x9 = +� +n∈N +(1x2n+12x2n+2) +(2) +In [12], R. Oldenburger refers to sequences over an alphabet Σ as trajectories and refers +to the sequence ∆(w) as the exponent trajectory of the trajectory w. He stated that « +a periodic trajectory is distinct from its exponent trajectory » (Theorem 2, [12]) and, +therefore, the Oldenburger-Kolakoski sequence is not periodic. +The Oldenburger-Kolakoski sequence is also connected to differentiable words, C∞- +words and smooth words [2, 4, 8]. A sequence w over the alphabet Σ ⊂ N⋆ is differentiable +if and only if ∆(w) is also defined over the same alphabet Σ. The sequence ∆(w) is called +the derivative sequence of w [8]. A C∞-word, or smooth word, is an infinitely differentiable +sequence. Obviously, the Oldenburger-Kolakoski sequence is a C∞-word since it is a fixed- +point of the run-length encoding operator ∆. +Although not answering Keane’s question fully, F.M. Dekking established connections +between possible combinatorial properties of the Oldenburger-Kolakoski sequence [8]: if +the Oldenburger-Kolakoski sequence is closed by complementation (that is, if w occurs in +O then so does �w with �1 = 2 and �2 = 1) then it is recurrent (any word that occurs in O +does so infinitely often) (Prop. 1, [8]). Moreover, the Oldenburger-Kolakoski sequence is +closed by complementation if and only if it contains every finite C∞-word (Prop. 2, [8]). +A few years later, A. Carpi stated that the Oldenburger-Kolakoski sequence contains +only a finite set of squares (words of the form xx where x is not empty) and does not contain +any cube (word of the form xxx where x is not empty) [6]. Hence, since O contains only +squares of bounded length then it cannot be the fixed point of a non degenerated morphism: +the image of a square w = xx by such a morphism is still a square longer than w. +There exist several ways to extend the definition of the Oldenburger-Kolakoski sequence, +depending on whether one wants to preserve the fixed point property or to follow the +construction scheme without requiring the resulting sequence to be a fixed point for the +run-length encoding operator ∆. +For instance, one can deal with other alphabets and +thus construct Generalized Oldenburger-Kolakoski sequence (GOK-sequence for short) as +follows: for any pair (a, b) of non-zero natural numbers, there exists a unique fixed point +Oa,b of ∆ over the alphabet {a, b} starting with a. Also, according to this notation, the +original Oldenburger-Kolakoski sequence is O1,2. For instance, if a = 1 and b = 3, the first +terms of O1,3 are: +O1,3 = 1�� +1 +333 +� � +3 +111 +� � +3 +333 +� � +3 +1�� +1 +3�� +1 +1�� +1 +333 +� � +3 +· · · = 1133133311311133 . . . +(3) +A significant result is, unlike the case of the original Oldenburger-Kolakoski sequence, +that the densities of 1’s in O1,3 and O3,1 are known and approximately 0.3972 [1]. +2 + +Generalized Oldenburger-Kolakoski sequences are also connected with smooth words +over arbitrary alphabets [3, 5]. As for the (Generalized) Oldenburger-Kolakoski sequences, +the properties of smooth words are better known for alphabets with letters of the same +parity: for instance, while the frequency of letters in an infinite smooth word over {1, 2} is +still unsolved, in [5] the authors showed that the frequency of letters for extremal smooth +words (for the lexicographic order) over the alphabet {a, b}, where a and b are both even, +is 0.5. They also computed the frequency for extremal smooth words over alphabets of +type {1, b}, where b is odd. Moreover, if a and b have the same parity, then every infinite +smooth word over the alphabet {a, b} is recurrent [5]. Also, if a and b are both odd, then +every infinite smooth word is closed under reversal but not under complementation [5]. +On the other hand, if a and b are both even, then the extremal smooth words over the +alphabet {a, b} are neither closed under reversal nor closed under complementation [5]. +For a more detailed survey on the Oldenburger-Kolakoski sequence and on generaliza- +tions over arbitrary two letter alphabets see [13]. +2 +Extending the construction scheme to any directing +sequence +2.1 +Notion of directing sequence. +In the construction scheme of a Generalized Oldenburger-Kolakoski sequence, the blocks +of Oa,b are composed, alternatively, of a’s and b’s as shown in (1) when a = 1 and b = 2 +and in (3) when a = 1 and b = 3. In other words, if t1 = a and t2 = b, then « the ith block +of Oa,b = (xi)i∈N⋆ is of length xi and is filled with the letter ti mod 2 ». +This construction scheme is clearly extendable to any finite sequences T = (t1, t2, . . . ) +over {a, b} as follows: « the ith block of OT = (xi)i∈N⋆ is of length xi and is filled with the +letter ti » (see Program 1). +1 def O(T): +2 +X = [] +3 +k = 0 +4 +for x in T: +5 +X += [x] # concatenate ’x’ at the end of X +6 +X += [x]*(X[k]-1) # concatenate ’X[k]-1’ copie(s) of x +7 +k += 1 +8 +return X +9 +Program 1: Python function: O is an operates on sequences over N⋆. +We say that the sequence OT is directed by the sequence T and the sequence T is a +directing sequence of OT. For instance, the sequence Oa,b is directed by T = (ab)ω while O +is directed by (12)ω. Notice that the directed sequence OT may no longer be a fixed point +of the operator ∆. +Let us now take a closer look at how the construction of OT provides a little more +3 + +information than the sequence itself. For instance, let T = (tn)n∈N⋆ = 21122 . . . be a +sequence over {1, 2}: +Step 1: Ot1 = 22 and the second block is of length 2: hence the 3rd and 4th letters are in a +same block of length 2. Let us denote Ot1 = 22 ??. +Step 2: Ot1t2 = 22 11 ? ?: the 5th and the 6th letter are respectively in blocks of length 1. +Step 3: = 22 11 1 ? ?: the 7th letter is in a block of length 1. +Roughly speaking, t1 gives the length of all the blocks that contain up to the 4th letter, t1t2 +gives the length of all the blocks that contain up the 6th letter and so on. . . Let wn = Ot1...tn +for each n ∈ N⋆ , then �|w1| +i=1 [w1]i = 4 , �|w2| +i=1 [w2]i = 6, �|w3| +i=1 [w3]i = 7. . . +2.2 +Partitions of the set of directing sequences +We now introduce some subsets of directing sequences that will be crucial in the following +sections. For this purpose, let us classify the sequences T according to the information +they provide on the length of the blocks of OT: let k and n be two integers such that +1 ≤ k ≤ n and let Sn,k be the set of sequences T = (tn)n∈N such that the length of the +block of OT containing its nth letter is known when reading tk but not before. Formally, if +wn = Ot1...tn, then we have +Sn,k = +� +(t1, ..., tn) ∈ {1, 2}n : min{j ∈ �1; n�, +|wj| +� +i=1 +[wj]i ≥ n} = k +� +. +(4) +Let us give a short example to illustrate the latter definition: let n = 5 and let T = +(t1, t2, t3, t4, t5) = (2, 1, 1, 2, 2). After the first step and the reading of t1, we only know that +t1 = 2 and still do not know the length of the block that will contain the 5th letter of OT. +On the other hand, after having taken knowledge of the value of t2, we know that the 5th +letter of OT will be written in a block of size 1. Hence, T ∈ S5,2. More generally, +Step 1: Ot1 = 22 ?? and T ∈ S3,1 ∩ S4,1 since t1 provides the length of the block containing +the 3rd and the 4th of OT. +Step 2: Ot1t2 = 22 11 ? ?: the 5th and the 6th letter are respectively in blocks of length 1. +Hence T ∈ S5,2 ∩ S6,2. +Step 3: Ot1t2t3 = 22 11 1 ? ? and T ∈ S7,3. +Step 4: Ot1t2t3t4 = 22 11 1 2 ? ?? and T ∈ S8,4 ∩ S9,4. +Step 5: Ot1t2t3t4t5 = 22 11 1 2 2 ?? ?? and T ∈ S10,5 ∩ S11,5. +4 + +The set {Sn,k : k ∈ �1; n�} is a partition of {1, 2}n. Indeed, the length of the word wn +is at least n, since at each step, after reading ti, we write at least one letter of OT. And +each letter is 1 or 2, so �|wn| +i=1 [wn]i ≥ n. +Let us notice that Sn,n = {(1, ..., 1), (1, ..., 1, 2)}. Indeed, the length of wn−1 is at least +n − 1, and is exactly n − 1 if and only if the written letters are all 1’s. Moreover Sn,n−1 = +{(1, ..., 1, 2, 1), (1, ..., 1, 2, 2)}. Indeed, one easily check that (1, ..., 2, 1), (1, ..., 2, 2) ∈ Sn,n−1. +Reciprocally, if (t1, . . . , tn) ∈ Sn,n−1, then there exists i ∈ �1; n − 1� such that ti = 2. Let +us note that the first 2 in T is written twice. Thus, if there were such a i in �1; n − 2� we +would have �|wn−2| +i=1 +[wn−2]i ≥ n. +2.3 +Extension of the definition to infinite directing sequences +Now that we have started studying the notions of directing and directed sequences, a nat- +ural question arises: « Can one extend the definition of OT sequences to infinite (possibly +not periodic) sequences T? » Let A ⊆ N⋆ be an alphabet and let T = (tn)n∈N⋆ be an +infinite sequence over A. By construction, Ot1...tn is a prefix of Ot1...tn+1for each n ∈ N⋆. +One thus defines OT as the limit of Ot1...tn when n tends to infinity: OT = lim +n→∞ Ot1...tn. +The present work deals with the densities of letters in OT when T = (tn)n∈N⋆ is an +infinite sequence over A. Do these densities exist? If so, how much are they value? +We are especially interested in the case where the directing sequence is random. Let +T = (Tn)n∈N⋆ be a sequence of random variables. By definition, the sequence directed by +T is the random sequence X = (Xn)n∈N⋆ defined by X = OT. +The present paper is organized as follows. In section 3, we consider the case where +the directing sequence is made of independent and identically distributed (i.i.d.) random +variables over a two-letter alphabet. In section 4, we treat the case where the random +sequence X is directed by a Markov chain. +3 +Sequence directed by independent random variables +In the present section, T = (Tn)n∈N⋆ is a sequence of independent and identically distributed +random variables (i.i.d. for short) over the two-letter alphabet A = {1, 2}, with P(Tn = +1) = p and P(Tn = 2) = 1 − p for each n ∈ N⋆, where p is a fixed parameter in ]0, 1[. The +sequence T is thus distributed according to the product distribution (pδ1 + (1 − p)δ2)⊗N⋆. +We denote T ∼ ((pδ1 + (1 − p)δ2)⊗N⋆. +Let X = (Xn)n∈N⋆ be the sequence directed by T. The sequence X is a random sequence +with a priori unknown distribution. Assume that one wants to compute the nth letter Xn +of X, for some large integer n. Unless the sequence T begins with a long succession of +1’s (an event which has a low probability to occur), one just has to read the first terms +T1, . . . , Tk of T, until knowing the length and position of the block containing Xn, and to +fill that block by 1’s with probability p, or by 2’s with probability 1 − p. The resulting +value of Xn obtained that way will have the desired distribution. This leads to the fact that +5 + +limn→∞ P(Xn = 1) = p, and we can even use this observation to compute more precisely +the value of P(Xn = 1), as shown in the following proposition. +Proposition 1. If T ∼ (pδ1 + (1 − p)δ2)⊗N⋆ with p ∈]0, 1[, then for any n ≥ 2, +P(Xn = 1) = p(1 − pn−2 + pn−1). +Proof. We will decompose the event {Xn = 1} according to the partition {Sn,k : k ∈ �1; n�} +of {1, 2}n introduced in the previous section, and use the observations (??) and (??). The +following two particular cases are obvious: +P(Xn = 1 | (T1, ..., Tn) ∈ Sn,n) = p, +and +P(Xn = 1 | (T1, ..., Tn) ∈ Sn,n−1) = 0. +Let us now consider some (T1, ..., Tn) ∈ Sn,k, with k < n − 1. Let k′ be the unique +integer such that |wk′−1| < n and |wk′| ≥ n. Concretely, the nth letter Xn is written during +the k′th step, and it is equal to Tk′. By definition of Sn,k, we have k′ ≥ k. Let us show that +k′ ̸= k. We reason by contradiction and assume that k′ = k. Let us now look at what we +know at the end of the k − 1th step. +• By definition of k′, Xn will be written in the next block to the right of wk′−1. +• By definition of k, we do not know the length of this block yet. +Consequently, we do not know the length of any (empty) block to the right of wk′−1. It +directly implies that wk′−1 is made of 1’s (and thus, that wk′−1 = 1k′−1). It follows that +n ≤ |wk′| ≤ |wk′−1|+2 = k′ +1. Since k′ = k < n−1, we get a contradiction, which means +that k < k′. +The fact that (T1, ..., Tn) belongs to Sn,k only depends on the beginning (T1, ..., Tk) of +the sequence. Recall that T has a product distribution (pδ1 + (1 − p)δ2)⊗N⋆. Since k′ > k, +and Xn = Tk′, we deduce that +P(Xn = 1 | (T1, ..., Tn) ∈ Sn,k) = p. +Finally, we use the formula of total probability: +P(Xn = 1) = +n−2 +� +k=1 +P((T1, ..., Tn) ∈ Sn,k) × P(Xn = 1 | (T1, ..., Tn) ∈ Sn,k) ++ P((T1, ...Tn−1) = (1, ..., 1)) × P(Xn = 1 | (T1, ..., Tn−1) = (1, ..., 1)) ++ P((T1, ...Tn−1) = (1, ..., 2)) × P(Xn = 1 | (T1, ..., Tn−1) = (1, ..., 2)) += �1 − (pn + 2(1 − p)pn−1 + pn−2(1 − p)2)� × p + pn × 1 + 0 += p(1 − pn−2 + pn−1). +As a corollary, we obtain the following convergence of the proportion of 1’s in X. +6 + +Corollary 1. If T ∼ (pδ1 + (1 − p)δ2)⊗N⋆ with p ∈]0, 1[, then +lim +n→∞ +E(|X1 . . . Xn|1) +n += p +Proof. Since X has values in {1, 2}, we have |X1 . . . Xn|1 = 2n − (X1 + ... + Xn). It follows +that +E(|X1 . . . Xn|1) +n += 2 − +�n +k=1 E(Xk) +n +. +By Proposition 1, we have limn→∞ P(Xn = 1) = p and limn→∞ P(Xn = 2) = 1 − p. We +deduce that limn→∞ E(Xn) = 2 − p. By Cesàro lemma, we obtain: +lim +n→∞ +E(|X1 . . . Xn|1) +n += 2 − (2 − p) = p. +Each time we run a simulation with T ∼ (pδ1 + (1 − p)δ2)⊗N⋆, the frequency of 1’s +in X seems to converge to p. +We thus expect the sequence |X1 . . . Xn|1/n to converge +almost surely to p, and not only in expectation. Since the random variables (Xn)n∈N⋆ are +correlated, we can not directly apply the strong law of large numbers (SLLN) to prove the +almost sure convergence of (X1 + ... + Xn)/n. However, the correlations being sufficiently +weak, we can apply the following stronger version of the SLLN. +Theorem 1 (Lyons [11]). Let (Yn)n∈N⋆ be a sequence of real-valued random variables such +that for all n ∈ N⋆, |Yn| ≤ 1 and +∀n, m ∈ N⋆, E(YmYn) ≤ Φ(|n − m|), +with Φ ≥ 0 and +� +n≥1 +Φ(n) +n +< ∞. +Then lim +n→∞ +1 +n +n +� +k=1 +Yk = 0 almost surely. +In order to apply Theorem 1, let us first prove the following lemma. +Lemma 1. If T ∼ (pδ1+(1−p)δ2)⊗N⋆ with p ∈]0, 1[, then for any m ≥ 1 and any n ≥ m+2, +P(Xm = 2 and Xn = 1) = p × P(Xm = 2). +Proof. Since Sn,n ∩ (Xm = 2) = ∅ and Sn,n−1 ∩ (Xm = 2) = ∅, we have +P(Xm = 2 ∩ Xn = 1) = +n−2 +� +k=1 +P(Xm = 2 ∩ Xn = 1 ∩ (T1, ..., Tn) ∈ Sn,k). +7 + +It follows that +P(Xm = 2 ∩ Xn = 1) = +n−2 +� +k=1 +P((T1, ..., Tn) ∈ Sn,k) × P(Xn = 1 | (T1, ..., Tn) ∈ Sn,k) +× P(Xm = 2 | Xn = 1 ∩ (T1, ..., Tn) ∈ Sn,k) +First, observe that for k ≤ n − 2, P(Xn = 1 | (T1, ..., Tn) ∈ Sn,k) = p. Let us now prove +that +P(Xm = 2 | Xn = 1 ∩ (T1, ..., Tn) ∈ Sn,k) = P(Xm = 2 | (T1, ..., Tn) ∈ Sn,k). +It is equivalent to proving that when Xm = 2 and (T1, ..., Tn) ∈ Sn,k are not incompatible, +P(Xn = 1 | Xm = 2 ∩ (T1, ..., Tn) ∈ Sn,k) = P(Xn = 1 | (T1, ..., Tn) ∈ Sn,k). +Let i be the integer such that the letter Xm is given by Ti. We can decompose the event +(T1, ..., Tn) ∈ Sn,k into the two following cases. +1. If i > k, then after reading (T1, . . . , Tk), we know the size of the blocks containing +Xm and Xn but not their content: +× × × × × × ×× +wk +?? ? 2 +Xm +?? ?? ? +Xn +. +In this case, the variables giving the values of Xm and Xn are independent, thus the +additional information that Xm = 2 does not affect the probability of having Xn = 1. +2. If i ≤ k, then reading (T1, . . . , Tk) already tells us whether Xm = 2, but does not +give us the content of the block containing Xn, which is drawn independently: +× × × × × × 2 × +wk contains Xm +?? ? ? ?? ?? ? +Xn +. +In all cases, we have +P(Xn = 1 | Xm = 2 ∩ (T1, ..., Tn) ∈ Sn,k) = P(Xn = 1 | (T1, ..., Tn) ∈ Sn,k) = p. +We deduce that +P(Xm = 2 ∩ Xn = 1) = +n−2 +� +k=1 +P(T1, ..., Tn) ∈ Sn,k) × p × P(Xm = 2 | (T1, ..., Tn) ∈ Sn,k) += p × P(Xm = 2). +8 + +We can now state the following theorem. +Theorem 2. If T ∼ (pδ1 + (1 − p)δ2)⊗N⋆ with p ∈]0, 1[, then +lim +n→∞ +|X1 . . . Xn|1 +n += p almost surely. +Proof. In order to apply Theorem 1, we need to center the random variables (Xn)n∈N⋆. +For n ∈ N⋆, we thus introduce the random variables ˜Xn = Xn − (2 − p), in order to have +| ˜Xn| ≤ 1 and limn→∞ E( ˜Xn) = 0. Now, let us exploit Lemma 1 to compute E( ˜Xm ˜Xn), for +n ≥ m + 2. We have +P( ˜Xm = p ∩ ˜Xn = p − 1) = p × P(Xm = 2), +P( ˜Xm = p ∩ ˜Xn = p) = (1 − p) × P(Xm = 2), +P( ˜Xm = p − 1 ∩ ˜Xn = p − 1) = 1 − P(Xn = 2) − p P(Xm = 2), +P( ˜Xm = p − 1 ∩ ˜Xn = p) = P(Xn = 2) − (1 − p) P(Xm = 2). +Gathering these values and using Proposition 1, we obtain +E( ˜Xn ˜Xm) = −(1 − p)2pn−1 ≤ 0. +Let us define a function Φ : N⋆ → R by Φ(0) = Φ(1) = 1 and for all k ≥ 2, Φ(k) = 0. +Then E( ˜Xm ˜Xn) ≤ Φ(|n − m|) for all m, n ∈ N⋆, and Φ satisfies obviously Φ ≥ 0 and +� +n≥1 +Φ(n) +n +< ∞. By Theorem 1, we deduce that +lim +n→∞ +1 +n +n +� +k=1 +˜Xk = 0 almost surely. +Consequently, lim +n→∞ +1 +n +n +� +k=1 +Xk = 2 − p a.s., and +lim +n→∞ +|X1 . . . Xn|1 +n += p almost surely. +To conclude on the case of a directing sequence following a product distribution, let +us mention that the previous results can be extended to other alphabets. In particular, +Proposition 1 is extended as follows. +Proposition 2. Let a, b ∈ N⋆ with 1 < a < b, and let p ∈]0, 1[. +1. If A = {1, a} and T ∼ (pδ1 + (1 − p)δa)⊗N⋆, then +∀n ≥ a, +P(Xn = 1) = p �1 − pn−a + pn−1� +9 + +2. If A = {a, b} and T ∼ (pδa + (1 − p)δb)⊗N⋆, then +∀n ≥ b + 1, +P(Xn = a) = p. +Proof. +1. We use the same partition as in the proof of Proposition 1, but we now +distinguish the sets Sn,k for n − a + 1 ≤ k ≤ n. We have Sn,n = {(t1, . . . , tn) ∈ +{1, a}n : (t1, . . . , tn−1) = (1, . . . , 1)}, and for n − a + 1 ≤ k ≤ n − 1, +Sn,k = {(t1, . . . , tn) ∈ {1, a}n : (t1, . . . , tk) = (1, . . . , 1, a)}. +If n−a+1 ≤ k ≤ n−1, then for the same reason as before, P(Xn = 1 | (T1, . . . , Tn) ∈ +Sn,k) = 0. In all the other cases, P(Xn = 1 | (T1, . . . , Tn) ∈ Sn,k) = p. Thus, +P(Xn = 1) = p(1 − pn−2(1 − p) − · · · − pn−a(1 − p)) = p �1 − pn−a + pn−1� . +2. Since a > 1, we have �|wk| +j=1[wk]j − |wk| ≥ (a − 1)|wk| > 0 for all k ∈ N⋆. This means +that we always know the length of at least one empty block after the kth step. Thus, +except if n ≤ b (in which case the nth letter might be written during the first step), +we are sure that we will know the length of the block containing the nth letter strictly +before filling it. As the Tj are independent, we deduce that P(Xn = a) = p. +4 +Sequence directed by a Markov chain +In order to get closer to the deterministic case where a 1 always follows a 2 and vice versa, +we are now interested in the case of directing sequences which are given by Markov chains. +In the present section, we assume that the directing sequence T = (Tn)n∈N⋆ is a Markov +chain over the alphabet {1, 2} with initial value T1 = 1 and whose transition probability +from 1 to 2 (and from 2 to 1) is p ∈]0, 1[. A large value of p encourages the alternation +between 1 and 2. The original case of the Oldenburger-Kolakoski sequence can be viewed +as a « limit » case of a Markov chain whose transition probability from 1 to 2 (and from +2 to 1) would be equal to 1. +Theorem 3. Let p ∈]0, 1[ and let T be a Markov chain over the alphabet {1, 2} with initial +value T1 = 1 and whose transition probability from 1 to 2 (and from 2 to 1) is p ∈]0, 1[. +Then +lim +n→∞ P(Xn = 1) = 1 +2. +Proof. Let us first note that for all integers s > r ≥ 1, P(Ts = 1 | Tr = 1) = 1 +2(1+(1−2p)s−r) +and P(Ts = 1 | Tr = 2) = 1 +2(1 − (1 − 2p)s−r). +Let ℓ ∈ N⋆ and let n ≥ 2ℓ. Consider the integer k ∈ N⋆ such that (T1, ..., Tn) ∈ Sn,k. If +we have at least 2ℓ occurences of 2 among T1, . . . , T⌊n/2⌋, then we have at least 2ℓ occurences +of 2 among (T1, . . . , Tk), and thus at least 2ℓ occurences of 2 in wk. This implies that +10 + +n − |wk| ≥ 2ℓ, meaning that when wk is written, we know the lengths of at least ℓ empty +blocks between position |wk| and position n. Similarly to the proof of Proposition 1, let +us consider the integer k′ such that Xn is given by Tk′. We also introduce D = k − k′, as +illustrated below (by definition, X|wk| = Tk and Xn = Tk′): +× × × × × × ×× +× +X|wk| +?? ? ?? ?? +� +�� +� +D − 1 blocks +? +Xn +. +By the above observations, we have +P(D < ℓ) ≤ P({ less that 2ℓ occurences of 2 among T1, . . . , T⌊n/2⌋ }), +and the probability on the right goes to 0 as n goes to ∞. Furthermore, +P(Xn = 1 | D ≥ ℓ) += +P(Tk′ = 1 | D ≥ ℓ) += +P(Tk′ = 1 | D ≥ ℓ ∩ Tk = 1) × P(Tk = 1 | D ≥ ℓ) ++ P(Tk′ = 1 | D ≥ ℓ ∩ Tk = 2) × P(Tk = 2 | D ≥ ℓ) +∈ +ï1 +2(1 − |1 − 2p|ℓ); 1 +2(1 + |1 − 2p|ℓ) +ò +thanks to the remark made at the beginning of the proof. We deduce that +lim sup +n→∞ +P(Xn = 1) ≤ 1 +2(1 + |1 − 2p|ℓ) +and +lim inf +n→∞ P(Xn = 1) ≥ 1 +2(1 − |1 − 2p|ℓ). +Then, by letting ℓ goes to infinity, we obtain limn→∞ P(Xn = 1) = 1 +2. +As a direct consequence of Theorem 3, we obtain the following result. +Corollary 2. Let p ∈]0, 1[ and let T be a Markov chain over the alphabet {1, 2} with initial +value T1 = 1 and whose transition probability from 1 to 2 (and from 2 to 1) is p ∈]0, 1[. +Then +lim +n→∞ +E(|X1 . . . Xn|1) +n += 1 +2 +We conjecture that the convergence also holds almost surely but we have been unable +to prove it so far, as the computation of the correlations is much more intricate in the +markovian case. +Conjecture 1. Let p ∈]0, 1[ and let T be a Markov chain over the alphabet {1, 2} with +initial value T1 = 1 and whose transition probability from 1 to 2 (and from 2 to 1) is +p ∈]0, 1[. Then +lim +n→∞ +|X1 . . . Xn|1 +n += 1 +2 almost surely. +Observe that Theorem 3 and Corollary 2 easily extend to other alphabets. In particular, +one obtain an identical result over the alphabet {1, 3}: if T is a Markov chain with transition +probability 0 < p < 1 from 1 to 3 (and from 3 to 1), then the average density of 1’s is 1/2. +11 + +Theorem 4. Let a ≥ 2 be an integer, let p ∈]0, 1[ and let T be a Markov chain over the +alphabet {1, a} with initial value T1 = 1 and whose transition probability from 1 to a (and +from a to 1) is p ∈]0, 1[. Then +lim +n→∞ P(Xn = 1) = 1 +2 and lim +n→∞ +E(|X1 . . . Xn|1) +n += 1 +2. +The statement of Theorem 4 is somewhat surprising and unexpected since we know that +the densities d1 and d3 of the letters 1 and 3 in the sequences O1,3 and O3,1 are respectively +d1 ≈ 0, 40 and d3 ≈ 0, 60. [1]. We will come back to this in the discussion of Section 6. +5 +Non conservation of the density +In previous sections, we have studied different cases where the directing sequences are ran- +dom. In all the cases we considered (sequences of independent and identically distributed +random variables, Markovian sequences), the densities of letters of the directed sequence +obtained are the same as those in the directing sequence, almost surely. +Simulations also suggest that for any (infinite) periodic sequence T, the density of 1’s +in directed sequence OT is well-defined and is equal to the density of 1’s in T, see Figure 1. +Figure 1: Evolution of the density of 1’s in increasingly large prefixes of OT for T = (122)ω +(left) and T = (2112111)ω (right). The densities seem to converge respectively to 1/3 and +to 5/7. +On Figure 1, we have chosen to represent only the data on short prefixes of OT so +that it remains usable, especially to distinguish the densities in the very first terms of the +sequence OT. However, further experiments have been carried out on a large number of +periodic sequences T and they seem to corroborate our first impression, namely that if the +sequence T is periodic then the densities in OT would be the same as those in T. This +leads us to state the following conjecture, that extends Keane’s conjecture. +12 + +1.0 -J +0.8 +0.6 +0.4 +0.2 +500 +100 +200 +300 +4001.0 +0.8 +0.6 +0.4 - +0.2 +100 +200 +300 +400 +500Conjecture 2. For any periodic sequence T over the alphabet {1, 2}, the density of 1’s in +the directed sequence OT is well-defined and is equal to the density of 1’s in T. +Then, a natural question arises: does there exist a directing sequence T over {1, 2} for +which the density of 1’s in T is not conserved in OT? Obviously, because of Conjecture 2, +we do not expect to find such a candidate of directing sequence among the periodic ones. +However, we answer this question partially and positively thanks to the fact that the +left-to-right reading of OT provides the size of the blocks even further to the right (see +Section 2). In a prospect of building step by step both sequences T and OT, the knowledge +of the length of not yet filled blocks of OT could allow us to choose, in a fully arbitrary +way, with which letter we will fill them and it could give us the opportunity to force the +sequence OT to contain relatively more 1’s than the sequence T. +The main idea of our simultaneous construction scheme of T and OT can be summarized +as follows: we initialize T1 to 2, then OT = 22 ?? and from now on, by reading OT from +left to right, we fill its blocks of size 2 with 1’s and its blocks of size 1 with 1’s and 2’s +alternatively. The first steps in the simultaneous construction of T and OT are thus as +follows (with the notation of Section 2): +Step 1: We set T(1) = (2) and then O(1) = 22 ?? +Step 2: +(a) The empty block of O(1) of size 2 must be filled with 1’s: O(2) = 22 11 ? ? +(b) We set T(2) = (2, 1) +Step 3: +(a) We fill the next block of O(2) +T +of size 1 with 1 : O(3) = 22 11 1 ? ? +(b) Hence T(3) = (2, 1, 1) +Step 4: +(a) We fill the next block of O(3) +T +of size 1 with 2 : O(4) = 22 11 1 2 ? ?? +(b) Hence T(4) = (2, 1, 1, 2) +Step 5: and son on. . . +For each n ∈ N⋆, we have O(n) = OT(n). Moreover, T(n) is a prefix of T(n+1) while O(n) +is a prefix of O(n+1), then we set T = limn→∞ T(n). It follows that OT = limn→∞ O(n). +Let us denote T = (ti)i∈N and OT = (xi)i∈N with ti, xi ∈ {1, 2} for all i ∈ N, then: +1. T(n) = (t1, . . . , tn) and |T(n)| = n. +2. O(n) = (txi +i )i∈[[1,n]] and OT = (txi +i )i∈N⋆. +3. |T(n)|1 = |x1 . . . xn|2 − 1 + 1 +2|x1 . . . xn|1 + Cn, with Cn ∈ {0, 1}: indeed, the number +of 1’s in T(n) is equal to the sum of the number of blocks of size 2 in O(n) (except +the first block of O(n) because of the initialisation of O(1)) and half of the number +of blocks of size 1 in O(n). By construction, the number of blocks of size 1 (resp. of +size 2) in O(n) is equal to the number of 1’s (resp. 2’s) in x1 . . . xn. The constant Cn +takes into account the cases where xn = 1 and is the first letter of a block of size 2 +in O(n). +13 + +Program 2 provides a Python function for the construction of OT and T. +1 +def +Sequences(n) : +2 +T = [2] +3 +O_T = [2, 2] +4 +d = 1 # digit to write in the next +block of size 1 +5 +for i in range(1, n) : +6 +if O_T[i] == 2 : +7 +T += [1] +8 +O_T += [1]*2 +9 +else : +10 +T += [d] +11 +O_T += [d] +12 +d = 3-d +13 +return (T, O_T) +14 +Program 2: Python function for the simultaneous construction of T and OT. +Theorem 5. Let T = limn→∞ T(n). The following properties hold: +1. If the density dT +1 of 1’s in T exists, then dT +1 = +1+ +√ +17 +8 += 0.640 . . ., dO +1 = +7− +√ +17 +4 += +0.719 . . . and so dT +1 ̸= dO +1 . +2. If the density dT +1 of 1’s in T exists, then the sequences T and OT are not periodic. +Proof. +1. For each n ∈ N⋆, we have: +��� |O(n)|1 − |T(n)|2 − 2(|T(n)|1 − |T(n)|2) +��� ≤ 1 +Indeed, to within one unit, each digit 2 of T(n) gives rise to a single 2 in O(n), and a +same quantity |T(n)|2 of 1’s gives rise to a single 1 in O(n), while the rest of them (so +|T(n)|1 − |T(n)|2) give rise to two 1’s in O(n). Moreover the first 2 of T(n) is the only +one to be written twice in O(n), so that we always have exactly |O(n)|2 = |T(n)|2 + 1. +Then, +|O(n)|1 +|O(n)| += +|O(n)|1 +|O(n)|1 + |O(n)|2 += +2|T (n)|1 − |T (n)|2 + o(n) +−1 + |T (n)|2 + 2(|T (n)|1 − |T (n)|2) + |T (n)|2 + 1 + o(n)) += +3|T (n)|1 − |T (n)| + o(n) +2|T (n)|1 + o(n) +−→ +n→∞ +3dT +1 − 1 +2dT +1 +We conclude that +dO +1 = 3dT +1 − 1 +2dT +1 +(5) +and the density of 1’s (resp. of 2’s) in OT exists. +14 + +Figure 2: Evolution of the densities of 1’s in OT (blue) and T (black), where the two +sequences are defined by Program 2. +We noticed above that, for each n ∈ N⋆, |T(n)|1 = |x1 . . . xn|2 − 1 + 1 +2|x1 . . . xn|1 + Cn, +with Cn ∈ {0, 1}. Moreover, if dT +1 exists then so do dO +1 and dO +2 and, by tending n +towards infinity, we have: +dT +1 = dO +2 + 1 +2dO +1 +(6) +By putting together equations (5) and (6), we deduce dT +1 = 1+ +√ +17 +8 +and dO +1 = 7− +√ +17 +4 +. +2. If the sequences T and OT were periodic, then their densities of 1’s and 2’s would be +rational, which is not the case. +Simulations suggest that the densities are indeed converging to these values, see Fig- +ure 2. +6 +Conclusion and discussion +Over the alphabet {1, a}, with a ∈ {2, 3}, we have shown that in almost all the sequences +directed by an infinite sequence T = (Tn)n∈N⋆ of i.i.d. random variables with P(Tn = 1) = +p ∈]0, 1[ and P(Tn = a) = 1 − p, the density of 1’s is equal to p. We have also shown that +the average density of 1’s among all sequences directed by a Markov chain with transition +probability p ∈]0, 1[ from 1 to a and from a to 1 is equal to 1/2. +Keane’s conjecture [9] states that this result can be extended to the deterministic case, +namely when p = 1, over the alphabet {1, 2}. On the other hand, over the alphabet {1, 3} +15 + +1.0 +0.8 +0.6 +0.4 +0.2 +50 +100 +150 +200this result is not extendable to the deterministic case since the density of 1’s in O1,3 is +close to 0.3972. [1]. +When T is a Markov chain, the closer its transition probability p is to 1, the more likely +the sequence OT is to share a long prefix with O1,3. Therefore, the closer the transition +probability p is to 1, the closer the density of 1’s in the sequence OT is to that in the +sequence O1,3 on a long prefix. However, computer experiments suggest that when the +first perturbations in the alternation of 1’s and 3’s appear in T, the density of 1’s in the +prefix of OT eventually approaches 0.5 as this prefix gets longer. +See Figure 3 for an +illustration with p = 0.99. +Figure 3: Evolution of the frequency of 1’s for a markovian directing sequence on the +alphabet {1, 3} of parameter p = 0.99: the frequency is first close to the one of O1,3 then +moves away from it to converge to 1/2. +This implies it seems difficult to derive information about the original Oldenburger- +Kolakoski sequence O1,2 by letting p tend to 1 in the Markovian case over the alphabet +{1, 2}. +Finally, the study of sequences directed by random sequences on alphabets of more +than 2 letters or by random sequences constructed from other distributions also seems +interesting. +References +[1] Michael Baake and Bernd Sing. Kolakoski-(3, 1) is a (deformed) model set. Canadian +Mathematical Bulletin, 47(2):168–190, 2004. +[2] Valérie Berthé, Srecko Brlek, and Philippe Choquette. Smooth words over arbitrary +alphabets. Theor. Comput. Sci., 341(1-3):293–310, 2005. +16 + +1.0 -J +0.8 +0.6 - +0.4 +0.2 +200 +400 +600 +BDO +1000[3] Valérie Berthé, Srecko Brlek, and Philippe Choquette. Smooth words over arbitrary +alphabets. Theor. Comput. Sci., 341(1-3):293–310, 2005. +[4] Srecko Brlek, Serge Dulucq, A. Ladouceur, and Laurent Vuillon. Combinatorial prop- +erties of smooth infinite words. Theor. Comput. Sci., 352(1-3):306–317, 2006. +[5] Srecko Brlek, Damien Jamet, and Geneviève Paquin. Smooth words on 2-letter al- +phabets having same parity. Theor. Comput. Sci., 393(1-3):166–181, 2008. +[6] Arturo Carpi. Repetitions in the Kolakovski sequence. Bull. EATCS, 50:194–197, +1993. +[7] Vašek Chvátal. Notes on the Kolakoski sequence. Technical report, DIMACS Technical +Report 93-84, December 1993. +[8] F. M. Dekking. On the structure of selfgenerating sequences. Séminaire de Théorie +des Nombres de Bordeaux, pages 1–6, 1980. +[9] Michael S. Keane. Ergodic theory and subshifts of finite type. In Ergodic theory, +symbolic dynamics, and hyperbolic spaces. Lectures given at the workshop "Hyperbolic +geometry and ergodic theory", held at the International Centre for Theoretical Physics +in Trieste, Italy, 17-28 April, 1989, pages 35–70. Oxford etc.: Oxford University Press, +1991. +[10] William Kolakoski. Self-generating runs, problem 5304. The American Mathematical +Monthly, 73(6):681–682, 1966. +[11] Russell Lyons. Strong laws of large numbers for weakly correlated random variables. +Michigan Mathematical Journal, 35(3):353 – 359, 1988. +[12] Rufus Oldenburger. Exponent trajectories in symbolic dynamics. Transactions of the +American Mathematical Society, 46(3):453–466, 1939. +[13] Bernd Sing. More Kolakoski sequences. Integers, 11B:A14, 2011. +17 + diff --git a/cNAzT4oBgHgl3EQfLfum/content/tmp_files/load_file.txt b/cNAzT4oBgHgl3EQfLfum/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29a54f5dd55d9275ad959dda8b2f23e7e1f87bc6 --- /dev/null +++ b/cNAzT4oBgHgl3EQfLfum/content/tmp_files/load_file.txt @@ -0,0 +1,688 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf,len=687 +page_content='On a probabilistic extension of the Oldenburger-Kolakoski sequence Chloé Boisson∗, Damien Jamet†and Irène Marcovici‡ chloe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='boisson@ens-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='fr, damien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='jamet@loria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='fr, irene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='marcovici@univ-lorraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='fr January 4, 2023 Abstract The Oldenburger-Kolakoski sequence is the only infinite sequence over the alpha- bet {1, 2} that starts with 1 and is its own run-length encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In the present work, we take a step back from this largely known and studied sequence by introducing some randomness in the choice of the letters written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This enables us to provide some results on the convergence of the density of 1’s in the resulting sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' When the choice of the letters is given by an infinite sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' random variables or by a Markov chain, the average densities of letters converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Moreover, in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' random variables, we are able to prove that the densities even almost surely converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1 Introduction The Oldenburger-Kolakoski sequence 122112122122112 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' introduced by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Oldenburger [12] and lately mentioned by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Kolakoski [10] is the unique sequence x1x2x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' over the alphabet {1, 2} with x1 = 1 and whose k-th block has length xk for k ∈ N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Keane asked whether the density of 1’s in this sequence is 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In [7], V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Chvátal showed that the upper density of 1’s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2’s) is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='50084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This bound has been slightly improved by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Rao but Keane’s question still stands: « Is the density of 1’s in Oldenburger-Kolakoski sequence defined and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='5?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' » By definition, the Oldenburger-Kolakoski sequence O = (xn)n∈N⋆ is a fixed point of the run-length encoding operator denoted ∆: ∗École Normale Supérieure de Lyon, 15 parvis René Descartes, F-69342, Lyon, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' †Université de Lorraine, Loria, UMR 7503, Vandœuvre-lès-Nancy, F-54506, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ‡Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='01116v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='DM] 3 Jan 2023 O ∆(O) = = 1�� 1 22 �� 2 11 �� 2 2�� 1 1�� 1 22 �� 2 1�� 1 22 �� 2 11 �� 2 · · = 112212211122112212 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' (1) O = 1x12x21x32x41x52x61x72x81x9 = � n∈N (1x2n+12x2n+2) (2) In [12], R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Oldenburger refers to sequences over an alphabet Σ as trajectories and refers to the sequence ∆(w) as the exponent trajectory of the trajectory w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' He stated that « a periodic trajectory is distinct from its exponent trajectory » (Theorem 2, [12]) and, therefore, the Oldenburger-Kolakoski sequence is not periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The Oldenburger-Kolakoski sequence is also connected to differentiable words, C∞- words and smooth words [2, 4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' A sequence w over the alphabet Σ ⊂ N⋆ is differentiable if and only if ∆(w) is also defined over the same alphabet Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The sequence ∆(w) is called the derivative sequence of w [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' A C∞-word, or smooth word, is an infinitely differentiable sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Obviously, the Oldenburger-Kolakoski sequence is a C∞-word since it is a fixed- point of the run-length encoding operator ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Although not answering Keane’s question fully, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Dekking established connections between possible combinatorial properties of the Oldenburger-Kolakoski sequence [8]: if the Oldenburger-Kolakoski sequence is closed by complementation (that is, if w occurs in O then so does �w with �1 = 2 and �2 = 1) then it is recurrent (any word that occurs in O does so infinitely often) (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1, [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Moreover, the Oldenburger-Kolakoski sequence is closed by complementation if and only if it contains every finite C∞-word (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2, [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' A few years later, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Carpi stated that the Oldenburger-Kolakoski sequence contains only a finite set of squares (words of the form xx where x is not empty) and does not contain any cube (word of the form xxx where x is not empty) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Hence, since O contains only squares of bounded length then it cannot be the fixed point of a non degenerated morphism: the image of a square w = xx by such a morphism is still a square longer than w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' There exist several ways to extend the definition of the Oldenburger-Kolakoski sequence, depending on whether one wants to preserve the fixed point property or to follow the construction scheme without requiring the resulting sequence to be a fixed point for the run-length encoding operator ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For instance, one can deal with other alphabets and thus construct Generalized Oldenburger-Kolakoski sequence (GOK-sequence for short) as follows: for any pair (a, b) of non-zero natural numbers, there exists a unique fixed point Oa,b of ∆ over the alphabet {a, b} starting with a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Also, according to this notation, the original Oldenburger-Kolakoski sequence is O1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For instance, if a = 1 and b = 3, the first terms of O1,3 are: O1,3 = 1�� 1 333 � � 3 111 � � 3 333 � � 3 1�� 1 3�� 1 1�� 1 333 � � 3 · · = 1133133311311133 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' (3) A significant result is, unlike the case of the original Oldenburger-Kolakoski sequence, that the densities of 1’s in O1,3 and O3,1 are known and approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='3972 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2 Generalized Oldenburger-Kolakoski sequences are also connected with smooth words over arbitrary alphabets [3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' As for the (Generalized) Oldenburger-Kolakoski sequences, the properties of smooth words are better known for alphabets with letters of the same parity: for instance, while the frequency of letters in an infinite smooth word over {1, 2} is still unsolved, in [5] the authors showed that the frequency of letters for extremal smooth words (for the lexicographic order) over the alphabet {a, b}, where a and b are both even, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' They also computed the frequency for extremal smooth words over alphabets of type {1, b}, where b is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Moreover, if a and b have the same parity, then every infinite smooth word over the alphabet {a, b} is recurrent [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Also, if a and b are both odd, then every infinite smooth word is closed under reversal but not under complementation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' On the other hand, if a and b are both even, then the extremal smooth words over the alphabet {a, b} are neither closed under reversal nor closed under complementation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For a more detailed survey on the Oldenburger-Kolakoski sequence and on generaliza- tions over arbitrary two letter alphabets see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2 Extending the construction scheme to any directing sequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='1 Notion of directing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In the construction scheme of a Generalized Oldenburger-Kolakoski sequence, the blocks of Oa,b are composed, alternatively, of a’s and b’s as shown in (1) when a = 1 and b = 2 and in (3) when a = 1 and b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In other words, if t1 = a and t2 = b, then « the ith block of Oa,b = (xi)i∈N⋆ is of length xi and is filled with the letter ti mod 2 ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This construction scheme is clearly extendable to any finite sequences T = (t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ) over {a, b} as follows: « the ith block of OT = (xi)i∈N⋆ is of length xi and is filled with the letter ti » (see Program 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1 def O(T): 2 X = [] 3 k = 0 4 for x in T: 5 X += [x] # concatenate ’x’ at the end of X 6 X += [x]*(X[k]-1) # concatenate ’X[k]-1’ copie(s) of x 7 k += 1 8 return X 9 Program 1: Python function: O is an operates on sequences over N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We say that the sequence OT is directed by the sequence T and the sequence T is a directing sequence of OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For instance, the sequence Oa,b is directed by T = (ab)ω while O is directed by (12)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Notice that the directed sequence OT may no longer be a fixed point of the operator ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us now take a closer look at how the construction of OT provides a little more 3 information than the sequence itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For instance, let T = (tn)n∈N⋆ = 21122 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' be a sequence over {1, 2}: Step 1: Ot1 = 22 and the second block is of length 2: hence the 3rd and 4th letters are in a same block of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us denote Ot1 = 22 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='. Step 2: Ot1t2 = 22 11 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' : the 5th and the 6th letter are respectively in blocks of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Step 3: = 22 11 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' : the 7th letter is in a block of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Roughly speaking, t1 gives the length of all the blocks that contain up to the 4th letter, t1t2 gives the length of all the blocks that contain up the 6th letter and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let wn = Ot1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='tn for each n ∈ N⋆ , then �|w1| i=1 [w1]i = 4 , �|w2| i=1 [w2]i = 6, �|w3| i=1 [w3]i = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='2 Partitions of the set of directing sequences We now introduce some subsets of directing sequences that will be crucial in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For this purpose, let us classify the sequences T according to the information they provide on the length of the blocks of OT: let k and n be two integers such that 1 ≤ k ≤ n and let Sn,k be the set of sequences T = (tn)n∈N such that the length of the block of OT containing its nth letter is known when reading tk but not before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Formally, if wn = Ot1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='tn, then we have Sn,k = � (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', tn) ∈ {1, 2}n : min{j ∈ �1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' n�, |wj| � i=1 [wj]i ≥ n} = k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' (4) Let us give a short example to illustrate the latter definition: let n = 5 and let T = (t1, t2, t3, t4, t5) = (2, 1, 1, 2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' After the first step and the reading of t1, we only know that t1 = 2 and still do not know the length of the block that will contain the 5th letter of OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' On the other hand, after having taken knowledge of the value of t2, we know that the 5th letter of OT will be written in a block of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Hence, T ∈ S5,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' More generally, Step 1: Ot1 = 22 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' and T ∈ S3,1 ∩ S4,1 since t1 provides the length of the block containing the 3rd and the 4th of OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Step 2: Ot1t2 = 22 11 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' : the 5th and the 6th letter are respectively in blocks of length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Hence T ∈ S5,2 ∩ S6,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Step 3: Ot1t2t3 = 22 11 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' and T ∈ S7,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Step 4: Ot1t2t3t4 = 22 11 1 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' and T ∈ S8,4 ∩ S9,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Step 5: Ot1t2t3t4t5 = 22 11 1 2 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' and T ∈ S10,5 ∩ S11,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 4 The set {Sn,k : k ∈ �1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' n�} is a partition of {1, 2}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Indeed, the length of the word wn is at least n, since at each step, after reading ti, we write at least one letter of OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' And each letter is 1 or 2, so �|wn| i=1 [wn]i ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us notice that Sn,n = {(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 1), (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 1, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Indeed, the length of wn−1 is at least n − 1, and is exactly n − 1 if and only if the written letters are all 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Moreover Sn,n−1 = {(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 1, 2, 1), (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 1, 2, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Indeed, one easily check that (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 2, 1), (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 2, 2) ∈ Sn,n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Reciprocally, if (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , tn) ∈ Sn,n−1, then there exists i ∈ �1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' n − 1� such that ti = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us note that the first 2 in T is written twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Thus, if there were such a i in �1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' n − 2� we would have �|wn−2| i=1 [wn−2]i ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='3 Extension of the definition to infinite directing sequences Now that we have started studying the notions of directing and directed sequences, a nat- ural question arises: « Can one extend the definition of OT sequences to infinite (possibly not periodic) sequences T?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' » Let A ⊆ N⋆ be an alphabet and let T = (tn)n∈N⋆ be an infinite sequence over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By construction, Ot1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='tn is a prefix of Ot1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='tn+1for each n ∈ N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' One thus defines OT as the limit of Ot1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='tn when n tends to infinity: OT = lim n→∞ Ot1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The present work deals with the densities of letters in OT when T = (tn)n∈N⋆ is an infinite sequence over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Do these densities exist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If so, how much are they value?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We are especially interested in the case where the directing sequence is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let T = (Tn)n∈N⋆ be a sequence of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By definition, the sequence directed by T is the random sequence X = (Xn)n∈N⋆ defined by X = OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The present paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In section 3, we consider the case where the directing sequence is made of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=') random variables over a two-letter alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In section 4, we treat the case where the random sequence X is directed by a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 3 Sequence directed by independent random variables In the present section, T = (Tn)n∈N⋆ is a sequence of independent and identically distributed random variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' for short) over the two-letter alphabet A = {1, 2}, with P(Tn = 1) = p and P(Tn = 2) = 1 − p for each n ∈ N⋆, where p is a fixed parameter in ]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The sequence T is thus distributed according to the product distribution (pδ1 + (1 − p)δ2)⊗N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We denote T ∼ ((pδ1 + (1 − p)δ2)⊗N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let X = (Xn)n∈N⋆ be the sequence directed by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The sequence X is a random sequence with a priori unknown distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Assume that one wants to compute the nth letter Xn of X, for some large integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Unless the sequence T begins with a long succession of 1’s (an event which has a low probability to occur), one just has to read the first terms T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , Tk of T, until knowing the length and position of the block containing Xn, and to fill that block by 1’s with probability p, or by 2’s with probability 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The resulting value of Xn obtained that way will have the desired distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This leads to the fact that 5 limn→∞ P(Xn = 1) = p, and we can even use this observation to compute more precisely the value of P(Xn = 1), as shown in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If T ∼ (pδ1 + (1 − p)δ2)⊗N⋆ with p ∈]0, 1[, then for any n ≥ 2, P(Xn = 1) = p(1 − pn−2 + pn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We will decompose the event {Xn = 1} according to the partition {Sn,k : k ∈ �1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' n�} of {1, 2}n introduced in the previous section, and use the observations (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=') and (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The following two particular cases are obvious: P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,n) = p, and P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,n−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us now consider some (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k, with k < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let k′ be the unique integer such that |wk′−1| < n and |wk′| ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Concretely, the nth letter Xn is written during the k′th step, and it is equal to Tk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By definition of Sn,k, we have k′ ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us show that k′ ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We reason by contradiction and assume that k′ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us now look at what we know at the end of the k − 1th step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By definition of k′, Xn will be written in the next block to the right of wk′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By definition of k, we do not know the length of this block yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Consequently, we do not know the length of any (empty) block to the right of wk′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' It directly implies that wk′−1 is made of 1’s (and thus, that wk′−1 = 1k′−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' It follows that n ≤ |wk′| ≤ |wk′−1|+2 = k′ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Since k′ = k < n−1, we get a contradiction, which means that k < k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The fact that (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) belongs to Sn,k only depends on the beginning (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tk) of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Recall that T has a product distribution (pδ1 + (1 − p)δ2)⊗N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Since k′ > k, and Xn = Tk′, we deduce that P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Finally, we use the formula of total probability: P(Xn = 1) = n−2 � k=1 P((T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) × P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) + P((T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='Tn−1) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 1)) × P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn−1) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 1)) + P((T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='Tn−1) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 2)) × P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn−1) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 2)) = �1 − (pn + 2(1 − p)pn−1 + pn−2(1 − p)2)� × p + pn × 1 + 0 = p(1 − pn−2 + pn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' As a corollary, we obtain the following convergence of the proportion of 1’s in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 6 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If T ∼ (pδ1 + (1 − p)δ2)⊗N⋆ with p ∈]0, 1[, then lim n→∞ E(|X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1) n = p Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Since X has values in {1, 2}, we have |X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1 = 2n − (X1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' + Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' It follows that E(|X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1) n = 2 − �n k=1 E(Xk) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By Proposition 1, we have limn→∞ P(Xn = 1) = p and limn→∞ P(Xn = 2) = 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We deduce that limn→∞ E(Xn) = 2 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By Cesàro lemma, we obtain: lim n→∞ E(|X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1) n = 2 − (2 − p) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Each time we run a simulation with T ∼ (pδ1 + (1 − p)δ2)⊗N⋆, the frequency of 1’s in X seems to converge to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We thus expect the sequence |X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1/n to converge almost surely to p, and not only in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Since the random variables (Xn)n∈N⋆ are correlated, we can not directly apply the strong law of large numbers (SLLN) to prove the almost sure convergence of (X1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' + Xn)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' However, the correlations being sufficiently weak, we can apply the following stronger version of the SLLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theorem 1 (Lyons [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let (Yn)n∈N⋆ be a sequence of real-valued random variables such that for all n ∈ N⋆, |Yn| ≤ 1 and ∀n, m ∈ N⋆, E(YmYn) ≤ Φ(|n − m|), with Φ ≥ 0 and � n≥1 Φ(n) n < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then lim n→∞ 1 n n � k=1 Yk = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In order to apply Theorem 1, let us first prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If T ∼ (pδ1+(1−p)δ2)⊗N⋆ with p ∈]0, 1[, then for any m ≥ 1 and any n ≥ m+2, P(Xm = 2 and Xn = 1) = p × P(Xm = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Since Sn,n ∩ (Xm = 2) = ∅ and Sn,n−1 ∩ (Xm = 2) = ∅, we have P(Xm = 2 ∩ Xn = 1) = n−2 � k=1 P(Xm = 2 ∩ Xn = 1 ∩ (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 7 It follows that P(Xm = 2 ∩ Xn = 1) = n−2 � k=1 P((T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) × P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) × P(Xm = 2 | Xn = 1 ∩ (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) First, observe that for k ≤ n − 2, P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us now prove that P(Xm = 2 | Xn = 1 ∩ (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) = P(Xm = 2 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' It is equivalent to proving that when Xm = 2 and (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k are not incompatible, P(Xn = 1 | Xm = 2 ∩ (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) = P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let i be the integer such that the letter Xm is given by Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We can decompose the event (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k into the two following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If i > k, then after reading (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , Tk), we know the size of the blocks containing Xm and Xn but not their content: × × × × × × ×× wk ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2 Xm ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In this case, the variables giving the values of Xm and Xn are independent, thus the additional information that Xm = 2 does not affect the probability of having Xn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If i ≤ k, then reading (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , Tk) already tells us whether Xm = 2, but does not give us the content of the block containing Xn, which is drawn independently: × × × × × × 2 × wk contains Xm ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In all cases, we have P(Xn = 1 | Xm = 2 ∩ (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) = P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We deduce that P(Xm = 2 ∩ Xn = 1) = n−2 � k=1 P(T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) × p × P(Xm = 2 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k) = p × P(Xm = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 8 We can now state the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If T ∼ (pδ1 + (1 − p)δ2)⊗N⋆ with p ∈]0, 1[, then lim n→∞ |X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1 n = p almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In order to apply Theorem 1, we need to center the random variables (Xn)n∈N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For n ∈ N⋆, we thus introduce the random variables ˜Xn = Xn − (2 − p), in order to have | ˜Xn| ≤ 1 and limn→∞ E( ˜Xn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Now, let us exploit Lemma 1 to compute E( ˜Xm ˜Xn), for n ≥ m + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We have P( ˜Xm = p ∩ ˜Xn = p − 1) = p × P(Xm = 2), P( ˜Xm = p ∩ ˜Xn = p) = (1 − p) × P(Xm = 2), P( ˜Xm = p − 1 ∩ ˜Xn = p − 1) = 1 − P(Xn = 2) − p P(Xm = 2), P( ˜Xm = p − 1 ∩ ˜Xn = p) = P(Xn = 2) − (1 − p) P(Xm = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Gathering these values and using Proposition 1, we obtain E( ˜Xn ˜Xm) = −(1 − p)2pn−1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us define a function Φ : N⋆ → R by Φ(0) = Φ(1) = 1 and for all k ≥ 2, Φ(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then E( ˜Xm ˜Xn) ≤ Φ(|n − m|) for all m, n ∈ N⋆, and Φ satisfies obviously Φ ≥ 0 and � n≥1 Φ(n) n < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By Theorem 1, we deduce that lim n→∞ 1 n n � k=1 ˜Xk = 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Consequently, lim n→∞ 1 n n � k=1 Xk = 2 − p a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', and lim n→∞ |X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1 n = p almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' To conclude on the case of a directing sequence following a product distribution, let us mention that the previous results can be extended to other alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In particular, Proposition 1 is extended as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let a, b ∈ N⋆ with 1 < a < b, and let p ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If A = {1, a} and T ∼ (pδ1 + (1 − p)δa)⊗N⋆, then ∀n ≥ a, P(Xn = 1) = p �1 − pn−a + pn−1� 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If A = {a, b} and T ∼ (pδa + (1 − p)δb)⊗N⋆, then ∀n ≥ b + 1, P(Xn = a) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We use the same partition as in the proof of Proposition 1, but we now distinguish the sets Sn,k for n − a + 1 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We have Sn,n = {(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , tn) ∈ {1, a}n : (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , tn−1) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , 1)}, and for n − a + 1 ≤ k ≤ n − 1, Sn,k = {(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , tn) ∈ {1, a}n : (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , tk) = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , 1, a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If n−a+1 ≤ k ≤ n−1, then for the same reason as before, P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , Tn) ∈ Sn,k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In all the other cases, P(Xn = 1 | (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , Tn) ∈ Sn,k) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Thus, P(Xn = 1) = p(1 − pn−2(1 − p) − · · · − pn−a(1 − p)) = p �1 − pn−a + pn−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Since a > 1, we have �|wk| j=1[wk]j − |wk| ≥ (a − 1)|wk| > 0 for all k ∈ N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This means that we always know the length of at least one empty block after the kth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Thus, except if n ≤ b (in which case the nth letter might be written during the first step), we are sure that we will know the length of the block containing the nth letter strictly before filling it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' As the Tj are independent, we deduce that P(Xn = a) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 4 Sequence directed by a Markov chain In order to get closer to the deterministic case where a 1 always follows a 2 and vice versa, we are now interested in the case of directing sequences which are given by Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In the present section, we assume that the directing sequence T = (Tn)n∈N⋆ is a Markov chain over the alphabet {1, 2} with initial value T1 = 1 and whose transition probability from 1 to 2 (and from 2 to 1) is p ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' A large value of p encourages the alternation between 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The original case of the Oldenburger-Kolakoski sequence can be viewed as a « limit » case of a Markov chain whose transition probability from 1 to 2 (and from 2 to 1) would be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let p ∈]0, 1[ and let T be a Markov chain over the alphabet {1, 2} with initial value T1 = 1 and whose transition probability from 1 to 2 (and from 2 to 1) is p ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then lim n→∞ P(Xn = 1) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us first note that for all integers s > r ≥ 1, P(Ts = 1 | Tr = 1) = 1 2(1+(1−2p)s−r) and P(Ts = 1 | Tr = 2) = 1 2(1 − (1 − 2p)s−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let ℓ ∈ N⋆ and let n ≥ 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Consider the integer k ∈ N⋆ such that (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', Tn) ∈ Sn,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If we have at least 2ℓ occurences of 2 among T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , T⌊n/2⌋, then we have at least 2ℓ occurences of 2 among (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , Tk), and thus at least 2ℓ occurences of 2 in wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This implies that 10 n − |wk| ≥ 2ℓ, meaning that when wk is written, we know the lengths of at least ℓ empty blocks between position |wk| and position n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Similarly to the proof of Proposition 1, let us consider the integer k′ such that Xn is given by Tk′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We also introduce D = k − k′, as illustrated below (by definition, X|wk| = Tk and Xn = Tk′): × × × × × × ×× × X|wk| ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' � �� � D − 1 blocks ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By the above observations, we have P(D < ℓ) ≤ P({ less that 2ℓ occurences of 2 among T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , T⌊n/2⌋ }), and the probability on the right goes to 0 as n goes to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Furthermore, P(Xn = 1 | D ≥ ℓ) = P(Tk′ = 1 | D ≥ ℓ) = P(Tk′ = 1 | D ≥ ℓ ∩ Tk = 1) × P(Tk = 1 | D ≥ ℓ) + P(Tk′ = 1 | D ≥ ℓ ∩ Tk = 2) × P(Tk = 2 | D ≥ ℓ) ∈ ï1 2(1 − |1 − 2p|ℓ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1 2(1 + |1 − 2p|ℓ) ò thanks to the remark made at the beginning of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We deduce that lim sup n→∞ P(Xn = 1) ≤ 1 2(1 + |1 − 2p|ℓ) and lim inf n→∞ P(Xn = 1) ≥ 1 2(1 − |1 − 2p|ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then, by letting ℓ goes to infinity, we obtain limn→∞ P(Xn = 1) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' As a direct consequence of Theorem 3, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let p ∈]0, 1[ and let T be a Markov chain over the alphabet {1, 2} with initial value T1 = 1 and whose transition probability from 1 to 2 (and from 2 to 1) is p ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then lim n→∞ E(|X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1) n = 1 2 We conjecture that the convergence also holds almost surely but we have been unable to prove it so far, as the computation of the correlations is much more intricate in the markovian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let p ∈]0, 1[ and let T be a Markov chain over the alphabet {1, 2} with initial value T1 = 1 and whose transition probability from 1 to 2 (and from 2 to 1) is p ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then lim n→∞ |X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1 n = 1 2 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Observe that Theorem 3 and Corollary 2 easily extend to other alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In particular, one obtain an identical result over the alphabet {1, 3}: if T is a Markov chain with transition probability 0 < p < 1 from 1 to 3 (and from 3 to 1), then the average density of 1’s is 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 11 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let a ≥ 2 be an integer, let p ∈]0, 1[ and let T be a Markov chain over the alphabet {1, a} with initial value T1 = 1 and whose transition probability from 1 to a (and from a to 1) is p ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then lim n→∞ P(Xn = 1) = 1 2 and lim n→∞ E(|X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Xn|1) n = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The statement of Theorem 4 is somewhat surprising and unexpected since we know that the densities d1 and d3 of the letters 1 and 3 in the sequences O1,3 and O3,1 are respectively d1 ≈ 0, 40 and d3 ≈ 0, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We will come back to this in the discussion of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 5 Non conservation of the density In previous sections, we have studied different cases where the directing sequences are ran- dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In all the cases we considered (sequences of independent and identically distributed random variables, Markovian sequences), the densities of letters of the directed sequence obtained are the same as those in the directing sequence, almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Simulations also suggest that for any (infinite) periodic sequence T, the density of 1’s in directed sequence OT is well-defined and is equal to the density of 1’s in T, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Figure 1: Evolution of the density of 1’s in increasingly large prefixes of OT for T = (122)ω (left) and T = (2112111)ω (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The densities seem to converge respectively to 1/3 and to 5/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' On Figure 1, we have chosen to represent only the data on short prefixes of OT so that it remains usable, especially to distinguish the densities in the very first terms of the sequence OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' However, further experiments have been carried out on a large number of periodic sequences T and they seem to corroborate our first impression, namely that if the sequence T is periodic then the densities in OT would be the same as those in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This leads us to state the following conjecture, that extends Keane’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='0 -J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='2 500 100 200 300 4001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='2 100 200 300 400 500Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For any periodic sequence T over the alphabet {1, 2}, the density of 1’s in the directed sequence OT is well-defined and is equal to the density of 1’s in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then, a natural question arises: does there exist a directing sequence T over {1, 2} for which the density of 1’s in T is not conserved in OT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Obviously, because of Conjecture 2, we do not expect to find such a candidate of directing sequence among the periodic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' However, we answer this question partially and positively thanks to the fact that the left-to-right reading of OT provides the size of the blocks even further to the right (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In a prospect of building step by step both sequences T and OT, the knowledge of the length of not yet filled blocks of OT could allow us to choose, in a fully arbitrary way, with which letter we will fill them and it could give us the opportunity to force the sequence OT to contain relatively more 1’s than the sequence T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The main idea of our simultaneous construction scheme of T and OT can be summarized as follows: we initialize T1 to 2, then OT = 22 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' and from now on, by reading OT from left to right, we fill its blocks of size 2 with 1’s and its blocks of size 1 with 1’s and 2’s alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The first steps in the simultaneous construction of T and OT are thus as follows (with the notation of Section 2): Step 1: We set T(1) = (2) and then O(1) = 22 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Step 2: (a) The empty block of O(1) of size 2 must be filled with 1’s: O(2) = 22 11 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' (b) We set T(2) = (2, 1) Step 3: (a) We fill the next block of O(2) T of size 1 with 1 : O(3) = 22 11 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' (b) Hence T(3) = (2, 1, 1) Step 4: (a) We fill the next block of O(3) T of size 1 with 2 : O(4) = 22 11 1 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' (b) Hence T(4) = (2, 1, 1, 2) Step 5: and son on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For each n ∈ N⋆, we have O(n) = OT(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Moreover, T(n) is a prefix of T(n+1) while O(n) is a prefix of O(n+1), then we set T = limn→∞ T(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' It follows that OT = limn→∞ O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let us denote T = (ti)i∈N and OT = (xi)i∈N with ti, xi ∈ {1, 2} for all i ∈ N, then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' T(n) = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' , tn) and |T(n)| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' O(n) = (txi i )i∈[[1,n]] and OT = (txi i )i∈N⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' |T(n)|1 = |x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' xn|2 − 1 + 1 2|x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' xn|1 + Cn, with Cn ∈ {0, 1}: indeed, the number of 1’s in T(n) is equal to the sum of the number of blocks of size 2 in O(n) (except the first block of O(n) because of the initialisation of O(1)) and half of the number of blocks of size 1 in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' By construction, the number of blocks of size 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' of size 2) in O(n) is equal to the number of 1’s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2’s) in x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The constant Cn takes into account the cases where xn = 1 and is the first letter of a block of size 2 in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 13 Program 2 provides a Python function for the construction of OT and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1 def Sequences(n) : 2 T = [2] 3 O_T = [2, 2] 4 d = 1 # digit to write in the next block of size 1 5 for i in range(1, n) : 6 if O_T[i] == 2 : 7 T += [1] 8 O_T += [1]*2 9 else : 10 T += [d] 11 O_T += [d] 12 d = 3-d 13 return (T, O_T) 14 Program 2: Python function for the simultaneous construction of T and OT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Let T = limn→∞ T(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The following properties hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If the density dT 1 of 1’s in T exists, then dT 1 = 1+ √ 17 8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='640 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', dO 1 = 7− √ 17 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='719 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' and so dT 1 ̸= dO 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If the density dT 1 of 1’s in T exists, then the sequences T and OT are not periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' For each n ∈ N⋆, we have: ��� |O(n)|1 − |T(n)|2 − 2(|T(n)|1 − |T(n)|2) ��� ≤ 1 Indeed, to within one unit, each digit 2 of T(n) gives rise to a single 2 in O(n), and a same quantity |T(n)|2 of 1’s gives rise to a single 1 in O(n), while the rest of them (so |T(n)|1 − |T(n)|2) give rise to two 1’s in O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Moreover the first 2 of T(n) is the only one to be written twice in O(n), so that we always have exactly |O(n)|2 = |T(n)|2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Then, |O(n)|1 |O(n)| = |O(n)|1 |O(n)|1 + |O(n)|2 = 2|T (n)|1 − |T (n)|2 + o(n) −1 + |T (n)|2 + 2(|T (n)|1 − |T (n)|2) + |T (n)|2 + 1 + o(n)) = 3|T (n)|1 − |T (n)| + o(n) 2|T (n)|1 + o(n) −→ n→∞ 3dT 1 − 1 2dT 1 We conclude that dO 1 = 3dT 1 − 1 2dT 1 (5) and the density of 1’s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' of 2’s) in OT exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 14 Figure 2: Evolution of the densities of 1’s in OT (blue) and T (black), where the two sequences are defined by Program 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We noticed above that, for each n ∈ N⋆, |T(n)|1 = |x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' xn|2 − 1 + 1 2|x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' xn|1 + Cn, with Cn ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Moreover, if dT 1 exists then so do dO 1 and dO 2 and, by tending n towards infinity, we have: dT 1 = dO 2 + 1 2dO 1 (6) By putting together equations (5) and (6), we deduce dT 1 = 1+ √ 17 8 and dO 1 = 7− √ 17 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' If the sequences T and OT were periodic, then their densities of 1’s and 2’s would be rational, which is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Simulations suggest that the densities are indeed converging to these values, see Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 6 Conclusion and discussion Over the alphabet {1, a}, with a ∈ {2, 3}, we have shown that in almost all the sequences directed by an infinite sequence T = (Tn)n∈N⋆ of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' random variables with P(Tn = 1) = p ∈]0, 1[ and P(Tn = a) = 1 − p, the density of 1’s is equal to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' We have also shown that the average density of 1’s among all sequences directed by a Markov chain with transition probability p ∈]0, 1[ from 1 to a and from a to 1 is equal to 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Keane’s conjecture [9] states that this result can be extended to the deterministic case, namely when p = 1, over the alphabet {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' On the other hand, over the alphabet {1, 3} 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='2 50 100 150 200this result is not extendable to the deterministic case since the density of 1’s in O1,3 is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='3972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' When T is a Markov chain, the closer its transition probability p is to 1, the more likely the sequence OT is to share a long prefix with O1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Therefore, the closer the transition probability p is to 1, the closer the density of 1’s in the sequence OT is to that in the sequence O1,3 on a long prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' However, computer experiments suggest that when the first perturbations in the alternation of 1’s and 3’s appear in T, the density of 1’s in the prefix of OT eventually approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='5 as this prefix gets longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' See Figure 3 for an illustration with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Figure 3: Evolution of the frequency of 1’s for a markovian directing sequence on the alphabet {1, 3} of parameter p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='99: the frequency is first close to the one of O1,3 then moves away from it to converge to 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' This implies it seems difficult to derive information about the original Oldenburger- Kolakoski sequence O1,2 by letting p tend to 1 in the Markovian case over the alphabet {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Finally, the study of sequences directed by random sequences on alphabets of more than 2 letters or by random sequences constructed from other distributions also seems interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' References [1] Michael Baake and Bernd Sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Kolakoski-(3, 1) is a (deformed) model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Canadian Mathematical Bulletin, 47(2):168–190, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [2] Valérie Berthé, Srecko Brlek, and Philippe Choquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Smooth words over arbitrary alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 341(1-3):293–310, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='0 -J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content='2 200 400 600 BDO 1000[3] Valérie Berthé, Srecko Brlek, and Philippe Choquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Smooth words over arbitrary alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 341(1-3):293–310, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [4] Srecko Brlek, Serge Dulucq, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Ladouceur, and Laurent Vuillon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Combinatorial prop- erties of smooth infinite words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 352(1-3):306–317, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [5] Srecko Brlek, Damien Jamet, and Geneviève Paquin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Smooth words on 2-letter al- phabets having same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=', 393(1-3):166–181, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [6] Arturo Carpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Repetitions in the Kolakovski sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' EATCS, 50:194–197, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [7] Vašek Chvátal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Notes on the Kolakoski sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Technical report, DIMACS Technical Report 93-84, December 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Dekking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' On the structure of selfgenerating sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Séminaire de Théorie des Nombres de Bordeaux, pages 1–6, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [9] Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Keane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Ergodic theory and subshifts of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' In Ergodic theory, symbolic dynamics, and hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Lectures given at the workshop "Hyperbolic geometry and ergodic theory", held at the International Centre for Theoretical Physics in Trieste, Italy, 17-28 April, 1989, pages 35–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Oxford etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' : Oxford University Press, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [10] William Kolakoski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Self-generating runs, problem 5304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' The American Mathematical Monthly, 73(6):681–682, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [11] Russell Lyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Strong laws of large numbers for weakly correlated random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Michigan Mathematical Journal, 35(3):353 – 359, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [12] Rufus Oldenburger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Exponent trajectories in symbolic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Transactions of the American Mathematical Society, 46(3):453–466, 1939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' [13] Bernd Sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' More Kolakoski sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' Integers, 11B:A14, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNAzT4oBgHgl3EQfLfum/content/2301.01116v1.pdf'} diff --git a/cdAyT4oBgHgl3EQf-foD/content/tmp_files/2301.00891v1.pdf.txt b/cdAyT4oBgHgl3EQf-foD/content/tmp_files/2301.00891v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fae1de095c681106b128170e0e2250de6a5384c4 --- /dev/null +++ b/cdAyT4oBgHgl3EQf-foD/content/tmp_files/2301.00891v1.pdf.txt @@ -0,0 +1,754 @@ +Understanding Political Polarisation using Language Models: +A dataset and method +Samiran Gode,1 Supreeth Bare, 1 Bhiksha Raj, 1 Hyungon Yoo, 1 +Carnegie Mellon University +Abstract +Our paper aims to analyze political polarization in US polit- +ical system using Language Models, and thereby help candi- +dates make an informed decision. The availability of this in- +formation will help voters understand their candidates’ views +on the economy, healthcare, education and other social is- +sues. Our main contributions are a dataset extracted from +Wikipedia that spans the past 120 years and a Language +model-based method that helps analyze how polarized a can- +didate is. Our data is divided into 2 parts, background infor- +mation and political information about a candidate, since our +hypothesis is that the political views of a candidate should be +based on reason and be independent of factors such as birth- +place, alma mater, etc. We further split this data into 4 phases +chronologically, to help understand if and how the polariza- +tion amongst candidates changes. This data has been cleaned +to remove biases. To understand the polarization we begin +by showing results from some classical language models in +Word2Vec and Doc2Vec. And then use more powerful tech- +niques like the Longformer, a transformer-based encoder, to +assimilate more information and find the nearest neighbors of +each candidate based on their political view and their back- +ground. +Introduction +Polarization among the two main parties in the US, Republi- +can and Democratic, has been studied for a long time ((Poole +and Rosenthal 1984),(KhudaBukhsh et al. 2021)). A lot of +the discussion online has become polarized(Jiang, Robert- +son, and Wilson 2020), and this discussion gets the most +traction online(Jiang, Robertson, and Wilson 2020). This po- +larization can affect the decision-making ability of a candi- +date if selected(Chen, Li, and Liu 2022). In such scenarios, it +is important for users to be able to separate the rhetoric and +understand how polar a candidate is. With this work, we set +out to ask exactly these questions, ”Can we measure how po- +larizing a candidate is?”, ”Can we measure how much this +polarity has changed over time?”, We try to answer these +questions using Natural Language based techniques and in +the process, create a dataset that will be useful for the re- +search community in trying to understand political polariza- +tion in the US. Though we have worked on the US politi- +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +cal system, the methods we suggest for measuring polariza- +tion would be useful for other countries with similar demo- +cratic elections in determining how polazised a candidate is. +We first try classical methods such as Word2Vec(Mikolov +et al. 2013) and Doc2Vec(Le and Mikolov 2014) to under- +stand if we can find polarization using the data we have +and gain more insights. We find that words that are politi- +cally sensitive(Center 2019) are related to other words which +are politically sensitive(Center 2019). We thus move on to +more recent and sophisticated models built using Transform- +ers(Vaswani et al. 2017) to gain more insight into the data. +We then use these, in particular, longformers(Beltagy, Pe- +ters, and Cohan 2020) to project candidate-specific data into +a particular embedding space and then use this data to find +the nearest neighbors of each candidate and provide one +metric to find how polarized a candidate is. +Related Work +(KhudaBukhsh et al. 2021) talks about political polariza- +tion online and uses machine translation to interpret politi- +cal polarization on the internet. (Bhatt et al. 2018) discusses +the impacts of hyper-partisan websites on influencing public +opinion as illustrated by their ability to affect certain events +in the 2016 US general elections. The authors then go on to +show how certain political biases are assumed for the pur- +pose of their study, namely overt support for either a Demo- +crat or Republican is taken to be an indicator of the site being +either Liberal or Conservative. This paper is fundamental to +our research as it looks into the political division and lays +the foundation for any following work in the domain of us- +ing specific features to classify an entity as being Liberal or +Conservative. The features they considered were transcripts +of the content being published or shared on these sites. In our +case, the features will simply be the Wikipedia page content +of the people. (KhudaBukhsh et al. 2022) shows the polar- +ization in TV media and fringe new networks and uses Lan- +guage model-based techniques to understand them further. +However, this polarization visible in the electorate stems +from the candidates. (DeSilver 2022) claims that the can- +didates become polarized and moved away from the center +over the years. With this paper, we release a dataset and a +few metrics that will help us understand if political polariza- +tion exists in political candidates and how we might be able +to measure this political polarization. The aim of this study +arXiv:2301.00891v1 [cs.CL] 2 Jan 2023 + +is to aid voters to make informed decisions before elections. +And we use language-based techniques on a dataset that is +classified into 4 eras and divided into 3 parts, mainly back- +ground, political and other. +(Belcastro et al. 2020) Demonstrates that Political Polar- +ization can be mapped with the help of Neural Networks. +This is almost a baseline idea as we are using attention net- +works and Longformer models for the same. The key differ- +ence lies in the data extraction and methodology. +(Khadilkar, KhudaBukhsh, and Mitchell 2022) goes in +depth towards finding gender and racial bias in a large sam- +ple of Bollywood (and Hollywood) movies. The author has +amalgamated several known NLP models while he tries to +create a reasonably robust model of his own. The portions in +which this particular study differs from those before is that +the sample size is fairly large. It then diverges further with its +rather innovative use of diachronic-word embedding associ- +ation tests (WEAT). Other techniques that are implemented +include count-based statistics dependent on a highly popu- +lar lexicon cloze test using BERT as a base model (an idea +we could consider after data attention) and bias recognition +using WEAT. The final model is a combination of the above +three. This paper is highly relevant to our project as it uses +a similar idea of our own. It uses aforementioned models +to predict bias, i.e. sentiment prediction. In our project, we +use data to predict political sentiment and attempt to classify +certain features as being precursors to classification. +(Rajani et al. 2019) tried to improve speech-based models +on their ability to verbalize the reasoning that they learned +during training. It uses the CAGE framework (Common- +Sense Auto-Generated Explanations) on the common sense +explanation dataset to increase the effectiveness by 10 per- +cent. It introduces improvements over the use of BiDAF++ +(augmented with self-attention layer) in these newer mod- +els. It further uses NLE as rationale generalization within +the second phase primarily as means for sentiment analysis. +In this paper, Mturk (from Amazon) is used to generate ex- +planations for the dataset. CAGE primarily uses a question- +answer format with 3 options, a label and the best expla- +nation for that label. Furthermore, other evaluation param- +eters affecting performance are tested and may be used in +our project either as verification models or otherwise. CAGE +is certainly an interesting choice for verification given the +higher accuracy it attains. A factor to be considered how- +ever is that the types of datasets and models are very dif- +ferent. Thus certain modifications will be made to the above +framework. +(Devlin et al. 2018) is the introduction paper for BERT, +a model that will be used extensively. It also shows the re- +sults of fine-tuning BERT. These indirectly or directly will +be used either as pre-trained constraints or as tuning meth- +ods. petroni2019language +(Petroni et al. 2019) Demonstrates the ability of pre- +trained high-capacity models like BERT and ELMo to be +used as knowledge repositories. This is mainly based on 3 +observations, (1) The relational knowledge of these models +is competitive to that of an NLP with access to certain oracle +knowledge. (2) The effectiveness of BERT in an open do- +main question answer test and (3) The fact that certain facts +are easily learnable. The Authors also demonstrate the us- +age of other models (unidirectional and bi-directional) in the +study, namely ’fariseq-fconv’ and ’Transformer-XL’. They +conclude by showing that BERT-Large is able to outperform +other models and compete even with supervised models for +the same Task. +(Palakodety, KhudaBukhsh, and Carbonell 2020) demon- +strates the ability of BERT and similar LM’s to track com- +munity perception, aggregate opinions and compare the pop- +ularity of political parties and candidates. This is demonstra- +tive of our work as we intend to use BERT for the purpose +of sentiment analysis. The authors conclude by stating that +the LM can be used as a pipeline for extracting Data in the +future. +In (Hamilton, Leskovec, and Jurafsky 2016) the authors +try to counter the problem of word meaning changing se- +mantically with context. They propose a robust method by +using embeddings. These are then evaluated with the ’Law +of Conformity’ and ’The Law of Innovation’. These display +the role of frequency and polysemy in the building struc- +tural blocks of language. These blocks will be crucial for +2 reasons, (1) The meaning changes may adversely affect +sentiment analysis and thus affect results. Thus frequency +and polysemy must be duly curtailed. (2) The embedding +research is fundamental as we are using embedding-based +models. Specifically Word2vec. +Dataset Description +Source +Our data is sourced from the individual pages of politi- +cians(Senators and Congress members) from the 58th to the +117th congress. We divide these into 4 phases, chronolog- +ically, with each phase consisting of about 14 congresses. +For each congress member, we scrape the section-wise data. +Data Collection and Processing +We scrape Wikipedia based on the list of politicians from +the Wikipedia page for each congress. For each congress +member in the list, we store the label, their party and the +metadata. For each instance, this includes their personal de- +tails and all the information from their page as a dictio- +nary, with the heading being the keys and the content be- +ing the value. This information helps with the downstream +task of cleaning. We annotate this data based on the exper- +iment, in our case we have manually annotated the data to +classify these keys into three separate categories. 1) Back- +ground data, 2) Political data, 3) Other; in our release, we +will be releasing both the annotated and raw versions to +facilitate custom use. Wikipedia page sections don’t have +a fixed format, each politician has different key sections. +For instance, Early Life and Background can be split into +many sections such as Education, Career, Family, Personal +History, etc. So all these sections are grouped into a sin- +gle category Background. Similarly, anything related to their +political affiliation, elections, campaigns and positions held +during their tenure are categorized into a single annotation +Political Career. All other categories such as Awards, Con- +troversies, Business related activity, Post political career are + +clubbed under the Others category. This way, only relevant +data is selected under each category by manually chang- +ing the annotation based on the content inside each cate- +gory. To conclude, for just Phase 4, a total of 1656 cate- +gories were merged into 3 categories for 1631 instances in +the first pass spread over roughly 26 years(1995-2021). This +data still contains information names, organizations, loca- +tions, numbers, etc. which need to be cleaned. We first run +a NER model on the data to remove the names and organi- +zation. However, we remove location names only from the +political section. The reasoning behind this is, to make sure +information from the political section is not influenced by +location information. However, for background, we want to +understand where a person was born and raised affects their +political views and for this only this was kept but others were +deleted. This information, after the NER, is passed to re- +move numbers and other irrelevant regular expressions. This +makes sure the data being passed for other downstream tasks +is clean and gives unbiased answers. +Figure 1: Webscraping based on each Tag +Figure 2: Background +Language Model +Natural Language Processing based applications have been +dominated by transformer-based language models where +models like BERT(Devlin et al. 2018) and RoBERTa(Liu +Figure 3: Political +et al. 2019) have been state of the art since 2018 but when +it comes to our dataset, these models have a drawback, that +is, their ability to process longer sequences since the cost of +attention grows on the order of O(N2). Longformer(Beltagy, +Peters, and Cohan 2020) and other variants are useful for +this task, they accept 4096 input tokens as opposed to 512 +for BERT. It reduces model complexity by reformulating the +self-attention computation. The performance of Longformer +against the current SOTA is represented by the table present +below on the raw data. +Experiments +Preliminary Experiments +Our initial experiments were aimed at gaining insights about +patterns or trends that might be present in our data, and +also questioning if polarization exists. We do these prelimi- +nary experiments using the Doc2Vec(Le and Mikolov 2014) +and Word2Vec(Mikolov et al. 2013) models. The Doc2Vec +model was built from scratch with the raw data, where each +Wikipedia page is considered to be a document. We first +use the Doc2Vec model with K-means clustering and get +a classification accuracy of 59.52% with political data and +61.846% with background data. We then used the same +Doc2Vec model with binary SVM classifier and achieved an +accuracy of 72.872% with political data and 63.564% with +background data. These results are summarized in the ta- +ble presented below. The Word2Vec tests were run on pre- +trained models as well as models we built from scratch and +trained using the data we collected. We used the Word2Vec +approach to find approximate nearest neighbors and exact +nearest neighbors for certain words on both the Democratic +and the Republican sides. This nearest-neighbor approach +led to some interesting insights. We expected to see some +disparity in the nearest neighbor searches for the Repub- +lican data and Democratic data basis the assumption that +there is polarization. However using the simple Word2Vec +models the 15 nearest neighbors we got were quite simi- +lar but as there were certain words for whom the order of +the neighbors changed based on the party, for example, for +the word ’GUN’ , ’VIOLENCE’ is the 2nd nearest neigh- + +Welcome +Elements +>》 ++ 52 +g +... +Tommy Tuberville + +From Wikipedia, the free encyclopedia +.. +- +coach at Auburn University from 1999 to 2008. He was also the head football coach at the + +University of Mississippi from 1995 to 1998, Texas Tech University from 2010 to 2012, and +
+
+the University of Cincinnati from 2013 to 2016. +id="top +a +
..
+Tuberville received the 2004 Walter Camp and Bear Bryant Coach of the Year awards after +
+Auburn's 13-0 season, in which Auburn won the Southeastern Conference title and the +.. +-

== $o +: :before +Sugar Bowl, but was left out of the BCs National Championship Game. He earned his +"Tommy Tuberville" +100th career win in 2007. Tuberville is the only coach in Auburn football history to beat in- +

+state rival Alabama six consecutive times. In 2015, he was the president of the American +
..
+
+Football Coaches Association. He worked for EsPN as a color analyst for its college +
...
+football coverage during 2017.[2] +
...
+